Preface to the first edition
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Preface to the first edition
the two most fundamental concepts in our world because all else is unimaginable without assuming that space (or time) exists. It is therefore not surprising that the sophisticated Euclidean
Space and
time
model of space was
a common
are
already existed more than 2000 years. For centuries it belief by scientists and philosophers alike that the Eu-
clidean structure of space was one of the very few eternal truths. It was only at the beginning of the 20th century that this belief was shat-
special and general relativity. Today, Einstein's theory of general relativity is completely established, and there are many textbooks which explain it at all levels of mathematical sophistication. What is missing, however, is a modern textbook on general relativity for mathematicians and mathematical physicists with emphasis on the physical justification of the
tered with the introduction of Albert Einstein's theories of
mathematical framework. This book aims to fill this gap. Knowledge of physics is not assumed. While physical and heuristic
proofs. The book is also suitable as an introduction to pseudo-Riemannian geometry with emphasis on the intuition for geometrical concepts. arguments
The
are
given, they
physical
are
not used
as
substitutes for any
theme of the book
general relativity typically start with a more or less pseudo-Riemannian geometry. In such textbooks of some knowledge special relativity is usually assumed, and the reader is expected to accept the geometrical framework presented on trust. This approach is very economical but obscures the extent to which classical general relativity succeeds in describing our universe, and also where it those attemptmay fail. This is a point that is of particular relevance to of view it is Rom a to important to physical point quantise gravity. ing realise which parts of the theory reflect genuine physical insights, and which are dispensible. One way this can be achieved is through a criti-
Modern textbooks
on
formal introduction to
cal introduction that stresses foundational matters. There
textbooks
taking
this
approach, and
I
hope
are no
modern
to fill this gap with my -book.
V111
Preface to the first edition
One of the most
exciting aspects of general relativity is the predicBig Bang. Such predictions gained weight through the singularity theorems pioneered by Penrose. In various textbooks on general relativity singularity theorems are presented and then tion of black holes and the
used to argue that black holes exist and that the universe started with a big bang. To date what has been lacking is a critical analysis of what
really predict-' We give a proof of a typical singularity theorem and use this theorem to illustrate problems arising through the possibilities of "causality violations" and very weak "shell crossing singularities". These problems add weight to the point of view that the singularity theorems alone are not sufficient to predict the existence of physical singularities. these theorems
The mathematical theme of the book In order to gain both a solid understanding of and good intuition for any mathematical theory, one,should try to realise it as a model of a familiar non-mathematical concept. Physical theories have had an especially important impact on the development of mathematics, and conversely various modern physical theories require rather sophisticated mathematics for their formulation. Today, both physics and mathematics are so complex that it is often very difficult to master the theories in both subjects. However, in the case of pseudo-Riemannian differential geometry or general relativity the relationship between physics and mathematics is especially close, and it is therefore possible to profit from an interdisciplinary approach. Euclidean geometry had its origins as the description of shapes in physical space. It is generally considered a mathematical discipline rather than a physical theory, because it is possible to derive it from a small set of physical postulates, which can alternatively be viewed as mathematical axioms. Since the concept of space is basic to our everyday experience, Euclidean geometry combines mathematical rigor with intuitiveness a combination which has proved to be extremely fruitful for both mathematics and physics. Riemannian geometry is abstracted from the study -
of surfaces in Euclidean space and inherits much of the intuitiveness of Euclidean geometry. Hence Riemannian geometry is very well developed,
and even
growing number of geometers have branched out to Lorentzian or pseudo-Riemannian geometry. In my experience, these fields (and
a
Since I had written this passage a review article (Senovilla 1998) which a very similar theme has been pointed out to me. This article pro-
has
vides many very illuminating examples of spacetimes as well as discussions which reinforce our sceptical approach towards the physical interpretation of singularity theorems.
Preface to the first edition
even
Riemannian
geometry)
appear
quite abstract
to the
Ix
majority of
students. A careful mental
analysis
of space,
time, and free fall
(classical) physical concepts
-
the most funda-
-
leads almost
automatically
to
Lorentzian geometry. With respect to Lorentzian geometry, we are therefore in a similar situation as ancient geometers were with respect to
Euclidean geometry. What's more, virtually no physical background is required for this approach. Since Riemannian geometry comes to play in the
study of submanifolds representing an instant in time, it is completely straightforward to extrapolate pseudo-Riemannian geometry from the special and physically motivated cases of Lorentzian and Riemannian
geometry. While
some
modern textbooks present
pseudo-Riemannian geometry
(and general relativity) to mathematicians (an example of this is that by O'Neill (1983)), they have not motiv,ated the geometry from basic properties of
space and time. Instead
stract mathematical
theory.
To
they have developed
it
as
an
ab-
that the mathematical
description physical concepts, all definitions have a justification in this book. This approach also leads to a careful treatment of the structural aspects of the mathematics. ensure
mirrors the
How to read this book This book is not
designed
at page 1 and then to read
who take this
approach
so on
that it is necessary for the reader to start until she or he arrives at page 424. People
will very
likely give
14! The material is ordered in such used the
as a
reference
theory
that
a
shortcut.
ning of
a
a
urged As
It is
way
are
page
be
reader who is interested in space and time, and so to follow the guides in the margins, which provide
example, the text in the belonging to the shortcut:
margin denotes the
indicated
by --->2, where
corresponding footnote. The larly. Having understood the
begin-
denotes the page p. 222] the page
p. 111
number where the last shortcut passage ended and number where the present passage will end. Additional the footnotes
they reach
to allow the text to
unfortunate fact that many parts of to the preliminaries are not of imme-
an
passage
up before as
an
logically belong
diate interest to the reader is
source.
a
explanations
in
2 refers to the number of the
end of shortcut passages is marked simimaterial leading to Einstein's equation it
is then not difficult to return to the parts that have been skipped on an earlier reading. In addition, hints are -given at the beginning of most
sections
'
as
to what is
important and should be read
Explanations referring
to the
guide
in the
margin.
.
P. 111 -2
[I
p.
1
222]
X
Preface to the first edition
This
and
book,
with its 424 pages is meant to cover both general relativity differential geometry. It is therefore clear that
pseudo-Riemannian
important topics had to be omitted. mathematicians, the most important omissions are certainly some topics peculiar to Riemannian geometry, such as the Hopf-Rinow theorem (O'Neill 1983, Theorem 5.21) and the Myers theorem (O'Neill 1983, Theorem 10.24). Because these results are contrary to intuition one should obtain for Lorentzian (or general pseudo-Riemannian) geometry and since they are not needed for the description of space and time, they have been omitted from this book. Physicists may find that the presentation of this book is only loosely linked to other physical theories. This loose linkage is possible since the theory of space and time is fundamental to any other physical theory. some
For
The book is therefore accessible to mathematicians and
physicists alike. applications to astrophysics may wish to consult the book by Weinberg (1972). Weinberg's approach is opposite to the one used in this book, and personally I believe that it should ideally be read after the reader has a solid knowledge of the conceptional aspects of relativity as presented in this book. Most other books on general relativity also present the "Kerr solution", which is supposed to model the exterior of a rotating black hole. It has been omitted since it is not essential to unders,tanding general relativity. Moreover, it is well described in other books. People interested in this solution should probably first read Chap. 12 of the book by Wald (1984). The purely mathematical aspects of this solutions are clearly presented in O'Neill's Physicists
book
who
are
interested in
(1995).
Acknowledgements The reader will
undoubtedly
notice that this book
owes
much to excellent
text books and survey articles. For the
I wish to mention
philosophical aspects of this book especially the classic book by Weyl (1923) and the
by Ehlers (1973). freely used material which appears elsewhere (O'Neill 1983; Wald 1984; Beem and Ehrlich 1981; Hawking and Ellis 1973; Sachs and Wu 1979; Karcher 1994; De Felice and Clarke 1990; Abraham and Marsden 1978; Garabedian 1986) without always acknowledging this
survey article
1 have also
fact. I
warmly thank
mended the book
Bernd
project
Wegner, who encouraged me and recomSpringer-Verlag. I wish especially to thank introducing me to relativity but also for read-
to
only for ing through the whole manuscript and for his
Volker Perlick not
ments.
many
important improve-
Preface to the first edition
xi
This book is dedicated to two Australian way to
relativity students who on their gaining their doctorates courageously stood up against the immoral behaviour of their supervisor and the highhandedness of their university.
G6ttingen, 19th July
1999
M. Kriele
Table of Contents
1.
Local 1.1
theory Space 1.1.1
1.1.2
of space and time
.......................
I
..............................................
1
Affine space The fundamental theorem in affine geometry and doubly ruled surfaces
.................................
.....................
3
...........................
14
1.2
Euclidean geometry Absolute space and absolute time
1.3
Non-relativistic particles Galilei's theory of relativity
1.4
Einstein's
1.1.3
.....................
17
......................
20
1.2.1
..........................
special theory of relativity Causality in special relativity Length contraction and time dilatation Relativistic particles and photons
..................
1.4.1 1.4.2 1.4.3 2.
Analysis 2.1
2.1.1 2.1.2 2.2
on
39
..........
40
...............
43
................................
47
.........................................
48
Construction of manifolds Partition of
unity
.....................
54
.............................
57
Vector bundles and the tangent bundle 2.2.1 Construction of the tangent bundle
................
61
.............
63
The derivative of maps between manifolds Tensors and tensor fields 2.2.2
2.3
2.4 2.5
2.3.1
Algebraic preliminaries:
2.3.2
Tensor fields
Vector fields and
2.7
tensors
................
69 84
The lemma of Poincar6
........................
2.5.2
The theorem of Frobenius
2.5.3.
Orientable real manifolds on
87 94 102
.....................
106
......................
109
real manifolds
..................
Connections and projective structures Examples of connections
112
.................
121
.............................
132
2.7.1
The Levi-Civita connection
2.7.2
The
Weyl
........
...................................
Integration
67 68
..................................
ordinary differential equations
Differential forms
2.5.4 2.6
.......
............................
2.5.1
22 27
..................
manifolds
Manifolds
2
connection
....................
132
.........................
135
XIV
Table of Contents
2.8
Curvature
2.9
Applications to Weyl Variation of geodesics
.........................................
2.8.1
3.
4.
structures
................
...............................
143
Space and time from a global point 3.1 Light rays: the conformal structure
of view
...........
151
...................
151
3.2
Inertial observers: the
3.3
Compatibility: Weyl
3.4
Reduction to the Lorentzian structure
projective
structure
Pse udo-Riemannian manifolds
structure
.............
........................
........................
171
4.2
The volume form and the
.....
175
.........
176
............
184
Hodge operator Curvature of pseudo-Riemannian manifolds 4.3.1 2-dimensional pseudo-Riemannian manifolds Submanifolds
star
.....
......................................
4.4.1
Hyperquadrics 4.4.2 Umbilic and totally geodesic submanifolds 4.4.3 Warped products Isometries and Killing vector fields Length and energy functionals 4.6.1 Variation of length and energy 4.6.2 Conjugate and focal points 4.6.3 Existence of focal points
...............................
.......
.............................
4.5 4.6
5.
General
Matter
5.2
Some
5.3
5.4
6.
6.2
.......................
.................
216
....................
227
241
.....................................
255
............................................
255
specific
matter models
perfect fluid
.........................
264
.............................
264
The collisionless gas The electromagnetic field
..........................
Einstein's
equation
The
The Einstein
267
.................................
268
equation
as a
cosmology
Homogeneity and isotropy The initial value problem
equation 271
.......................
279
.........................
287
...........................
287
infinitesimally isotropic spacetimes
................
294
..............................
297
6.3
Geodesics and redshift
6.4
The age of the universe and the A simple model for the universe
6.5
of Einstein's
system
partial differential equations
for
265
......................
Lagrangian formulation
Robertson-Walker 6.1
205 209
5.2.2
of
204
213
The
5.3.1
202
....................
5.2.1
5.2.3
191
193
.......................
relativity
5.1
160 166
Existence of Lorentzian and Riemannian manifolds
4.4
158
.................
4.1
4.3
137 142
big bang we
live in
..............
.............
300 304
Table of Contents
7.
Spherical symmetry
XV
..................................
7.1
Pseudo-Riemannian manifolds with
7.2
The Schwarzschild solution
spherical symmetry
308
..........................
315
Experimental tests for the Schwarzschild solution. Quasi-linear hyperbolic systems of equations in two independent variables The initial value problem for spherically symmetric perfect fluid spacetimes with non-interacting electromagnetic fields Static perfect fluid stars 7.2.1
7.3
7.4
7.5 8.
9.
.........................
328
.............
337 348
.............................................
357
................................
358
8.2
Cluster and limit
8.3
Achronal submanifolds and
curves
.............................
Cauchy developments
Singularity theorems 9.1 Energy conditions Closed trapped surfaces 9.2 The singularity theorem of Hawking and Penrose 9.3 9.3.1 Applications of the singularity theorem 9.3.2 General problems with Theorem 9.3.1 9.4 Singularities and causality violations
.......
365
374
..................................
383
..................................
384
.............................
389
.......
390
..........
395
...........
397
..................
397
9.4.1
The G6del solution
...........................
9.4.2
Newman's
...........................
405
example Strength of singularities and cosmic censorship A simple, 3-dimensional example 9.5.1
397
.........
409
...............
412
................................................
425
.....................................................
429
References Index
322
.............................
Causality 8.1 Causality conditions
9.5
307
.
List of
Figures
1.1.3
Additivity of f Additivity of k Multiplicativity
1.1.4
A ruled surface
1.1.5
Proof of Theorem 1.1.2
1.1.6
Proof of Theorem 1.1.2
1.2.1
A
1.2.2 1.3.1
Absolute space, absolute time Parallaxis effect
1.3.2
Tower
1.3.3
1.4.1
Relative space, absolute time A flash of light at times to 0, tI
1.4.2
A
1.1.1
1.1.2
1.4.3 1.4.4
curve
in
8
........
.....................................
of k
10
.....................................
11
-
-
first
case
second
spacetime diagram
a
example
....................
13
.......................
18
........................
19
.....................................
23
......................................
23
t2
=
..................
consisting of linked oscillations Superposition of waves Future light cone and Galileian relativity. The observer moving with spatial velocity U measures a different centre OV of the flash of light and therefore different radii dj, d2 front
25 27
................
28
...............................
29
.....................................
29
wave
wave
12
.................
case
.........................
for its
9
.................................
30
1.4.7
Michelson-Morley experiment, at rest relative to the ether Michelson-Morley experiment, moving relative to the ether Length contraction ..................................
41
1.4.8
Time dilatation
.....................................
41
1.4.9
Twin
1.4.10
The
2.0.1
The torus T2
2.1.1
A
1.4.5 1.4.6
paradox twin paradox
......................................
in
43
.......................................
48
universe
locally homeomorphic
to R
.............................
R 2 which is not
submanifold of R 2
2.1.2
A manifold M
2.1.3
The construction of
2.1.4
The
2.8.1
The immersed surface in Theorem 2.8.1
3.4.1
C
a
and
finally
a
M6bius band
of Lemma 2.1.7
The world lines from
tially
42
.............
cylindrical
a
topological space which is but fails to be Hausdorff
proof
30
x
...
52
....................
53
............................
................
to y of two atoms which
at rest with
51
are
respect to each other
59 138
ini-
......
167
XVIII
4.6.1
List of
Figures
A broken curve
4.6.2
A
of
curve
lightlike geodesic can be smoothed out by a arbitrarily small length minimising the distance between two spacelike
submanifolds Z1 and Z2 4.6.3
Conjugate points
the
on
5.1.1
A localised congruence
5.1.2
Transformation of the
7.2.1
.......................
216
.............................
217
sphere
.......................
232
...............................
258
density
mass
in
special relativity
.
.
Schwarzschild spacetime in Schwarzschild coordinates Schwarzschild spacetime. Radial null geodesics are the
....
7.2.2
const and Y const. The region straight lines X Schwarzschild coordinates is shaded by =
=
............
The size of
central star in Schwarzschild spacetime
a
....
8.1.1
A strip of two-dimensional Minkowski space where future and past boundaries are identified
8.1.2
Misner's spacetime (S' x R, 2dtdW + td W2) A gedanken experiment to disprove causality violation
.
...................
.............
8.1.3 8.1.4
8.1.5
small
8.2.2 8.2.3
319 324
359 360
361 A spacetime which is causal but fails to be strongly causal 364 A spacetime which is strongly causal. An infinitesimally
with 8.2.1
317
cov-
ered 7.2.3
263
The
perturbation of the chronology violation
proof of Lemma
A limit
curve
-y of
a
8.2.1
metric results in
...
spacetime
a
............................
365
............................
366
sequence of
curves
7
..............
367
An example where a limit curve from x to y is not a cluster curve for a sequence of points from x to y 367 A spacetime which is causal but fails to be strongly causal 368 .......
8.2.4 8.2.5
Assume that
b1c
is rational and
a/b
is irrational. Then
the
projection of the line with slope c/a from torus depicted in the figure is a dense curve 8.3.1
is
cluster
of -y The definition of null boundary and achronal every
curve
a
8.3.2
The
Cauchy
9.5.1
The
singularity
curve
horizon for
a
R 2 to the -y. Hence
.....................
boundary
structure of
at
A the
given by Vo + tq
direction and that at B
375
spacetime. The case kI, k2 < suppressed. The sin-
(Wo/i9y)-l (o9q/Oy)
is
singularity B the singularity
368
set which fails to be achronal.. 381
0. The y-component of spacetime is gularity A is given by I + t
the
..
=
=
0 and
0. Observe that
light cone degenerates in the ydegenerates in the x-direction.
Hence there exist future directed timelike
ing from the singularity and
cosmic
curves
censorship
emanat-
is violated 416
List of
postulates
1.3.1
Galileian
1.4.1
Invariance of the future
3.1.1
Existence of
3.1.2
Light
relativity
........................................
light
cones
conformal structure
........................
25 32
.......................
152
...............................................
153
a
3.2.1
rays Existence of inertial observers
3.2.2
Law of inertia
3.3.1
Compatibility
3.4.1
No second clock effect
5.1.1
5.1.2
Tensorial. character of energy momentum Infinitesimal conservation law
5.3.1
Gravitation is determined
...........................
158
...........................................
158
with the causal structure
..................
...................................
................
............................
by
a
2nd-order
pde
............
160 168 259 264 269
1. Local
theory
of space and time
This book is not meant to be read in the order the
presented. Please follow the guide in the margins or skip material as proposed in the italic text at the beginning of most sections. material is
P*
chapter we will develop those aspects of space and time which can be locally observed, say in a laboratory. We will start with Euclid's description of space and then incorporate time in to the picture. The path we take is rather historical. It starts with intuitive but surprisingly complicated concepts (Newton's theory of absolute space and absolute time) and ends with the not so intuitive but mathematically simpler theory of special relativity. The guiding principle of this book will not be mathematical simplification, but the solution of problems occurring In this
in earlier theories.
description in this chapter seems to be global and extrapolations which are not validated by any experiments and which are not generally true. In the following chapters we will take up this point again, and show that the description given in this chapter should be considered infinitesimally rather than globally. This is the theme of The mathematical
leads to
the book.
1. 1
Space
In this section
we
consider space and introduce Euclidean geometry. This
material is assumed to be familiar to the reader and is therefore in
a
presented
rather concise way.
Readers who wish to learn the essentials of the
quickly and do not mind skipping guide in the margins.
M. Kriele: LNPm 59, pp. 1 - 1, 1999 © Springer-Verlag Berlin Heidelberg 1999
some
theory
mathematical
of space and time can use the
proofs
1
U
p.
3]
Local
1.
theory
of space and time
1.1.1 Affine space In this section
we
introduce
scription of space. Affine of 0 E R7 are ignored.2
affine
space
as
our
most
elementary de-
just R' where the special properties
is
space
basic experience that we can uniquely describe any point in space three real numbers. This seems to be the idea of Descartes (1637)
It is
by
a
who
developed analytic geometry as an example of his Discours de la plausible to identify R 3 with (physical) R3 contains a distinguished point 0 whereas space apparently does space, 3 Hence not. by using R as a description of space we introduce a mathematical structure which has no physical counterpart. This would lead M6thode. While it is therefore
to constructions which cannot be realised in space. For
instance, there is 3 negative of a vector v e R but there is no way to assign the negative to a point in space. As another example, addition of vectors has no direct interpretation in terms of points in space. If we want to have a reliable description of space with the property that all phenomena exhibited in this description are mirrored by physically verifiable phenomena,
the unique
we
have to abstract from these additional structures.
We will
meaning
isolate those structures of R 3 which have
now
in terms of space. Given two
points
we can
x, y
an
intuitive
construct
an
which points from x to y. This arrow induces a map from space to space. We just move the arrow (without rotating) such that its untipped end coincides with a given point z. The point z is then mapped to the arrow v
It appears that as long definition is independent of the path which
tip of the z.
In
z
-4
R3, z
arrow.
this
+ y
parallel transport Observe that y
x.
-
-
as we
we use
just given by the
is -
x
don't rotate to
move v
map
stands here for the
v
Ry-x: R3 arrow v
this
-
from
x
--->
to
R3,
which is
not a point. Thus the geometric interpretation is different from a simple addition of vectors. In order to separate the concepts involved we define
the concept of real affine space. We do not yet know what "rotating" should terms
-
following of
our
so
far
we
have
definition
(for
simply
mean
physical picture
a
the n-dimensional
case)
reflects
naive notion of
additional
in mathematical
in mind.
However, the some properties
parallel transport and does not introduce The set of points is denoted by A'. properties.
any
Definition 1.1.1. An n-dimensional real affine space is a set A' and v E Rnj of bijective maps such that the a collection fRv: A' --> A' :
following
holds.
N R,+,, 2
=
Readers who
R,
R,,
Vv,
already have
and connections considered
o
as a
(cf.
Sect.
manifold
w
some
2.6)
G
Rn,
knowledge
together
of manifold
theory (cf. Chap. 2)
take affine space to be the usual R7 with the flat connection.
can
1.1
(ii) for all Rvx,, (x)
R,(x) by
We denote
allow for
see
A"
there exists
+
or v
x
v
+
and v.,,Y
x
below that this definition is
more
unique v,,,y
a
R' such that
c
y.
=
We will
E
x, y
Space
affine spaces than there
by
or
y-x
x--y'.
restrictive that it does not
so
vector spaces.
are
Remark 1. 1. 1. Of course, space is 3-dimensional. We work with general n for several reasons. Firstly, most of the theory we will be
dimension
with the exception of developing does not depend on the dimension a key-result whose proof, however, is too complicated to be presented in this book (cf. Theorem 5.3. 1). Secondly, it is often convenient to compare the theory with lower dimensional analogues which are easier to visualise. -
It is therefore advisable to formulate it in
analogues
a
way which encompasses these
and also shows the restriction of the
exist modifications of Einstein's
dimensions
general theory
analogy. Thirdly, there of relativity to higher
("Kaluza-Klein theories").
Another way to generalise the theory would be to allow for complex vector spaces as well. We refrain from doing so since the main result of Section 1.1.2 Note that for
only x
E
holds for affine spaces
An,
u,
x
R' the associative law
v c
(U
+
R.
over
+
V)
(X
=
U)
+
+
V
holds. It is easy to see that all n-dimensional real affine spaces are isomorphic, and can be realised by R' in the following way. Choose E
A' and define
any
o
tify
An with
Rv(x)
=
0,, (A'
An
0,,:
R',
--
and define for
oo(Rv(oo- (x))). Clearly, x,,y
these definitions
are
Oo(x) OO(A'), v
x
E
x
F-+
Observe that
=
recover
the structure of R'
1.1.2 The fundamental theorem in affine
doubly
o.
Con-
by
iden-
geometry
ruled surfaces
In this section
we
be needed in the
results
of affine geometry which will 1-4.1. This section is very technical first reading.
present
proof of
and should be omitted
some
Theorem
on
Let o, xj,..., xk E An and a',..., the barycentre with masses a 1 3
vo,x. Now idenRn the bijection
independent of the arbitrarily chosen point
versely, choosing an o E An we can tifying o with the zero vector 0.
and
=
c
ak
E
R such that
Ek
j=1
a'
=
1. Then
k a
Section 1.1.2 is needed for the proof of Theorem 1.4.1 which is central to our interpretation of the Michelson-Morley experiment. However, the reader is strongly advised against reading this part now.
[
[P_
7 311]
4
1. Local
theory
of space and time
k
al X, +a 2X2+
-
-
-
+a kXk
:
0
`:::
+
Ea
(xi
-
0),
i=1
is
independent
of
defined via the
and therefore
o
an
hand side and
right
affine invariant. The
binations" where the real factors add to 1. is
set of
a
X11
...
is k
7
xk
symbol + is applied to "linear comAn affine subspace Bof A'
only
can
be
1
2 k a Xl+a X2+ points f x +01 Xk : Eki=1 a i 1, where are pairwise different, fixed points. The affine dimension of B
1. It follows that
an affine subspace is an affine space. An affine subspace of dimension 1 is called an affine line. We call points lying on a single line collinear. Observe that lines are the smallest sets which are invariant under parallel transport. -
Lemma 1. 1. 1. Let x, y, z E An. Then x, y, z lie on an only if there exists a A E R such that x y + A(z-y).
affine line if
and
==
Proof
x
R with
lies
x
=
the line
on
generated by
Oy+(l -O)z
=
z
y,
if and
only
if there exists
y+O(y-y) + (1 -0)(z-y)
=
then
f
is called
A collineation is collinear points into Consider
given by
a
an
biJ, ection
n
f (x) If A
=
is
A' which maps any three
--
line 1 and three points X 1) X2 i X3 A(X2 -X1) is denoted by
X3 -X1
G
affine transformation.
f: A collinear points. a
0
y+ (I -0)(z-y). I
Definition 1.1.2. An affine map is a map f: A' -4 An, A(x-o) + b, where A is a linear map, o E An, and b c Rn.
bijective
an
on
1. Then the number A
=
X3-XI
X2-XI
The the
following lemma is the classical theorem of Thales. It will be used in proof of the fundamental theorem in affine geometry (Theorem 1.1.1
below). Lemma 1.1.2. Let a
H1, H2, H3
line which intersects these
C Rn
be
parallel hypersurfaces and 1 hypersurfaces. Let xi(l) Hi n 1. Then
be
=
X3(l)-XI(l) X2(l)-X1(l)' does not
Proof.
depend
Denote
on
1.
by R
the
subspace
of Rn which is the associated vector
space to the affine space
H1 (and since H1, H2, H3
H2, H3).
quotient
We consider the
space A
nIft
are
defined
parallel also by
to
1.1
This space has Rn / fJ given by to the
a ir
5
y-xEfl.
ifandonlyif
x-y
Space
natural affine structure with associated vector space Jf (x z) where 7r, f denote the proj ections 7r (z)
(x)
=
-
-
equivalence classes. We have
7r(X3(l))-7r(X1(l)) ::_-';f(X3(l)-X1(l)) Jf
X3(l)-Xl(l) (X2 (1) (X2(l)-XI(l)
-XI
(1))
X3(l)-XI(l) f (X2 X1 X2(l)-Xl(l) X3(l)-Xl(l) (7 (X2 (1)) 7F (X 1 (1))) X2 (1) -X1 -
-
which
implies
that
(1) -XI X2(l)-XI(l)
(X3 (1)) -7r (XI 7r (X2 (1)) IT (X 1 (1)) Ir
X3
only depends
ir(xi(l))
that
Hi
x, y E
==> 7r
-
the projected values. Now it is sufficient to observe independent of I since all points in Hi are equivalent:
on
is
(x)
=
7r
(y).
I
It is easy to see that all bijective, affine maps are collineations. Conversely, the fundamental theorem in affine geometry asserts that any
collineation must be affine: Theorem 1.1.1. Let A' be 0
E
A. Let
points
f:
An
--+
into collinear
invertible linear map
An be
an a
affine
over
space
R with
n
> 2
and
fix
bi ection which takes each three collinear
points. Then there f such that f (x)
=
exists
a
f(x-o)
point b E An and for all x E An.
an
+ b
The
proof is elementary but lengthy and requires some preparatory lemWe will follow (Berger 1987, p. 52-55) where one can also find a version of this theorem which holds in the complex case. Observe that the following proof makes heavy use of the assumption n > 2. The themas.
orem
does not hold for
points
into collinear
n
=
I since in this
any map maps collinear
case
points.
Lemma 1. 1.3. Let o, x,.... iXk E R, and
An, f
be
a
collineation,
k
x==o+EA'(xi-o)EA
n
i=1
Then there exist
fil.... Ak I
E R
such that k
fW
=
f(o)
+
E Ai ff (xi) i=1
-
f (0))
-
A
Ak E
6
1. Local
Proof.
For k
theory
of space and time
1 the claim is clear
=
by the
Assume now, the assertion is true for all
definition of
1,
m
M+1 X
Then
we
.
.
k
,
collineation.
a
-
11.
For
M
Ai (Xi-0)
+
0
=
.
X,
let
=
0
Ai (Xi-0).
+
have X
=
X/ + Am+, (XM+1 -0)
(1.1.1)
p"
by induction hypothesis there are real numbers y' f (x') f (o) Eim. 1 M" (f (xi) f (o)). We define also and
=
-
with
-
Y
=
z
=
A'+'(xm+,-o),
+
0
2
Y+
(1.1.2)
X
2
The triples I z, x', y 1, 1 y, o, xm+ 1 1, and f z, o, x I consist each of collinear points. This is clear for the first triple and follows from Lemma 1.1.1 for the second triple. To see this for the third triple observe that y-o 1 x1. z x + 12 x' is the centre of the parallelogram defined by o, y, x, x' 2y and therefore the intersection of the line connecting y with x' and the line connecting o with x. Since each of these three triples consists of collinear points there exist a, 0, -/ such that =
-
This
=
f (Z)
=
af (X')+ (I
fW
=
Of (o)+ (I
f (Y)
=
f (o)+-Y(f (x.+i)-f (0)).
-
-
a) f (y),
O)f (z),
implies
fW
=
Of (o)+(I
=
'3(f (0) (I
(I
-
-
-
-
O)f (Z)
f (0))
+
(I
-
'3) ((af W) + (I )3) (c, (f (x')
-
0) (f (Z)
-
-
00 f (y))
f (o))
+
(1
-
f (0))
-
f (0))
+
(I
-
/i
A
(f (Xi)
a)-Y(f (X-+1)
-
-
f (0))
f (0)))
M+1 =
E 'U
(f (Xi)
-
f (0))
+
f (0)
-
f (0)
+
a) (f (y)))
M
0) (01
+
+
f (0)
f (0) +
f (o)
Space
1.1
Lemma 1.1.4. Let o, xl,. Xn E An such that f X1 -0i iXn-01 is a basis of R'. If f is a collineation then f f (xi) f (o), if (Xn)-f(O)j is ...
-
...
also
Proof
ol
basis
a
is
of R'.
Let -3- E An be any point and let x f basis of Rn there exist ' E R such that
Since
=
a
Lemma 1.1.3
that there exist
implies
pl,...'
n
x-o
E
=
fxl-o,...
7
Xn-
Enj I V(xi-o).
R such that
k
:
-
f (0)
=
fW
-
f (0)
=
EA
(f (Xi)-f (0))
-
j=1
Since 1--
was
arbitrary the
assertion follows.
Lemma 1.1.5. A
bijection f affine lines.
is
a
collineation
if
and
only if
it 'Maps
affine
lines onto
Proof.
Let x, y E An and denote by 1 the line spanned by these points. a point on the line spanned by f (x), f (y). We have to show
Let i be that
z
=
f-'( )
G
1. If this
was
not true than the vectors z-x,y-x
would be
linearly independent. But then Lemma 1.1.4 would imply that f (z) f (x), f (y) f (x) where linearly independent as well. Contradiction to the construction of I f (z) -
-
=
Lemma 1.1.6. Let
parallel
f
be
collineation. Then f maps parallel lines into
a
lines.
otherwise 1, 1 be two parallel lines (which do not coincide nothing to prove). Since they are parallel they span a plane P rather than a 3-dimensional subspace of An. This plane is mapped into a plane P'. In order to see this consider a line i such that the lines 1, i intersect and span P. It is clear that any line which intersects both 1 and i is contained in P. Moreover, any point y E P lies on a line I which intersects both I and 1. Let P' be the plane generated by the (intersecting) lines f (1) and f (1). f (y) lies on the line f (1) which intersects f (1) and f (i). Hence f (T) (and therefore f (y)) lies
Proof.
Let
-
there would be
in P'.
Having established that f (P) 0. If there f (1) n f (1)
show that
=
would lie in both 1 and V which is
is
subset of
a plane we only have to point z c f (1) n f (1) then f -'(z) impossible since both lines are parallel.
a
was a
I
Lemma 1. 1. 7. Let k: R
and k (a +,3)
=
--->
k (a) + k (,3)
R
for
an
k (a) k ( 3) automorphism, i. e., k (ao) id oz,,8. If k 7 0 then k
all real numbers
=
=
theory of
1. Local
space and time
-
k (0 + 0) k (0) + k (0) implies k (0) 0. Assume, there is an Proof. k (0) 0 0. Then with a 4 k(a) k(o) k(oz)k(ola) 0 for all 0 and k must vanish. Hence k(a) =A 0 Va 34 0. k(l) k(l 1) k(l)k(l) implies 1. By induction we obtain k(n) n for all natural numbers. k(l) k(-n) k(O) k(n) -k(n). Similarly, we have k(l/n) k(O n) =
=
=
=
=
=
=
-
=
-
=
1/n. For n, m E Z we have now k(n/m) for all rational numbers. a < 0 implies
1/k(n) is
proved any positive number
2 we
7
number. Then there exists
k(-y 2)
have
=
n/m k(a) :5 k(,3)
since for
Let
-y be any
and the lemma
=
k(-y)k(-y)
> 0.
now
monotonically increasing sequence ai 'Y a monotonically decreasing sequence of Hence ai Oi which k(ai) < k(-y) < k(0j) a
-->
of rational numbers and likewise
rational numbers
implies k(-y)
Oi
--->
-y.
=
=
I
-y.
=
Observe that this lemma would be false if z
would be
i-->.
above)
a
counter
This is
example.
is not true for affine spaces
f:
had
we
why
replaced
R
theorem 1.1.1
by
C
as
stated
(as
the field C.
over
R', v -4 f(v) f (o + v) f (o). an automorphism k: R --> R such that proof k (A) f (v) + k (y) f (w) holds for all A, M G R and v, w c R'. f (Av + I-tw) We will use constructions based on parallel lines in order to represent vectors such as v + w, (A + A)v, Attv. Since f maps parallel lines into parallel lines (Lemma 1.1.6) these constructions will be preserved by f and can therefore be used in order to prove linearity and multiplicativity Proof of
Theorem 1. 1. 1.
The idea of
Let
R'
--->
=
-
is to construct
f, k.
of
We will first show that
fis
additive.
f
(o+
(o o
+
V
+
o
o,
of
1.1.1.
+
W)
(0)----Ie- f(lo + V)
Additivity of f
1w that contains
parallel
lines
o
+
=
v
and of
1, that
mapped analogously
f(o+V+W)-f(o)
=
contains
o
+
w
(cf. Figure
1. 1. 1).
f (o + v), f (o + w). Hence f(o+V+W)-f(0+V)+f(o+V)-f(o) into
are
is constructed
w) f(V+W) +
V
Let v, w E R' and consider the lines 1,, 1w spanned by o, o + v and + w. The point o + v + w is the intersection of the parallel translation
Since v
W) (o +
f
Fig.
+
W
lines
parallel f (o), f (o
from
we
know that
+
=
1. 1
Space
+ v) f(w) + f(v). Here we have used the f (o) connecting f (o) with f (o + w) and f (o + v) with f (o + v + w) are identical since they correspond to opposite sides of a parallelogram in a plane. Now we show that there is a well defined automorphism k: R --> R
f (o
+
w)
f (o)
-
f (o
+
=
-
fact that the vectors
such that
f(Av)
k (A)
=
f(v)
for all
v
E
Rn and A E R. We first fix
a
and consider the line 1
vector
R through o spanned by v. Denote by gi: 1 the map o + AV --> A and by gf the i--> Since + ttf(v) map f (o) f /-t. (1) maps the line through o which is spanned by v into the line through f (o) which is spanned by f (o + v) f (o) the map k: R R is well defined the From through relationship f (Av) k(A)f(v). v
-->
-
=
f (o) we see
f(v)
+ k (A)
that k is
f (o)
given by k (A)
+
=
f(Av) o
gf (1)
f (o f
+ 1
o
gi
Av)
f (gi
'(A))
(A).
f
/,00-
f (0
+
W)
W O+W
AV AV 0 0
Fig.
0
+ AV + [tv
+ ttv /,V
X" , AV +
+ AV +
/-tV)
of k
Additivity
1.1.2.
(*0
f (0) f (0 + AV) f (0 + /,tv) f (0
In order to prove additivity of k we use the fact that (A+M)v Av+,uv be constructed using parallel lines (cf. Figure 1.1.2) Let w E R' be =
can
linearly independent from points
o,
that the
the lines
o
+ w,
o
v
and consider the
+ Av. This
triangle defined by the parallely translated so (We simply parallely translate
triangle
o is mapped into generated by its sides
point
o
can
+ /-tv.
be
indicated in the figure). Since this translation preserves the vectors defined by the sides of the triangle we have obtained a geometric construction of the point o + Av + Aw. Since this construction only employs intersection points and lines it is as
p rallel
preserved by the
k(A)f(v)
+
k(A
map
k(p)f(v) +
p)
=
=
=
f
.
Hence
we
obtain
f ((A + /-t)v)
=
f (Av)
+
f (ttv)
and therefore
gf (1)
o
f
1 o
gi
(A
+
/-t)
==
gf (1)
o
f (o
f((A + tt)v)) gf (1) (f (o) + k (A) fi(v) + k (M) f(v)) gf (1) (f (o) +
+
(A
+
tz)v)
10
Local
1.
theory
of space and time
gf (1) (f (o) +
(k (A)
+ k (p))
f(v))
=
k (A) + k (y).
f
f (0 o
f (0) (0
A> 0
f 1(o
+
AW)
+ AW
W w 0+W
0 + 01
W)
+
0 + o4
A[IV f (o + V) f (o (0, + /\V) AV) ff (0
'+U Vlf
()o + pv)
'v 0 + /_tV 0+
\v + AV
f f(o+ (0 + A/,tV)
V
-
Fig.
1.1.3.
Multiplicativity of k
proof of multiplicativity is similar and employs a slightly difgeometrical construction (cf. Figure 1.1.3) which is justified by Lemma 1.1.2. The configuration in the first part of Figure 1.1.3 lies in a plane whence hypersurfaces are simply lines. Denote by H2 the line which connects o + v with o + w, by I-I, its parallel translation through o, and by H3 its parallel translation through o + Av. Further denote the line through o and o + v by 1 and the line which connects o with o + w by 1'. Using the notation of Lemma 1.1.2 we have The
ferent
(0 + AV)-0 (0 + V), 0
A
X3(1)-X1(0 X1 (1) X2 (1) -
implies that the intersection of H3 and 1' is really depicted in the figure. We apply this lemma a second time where the three parallel hypersurfaces H2, H1, H3' are now given by the line connecting o + /-tv with o + w, its parallel translation through o, and its parallel translation through o + Aw. It follows that the intersection of H3' with 1 is o + y(Av) o + A/-tv. Since this construction only em1 loys intersections and parallel lines it is preserved by f and we obtain f (Apv) k(A)k(p)f (v). This implies Hence Lemma 1.1.2 o
+ Aw
as
=
k(Att)
=
=
Hence k is
metrically o.
gf (1)
o
+
Aix)
=
gf (1) (f (o) +
gf (1) (f (o) + k (A) k (y) f(v))
really
our case
=
&yv))
k (A) k (y).
automorphism of the real line. One can geoautomorphism neither depends on v nor on this automorphism is trivially well defined since
an
show that this
However in
f (o
1. 1
Space
11
already know that the only non-zero automorphism of R is the identity. This also implies f (,\v) \f (v) for all X E R, v E Rn Hence the theorem is proved. I we
=
-
We will
now
turn
attention to
our
special subsets of affine
space which
become important in the proof of Theorem 1.4.1. Definition 1.1.3. Let U C R 2 be
an open set and x: U --, An be a C' map such that at each point (s, t) E U the differential Dx(s, t) is injective. Then x is called an immersed surface. If x is also injective
then it is
simply called
surface.
a
A surface should be envisaged by its image, a two-dimensional, smooth subset. An immersed surface may have self-intersections. Since lines have such a fundamental meaning in affine geometry, surfaces which are generated by lines are of special interest.. Definition 1.1.4. A ruled surface is zed
by
sation
a surface which can function of the form x(s, t) c(s) + tw(s). Such is called a ruling of the surface. a
=
Fig.
1.1.1. The
Example different
(i) (ii)
be parametria
parameteri-
1.1.4. A ruled surface
simplest ruled surfaces
those which admit two
are
rulings.
A trivial
example would be any plane. slightly more sophisticated example is given by the rotational hyperboloid. Let c(s) (COS(S), Sin(S), O)T be the unit circle in A
==
R' and consider is
a
Xhyp(S,t)
ruled surface and
t sin (s), sin (s) + t 2
(X hy P)2
-
(X3hy P)2
COS =
surface is described
C(S)
=
(8), t) T.
a
different ruling.
t(6(8)
+
(0, 0, I)T). Clearly, (COS(S) =
Since it satisfies the equation
I it must be
by -Zhyp (8 t) 7
is
+
explicitly given by Xhyp(S,t) a
rotational
=
C(S)
(xlhy P)2 +
hyperboloid. The
+ t (_ 6(8) +
x -
(1, 0, 0) T)
same
which
Local
1.
12
(iii) A c(s)
theory
third
=
of space and time
is
example
(s, 0, 0)
'
and
w
given by the hyperbolic paraboloid. Let 1, ks)T. Then Xpar(S7 t) 2 (0, 1_k1-2
(s)
=
v
+1
8-
X 3ar and Xpar parameterises a p I hyperbolic paraboloid. We can interchange x par and Apar to obT tain a different ruling of the same surface, ;r-par (8) t) (0, S, 0) +
c(s)
+
satisfies
tw(s)
kxlarx2ar p P
=
=
I
(1,0,kS)T. a surface which admits two different M is a subset of affine transformation rotational hyperboloid, or a hyperbolic paraboloid.
Theorem 1.1.2. Let M C A' be
rulings.
Then
either
plane,
a
-
a
up to
an
-
Proof. It is easy to see that any surfaces M C A' with two rulings can locally be embedded into A'. One just has to consider -a line 11 of the first ruling which intersects a line l'2 of the second ruling. Choose another line l'3 of the second ruling which also intersects 11. Then all three lines further line of span a 3-dimensional affine subspace. At least locally, any the first ruling must intersect both l'2 and l'3 whence it is contained in the same affine subspace. Since M is generated by the lines of the first ruling we have proved the assertion.
1.1.5. Proof of Theo-
Fig. rem
If there
are
any two
1.1.2
-
first
generators of the first ruling which lie
then the ruled surface must be this
Hence
plane. linearly independent. There
case
in
we can assume
a
plane
that any
possibilities. parallel to a single plane or any three generators are linearly independent. In the first case let 111 12,13 be different generators of the first ruling which are all parallel to a single plane. We can now find linear coordinates I X ,X2,X31 such that the x1-Axis coincides with 11 and the x 2-axis is 2 parallel to 12. By choosing the origin appropriately, 11 is given by x 3 3 a2, X1 +a 3X2 0, a', and 13 is given by x X 0, x 0, 12 by x1 3 contains which be P Let 11. Then there R. a where a 1,a2,a E two-plane 2 SX3 0. P is that x + such R Any generator V exists an s E given by both 12 and intersect must P in contained is which second of the ruling intersection the for obtain We points: 13-
two
generators
are
Either there exist three generators which
are
are now
two
all
=
=
=
=
=
=
=
1. 1
1.1.6. Proof of Theo-
Fig.
1.1.2
rem
jPn12j:
X
jPn13j:
X
=
0,
=
a
1
It follows that 1' has the
2
1
X
a
X
8,
Since the ruled surface is a
eters s, t
2
X
s,
=
a
a2a3s
0
tained
case
,
3
a
second
parameter form
-als al
1/
a
==
2
2
3
2
X
s,
-
1.
3
a
13
Space
+ t
W
-S
a2
al)
_
:
t ER
al
_
1', we have obEliminating the param-
such these lines
generated by
x(s, t) parameterisation (s, t) 3 -a3a 2 2, whence the surface we obtain xlx of it.
-->
=
must be
a
hyperbolic paraboloid. For the second
case we
f x1, x2, X31
choose linear coordinates
thatthe X3 -Axis coincides with 13 , the X2-axis is
such
parallel to 12, and the x1
-
11. We can chose the origin 0 of the coordinate system parallel 0. Then such that it lies in 13 and such that 12 lies in the plane X3 X2 there exist numbers al, a2, a3 C R such that 13 01, JX : X1 a3 P be X2 Let x3 a and X1 : a2,X3 11 : a', 1. fx 12 01, fX X2 exists Then there 0. to not and is contains which 13 parallel plane 2 0. Any line 1' of the second sx an s c R such that P is given by x1 axis is
to
=
=
=
=
=
=
=
=
=
=
-
family which
11 and 12- We calculate
lies in P must intersect
JP
n
111: x1
fP
n
121:
2 =
sa
3
2
2 ,
X
a
,
3
X
a
1
This
1
I X
=
a
a
2 ,
X
3
=
,
X
=
0.
gives the line a'
a'
alls 0
+ t
2 -
alls
sa
2 -
-a
3
a
tGR
14
1. Local
It follows that
(s, t)
theory
(s, t)
-a
a
that this
1, (X 3)2 ( 2)2+ (X3)2 =
X
-
1, the
(X 2)2 that
cone
_
our
we
eliminate
3X1X2
2
+
a
xIx3
1x2x3
a
_
+
I a
a3x 2
=
0.
quadric. We could use the Gram-Schmidt-procedure to show quadric is affinely equivalent to a hyperboloid. But since any
in R 3 is
quadric
parameters of the ruled surface. If
are
obtain the equation
we
This is
of space and time
X3
==
affinely equivalent either
the two-dimensional
1, the rotational
=
(X 1) 0,
2
+
or a
(X2)
2 _
X3
to the
pseudo-hyperbolic
hyperboloid,
=
-
(x')
(X 1)2 +
space,
2
+
(X2)
plane, we can infer without affinely equivalent to a
(X2)2 + (x 1)2 +
-
2
0, the hyperbolic paraboloid
surface must be
1.1.3 Euclidean
sphere
2
+ -
(X3) (XI)
2
+
any further calculation rotational hyperboloid.
geometry
Euclidean geometry gives the local model of space. In the following sections we will obtain models of space & time which incorporate Euclidean geometry as description of space. Unless otherwise stated, here and in the following space has dimension n 1. We assume that Eu-
clidean geometry is known to the reader and a
therefore only
summarise
few facts.
In affine space, we have no definition for "length" or "angle". Since these are fundamental concepts for our perception of space, we must endow affine space with an additional structure. The first scientific and ex-
perimentally well tested description ing
and axiomatisation of space involv(ca. 340b.C.-
these notions culminated in the "Elements of Euclid"
In modern
270b.C.).
terminology, Euclid's theory
of space
tified with Euclidean geometry. The central object of Euclidean geometry is the scalar Definition 1.1.5. A scalar
product V
such that
for
any u, v,
w
E
V
--+
(u,v)
F-4
V, A,
A u, V) (i) (U Av + AW) V, U), (ii) (u, V) (iii) (u, U) > 0, OV) u, U) 0 == > U 0 =
,
=
=
hold.
=
x
+
on a
U
product. a
map
(1.1.4)
(U,V)
(u, W),
be iden-
real vector space V is
R
It c R the
can
properties
Space
1.1
We a
define
can now
scalar
Euclidean space
an
as an
affine space
15
equipped with
product.
Definition 1.1.6. An Euclidean space is
(i) A "
is the
(n
-
pair (A",
a
real
1) -dimensional,
where
space with associated
affine
vector space R n- 1,
(") (*C)R1-1 0: An'
A map
is
scalar
a
product
An-1 is
-->
an
R".
on
isometry if and only if
O(Yl)-0(X1)i 0(Y2)-0(X2))R71-1 for
An-1
all Y1) X1 7 Y2 7 X2 c:
The
notions
physical
initions x, y E
It is
wish to capture with
we
"angle". only depend on
A` should
plausible
mathematical def-
our
The distance between two points
"distance" and
are
YI-Xli Y2-X2)Rn-1
:--::
the connecting vector u y-x. x and x + \u is X times =
to demand that the distance of
the distance between
definition dist (x ,
x
y)
and X
x
+
Hence, given
u.
Y Rn-1
-
scalar
a
V(X_y1X_y)R11-1
:=
product, the to be
seems
a
reasonable choice. It is vectors
in order to
clear, however, that needs
one
map Rn
a
x
Rn
measure
of the
real number. The
directions u,
by
Z (u,
v)
=
(
arccos
between two
angle
symmetric in both vector is multiplied by a
entries and remains
defined
unchanged if one angle between two
the
R which is
-->
v
may therefore be
(U,V)R"-l IIUIIR"-l 11V11R11-1
priori clear that a scalar product is indeed the appropriate additional structure for defining lengths and angles. See (Weyl 1923, 19) It is not
for
a
a
theoretical
1.1.1. A map
Proposition ture
(A n-1, (7 *)RII-1)
A: R n-1 and
(Au, Au)R11-1
0: An-1
invariant
Rn- ' and
-4
of the usage of scalar
justification
points
if
only if there
I such that o, b E An-
U) U)Rn-1 for
:=
An-1 leaves the Euclidean
__
and
all
products.
u E
exist
V)(x)
struc-
linear map
a
A(x-o)
=
+b [p.
R".
I
Proof. Observe first that all
u
E R
for all v,
O(Yl)
n-1
w
-
c
application of Au, Au)R11-1 u v + w implies (Av, Aw) R11
an
to the vector
R n-1
.
:--
=
Hence the map
0(X1)i0(Y2)
-
In the
proof
of
O(x)
=
A(x-o)
-
I
(U7 U)R11-1 for (V W) R11 =
I
+ b satisfies
0(X2))R11-1
(A(yi-o) =
(A(y,
=
(Y1
-
x t-->
-
-
X1, Y2
Proposition
A(x1-o), A(Y2-0)
x1), A(Y2
1.1.1
-
-
A(X2_o))R11-1
X2))R11-1 X261-1 -
*
we
appeal
14
1]
-4
to Theorem 1.1.1.
-
p. 16
16
1.
Local
theory of
space and time
Conversely, any map V) which preserves the Euclidean structure preserves particular the affine structure. Hence Theorem 1. 1. 1 implies that there
in
linear map A and a point b such that O(x) A(x-o) + b for all Since 0(y) O(x) A(y x) it is clear that A must satisfy
is
a
x
E A.
=
=
-
Au, Au)R,,-'
(ul U)R,,,-'
=
-
for all
u c
R".
P.
Remark 1.1.2. At first
sight our definition of a Euclidean space may general. The reader may feel that in space there is a subset of physically distinguished scalar products: Let e be a vector which we use as measuring stick defining unit length and E a plane which contains el. Using a pair of compasses we can construct a line 1, C E which is orthogonal to el and therefore also seem
to be too
vector e2 of the
length as el but perpendicular to el. We may plane E,, by rotating,e-2 around el and a third plane E,, by rotating el around e2. The intersection E,1 n Ee2 is a line orthogonal to el and e2. Using again or pair of compasses we can construct a third vector e3 which is of unit length and orthogonal to el and e2- Our distinguished scalar product is now given by (ei, ej)R3 6ii' It follows from the Theorem of Pythagoras that the length of a vector u is given by JUIIR3- We can use a pair of compasses to approximately (but arbitrarily well) divide the circle into a fixed number of arcs thereby introducing an approximate measure of angle. From the definition of the cosine it is clear that (up to a constant factor depending on the number of arcs) the size of an angle is given by the definition above. However, this introduction of the standard scalar product is based on procedures which are intuitive but which cannot be defined in mathematical terms without having a scalar product in the first place. In fact, if we had started with any given scalar product (., -) and had defined a
now
construct
a
same
second
=
(i)
a
rotation
as a
linear map which leaves the scalar
product
invari-
ant and
(ii)
a
pair of
containing same
then using
a
length our
compasses
given as
construction
1.1.2. Let
Euclidean spaces. which satisfies
(O(Yl) for
-
a
e
device which for each given plane E produces all vectors e' c E with the
e,
lowing proposition gives Proposition
as
vector
a
we
would just have recovered
(An-1, (*1')R"1-1)
Then there exists
V)(X1)i'0(Y2)
all X1i Yli X21 Y2 E
The fol-
mathematical explanation of this fact.
An-1.
-
an
V)(X2))R11-1
and
affine
=
Yl
R11-1) map
-
0: A'-'
X1) Y2
-
-,
X2 R-l
be two
An-1
1.2 Absolute space and absolute time
Proof. Choose tively
points o,b
any
be
Corollary is
a
basis
(-)R11-1)
ei. Then 1) We define the linear map A by A i b desired is the isomorphism. A(x-o) +
(respectively
-
O(x)
=
is often referred to
0
A'-' and let lei,..., e,,_11 (respec-
E
orthonormal basis with respect to =
the affine map The map
an
17
as an
Euclidean
transformation.
*) *61-, be a scalar product of Rn-1. Then there 6ij, where en-11 of Rn-1 such that ei, ej )RI-1
1.1.1. Let
lei,
=
6ij is the Kronecker
I
for
i
0
otherwise
=
j,
symbol.
Today, Euclidean geometry is-often taught as a prime example for a closed and consistent mathematical theory. This obscures the fact that angles and distances are physically measurable and that therefore Euclidean geometry can be falsified as a physical theory. (For instance, one of the most influential philosophers since the time of enlightenment, Kant (1781), wrongly considered space as given "a priori"). In modern times, Carl Friedrich GauJ3 (1777-1855) seems to have been the first to realise the possibility that Euclidean geometry may not be the correct description of our world though the legend that he tried to verify Euclidean geometry by measuring the angles between three mountain summits is not true (Osserman 1995, page 66). He has developed a non-Euclidean geometry in which the parallel axiom does not hold but did not publish it. This geometry was also independently discovered by the Hungarian mathematician Jdnos Bolyai (1802-1860). Later in this book we will conclude that space should be described by geometries which are far more general than those considered by GauB and Bolyai. -
1.2 Absolute space and absolute time In this section
we
present the "naive" model of
space and time.
We
complicated it really is. We will also give a short account of Newton's theory of particles which is the main physical justification of this spacetime concept. will take
Time
seems
care
to show how
to have
striking similarities with
space but nevertheless to be
something which is very different. Like space time is a continuum. However, space is a 3-dimensional continuum while time is 1-dimensional. Moreover,
we
can
freely
move
in space but
merely
drift in time. It is
18
1.
Local
theory
of space and time
--------------
-----------_-------
movement
inovement
in tirrie
in
space-
tP1
time
t--------
Fig. 1.2.1. A curve in spacetime diagram
often
practical to treat space and time in spacetime diagrams are used to describe Hermann Minkowski
nobody
has
ever
unified
manner.
movements
(1864-1909) (Minkowski 1909)
experienced
space without time
This observation is borne out
following
a
For
instance,
(cf. Fig. 1.2.1).
has
pointed
out that
time without space. space & time in the
or
by characterising
way.
Definition 1.2.1. A primitive spacetime is set. The points time are called events.
of
a
space-
Of course, this definition does not tell anything about the relation of or even allows to distinguish between these concepts.
space and time
In order to do
so we must supplement the primitive spacetime with a geometrical structure. In the preceding section we have recalled that space can well be described by (n I)-dimensional Euclidean space. The fact that time is I-dimensional indicates that spacetime can be considered as an ndimensional affine space which is foliated by (n I)-dimensional subspaces each of them carrying a Euclidean structure. Any foliation with affine hyperspaces corresponds to a linear map T: R' -4 R, where x, y E A' are in the same hyperspace if and only T(x-y) 0. Denote by Ex fy E An : 7-(y-x) 01 the affine hyperplane through x and let o'o/ E An. Then the vector spaces associated with all these affine hyperplanes E., (x E An) are identical. In fact, they are given by -F-1(0) 0 1. Hence we only have to specify one single Euclidean f v E Rn : -r (v) scalar product .),-, (0) on the vector space -r- 1 (0) in order to get a -
-
=
=
=:
=
foliation of
with one
our
(n
-
l)-dimensional
Euclidean spaces. This is in accordance of space does not change from
experience that the geometry
instant of time to another.
The map -r can be interpreted between to events x and z is just
as a
T(z
world clock: The time difference -
x).
Observe that
defined up to a factor. This factor corresponds to the which time is measured. We still need to link events in different
spond
to the
same
point
in space. The
is
uniquely physical unit in T
hypersurfaces which simplest way to do so is to
corre-
intro-
1.2 Absolute space and-absolute time
duce
line
as a
x
by -r(t)
t
second structure
+ Rt
as
the
a
vector t and
interpret all points lying
in space at different times. If
point
same
I then the time difference between
=-
Definition 1.2.2. A Newton
spacetime
is
and y
x
=
we
x
where t E Rn is =
the
+ tt is
just
t.
quadruple
a
(1.2.6)
distinguished vector, -r: R' ---),_ (O) is a scalar product on
R
a
1, and (.,
on
normalise
(An, t',r'
,r(t)
19
linear map such that
a
the vector space
This definition is just the content of Isaac Newton's of absolute time and absolute space.5
7-'(0)..
(1642-1727) theory
1.2.2. Absolute space, abso-
Fig.
lute time
We
see
that
is fibred
spacetime
twice, By lines parallel
hyperspaces
of the form
may appear
quite cumbersome but
in
a
geometrical
One
as
defining
notion of rest, and think of
of v
o
it
is
some
to t and
by
fixed event. This structure
captures
naive
our
point of view
way.
think of t
can
where
E,+tt
-r
as
pair (-r, t) induces A +V and -r(,U) 0.
time. The
a
a
time axis and therefore
an
absolute
defining an absolute notion of instant projection -:: R' -->, R', v 1--* 6, where
=
=
A map which leaves the structure of a Newton spacetime invariant a spatial Euclidean transformation as given in Proposition
consists of 1.1.2 and
a
spacetime translation. 1.2.1. A map
Proposition
(An, t,,T, (.' 1 -r- (0)
A:
-4
-F-
1
(0)
and points o, b
(i) O(x) A(x-o) (ii) (Au, Au),-, (0)
+
=
5
V): A' --+ An leaves the Newton spacetime if and only if there exist a linear map
invariant
In the next section
=
we
E
An such that
T(x-o)t + b and u, u),-, (0) for all
will discuss
an
u
E T_
1
(0)
-
improved spacetime model
named after Galileo Galilei who lived before Newton. The is that Calilei
reason
which is for this
emphasised different points than Newton, points which are more important to us nowadays. However, what will be referred to as a Galilei spacetime also incorporates ideas due to Newton.
20
1.
Local
theory
of space and thne
It is easy to check that maps of this form are isomorphisms of NewConversely, observe that any affine map '0 which maps any affine hyperplane E,, into some affine hyperplaneE,,, is necessarily of the form
Proof
ton structures.
O(x) where A is
a
linear map of
That A satisfies
follows from
Proposition
(0)
into
-F(x-o)v itself,
=
+ b E
v
R'*
(u,u),-,(O)
=
1.1.1 and the fact that
0
and o, b c A'.
for all
-r-'(0)
E
u
restricted to Ex is
invariant of the Newton
an
implies 0 (o + t) 0 (o) this equation is equivalent to v
o+
t. But
1
T
+
an
spaces.
Since the vector t is t- o
-
(Au,Au),-,(O)
isometry of Euclidean tion
A(x-o)
=
t
Observe that the choice of
is
o
spacetime the
t and therefore
-
(-r (t) v
+
equa-
b)
-
b
t.
irrelevant,
it
always
can
be absorbed
by
b.
We call the set of all
isomorphisms 0
Newton group JV. Given a Newton spacetime respect to which T (1, 0,
we can
find
of the Newton
a
basis
spacetime the
lei,. e,,J of R' with 0) T, and (u, v) (0) ..,
t
=
.
=
(1, 0,
I
Eni,j=i 6ijuiVi.
1.2.1 Non-relativistic
particles
Here
we very briefly indicate elementary aspects of Newton's theory of particle mechanics. We will only touch on those features which are necessary for later sections. This section is included for the benefit of
mathematicians.
A particle is
thought
to be
a
exterior structure. This is of
small material
course a
object without
interior
gross idealisation of many
or
macro-
scopic objects, but for
some purposes surprisingly good. Billiard balls typical examples. On the other hand, one cannot neglect the internal structure of a football. It will be noticeably deformed when hit. This contributes to its springiness and at the same time shows that the particle model is not adequate. An American football has a shape which are
contributes to its movement when it rolls
particle description would
be
on
a
flat surface.
bad approximation. Newton observed that even if all its structure
material ment in
object does
carry
a
Again,
a
neglected,
a
a
can
be
parameter which characterises its
spacetime. This parameter is its
mass.
move-
1.2 Absolute space and absolute time
Definition 1.2-3. A non-relativistic particle with mass (m,,y) where m E R+ and -y: t --> - (t) G An satisfies T( -y is called its world line in
curve
21
is
m
(t))
=
a
pair
1.
The
'
spacetime.
It has been first
expressed by Galilei (cf. Sect. 1.3) that under ideal particle which is not subjected to any external force moves straight line. 7
conditions
along
a
a
Definition 1. 2.4. A non-relativistic inertial
particle (m, 7) which satisfies It is clear that x
t(t
+
7('Y)
+
Y)
is
a non-
relativistic
some
0.
=
Of
special
interest to
derstand under confined to in the
particle
0.
non-relativistic particle is inertial if and only if -Y(t) point x E A' and some constant vector U with
a
for
=
compact subset of spacetime. It is best to think of collisions
a
sense
collisions of inertial particles. We unparticles any interaction of them which is
us are
collision of
a
of
colliding
billiard balls. But
we
explicitly
allow that par-
ticles break up or stick together. Since the collision is confined in space and time it is possible to speak in connection with a collision of incoming
and outgoing inertial particles. Let (Tni, ^ i)i=1,_k denote the incoming the outgoing inertial particles. Then particles and (m 3, 7
inertial the
following
laws
experimentally well justified.'
are
I:k
(i)
Conservation
of
(ii) (iii)
Conversation
of spatial of kinetic
Conservation
mass.
Eli momentum. I:k i= 1
Mi
=
j,
i=1
M/.,Yf 3
Mi7i
energy.
k
1
1
E2
M,
Mi
1
oli"07_1(0)
=
i=1
1: 2
i( j, /
M
j=1
It is easy to see that these laws are invariant with respect to isomorin the Newton group A(. It is clear that most particles do not move along straight lines. In
phisms this
case an
to take
a
external
force
must act
on
the
particle
in order to force it
different path.
Definition 1.2.5. A
(time dependent)
force field
_P
is
a
map F: An
7-1(0). We
only need -r( (t))
> 0 in order to
its future. The normalisation
guarantee that the particle moves into I synchronises each particle with
-r( (t))
=
the world clock t. 7
8
It is not
absolutely clear whether Galilei really meant straight lines or more complicated curves which take into account the shape of the earth. These laws are intimately linked to the homogeneity of space and time. This is the content of the Noether Theorem. For further details cf. any textbook on
(theoretical)
mechanics,
1. Local
22
of space and time
theory
given force field f equation
In
a
particle
a
-y
according
moves
to the differential
my= F.
(1.2.7)
In
particular, vanishing force implies that -y is an inertial particle. According to the physical interactions under consideration a particle may also carry a variety of other parameters besides m. As an example consider an electrical field f : A' ---+ -F 1 (0). Every particle (m, -y) carries -
another parameter q which determines the force with which the electrical on the particle, f qf.
field acts
=
1.3 Galilei's
theory of relativity
In this section
drop
of the structure of Newton spacetime in theory of relativity. We also argue that his revolutionary given the paradigms of the time. we
some
order to arrive at Galilei's
theory Galilei's
was
of
theory
has been motivated
relativity
not feel that the earth
moves
into any
by cosmology.
We do
preferred direction. It is therefore
plausible to believe that the earth is at rest and that all objects at the sky are moving around it: The sun rises in the East and during the course of a day moves to the West, and there are analogous descriptions of the movements of the moon and the stars. It was already well known that planets are not moving along strictly circular orbits. In the traditional cosmology of the Greek astronomer Claudius Ptolemeaus (ca. 100-160) this was accounted for by an elaborate construction using epicycles. It was a revolutionary act of Nicolaus Copernicus (1473-1543) to assert that the
around the
did
so
sun
sun
is the centre of the universe and that the earth is
just like
any other
in order to arrive at
a
planet
or
star
moving
(Copernicus 1543).
He
model in which movements would be theo-
retically more uniform and which would therefore be in better accordance with the teaching of the ancient Greek philosophers Pythagoras (ca. 570 b.C.-500
b.C.)
and Platon
1984)). However, (using more epicycles
his model
of serious
(i)
If
than
(ca.
428 b.C.-347
was
not
b.C.) (cf. (Kanitscheider
only technically
Ptolemeaus)
more
complicated
but also encountered
a
number
problems.
Copernicus
was right one should be sphere of fixed stars. If
able to discover
a parallaxis sphere and the earth rotate both around the sun with different velocities, then one should observe different angles a, 0 between two neighbouring stars according to the time of the year. (Cf. Fig. 1.3.1). (ii) Some passages in the bible seem to contradict the theory of Copernicus. In particular, it states that Joshua stopped the sun for a few
effect at the
the fix star
hours. This statement would not make have moved before.
sense
if the
sun
would not
1.3 Galilei's
(iii)
The model of
Copernicus
sun
the
other exception
was
around the
in the
sky
seems
to be
moves more
theory
of
relativity
23
inhomogeneous. While all planets circle definitely moves around the earth. No known. According to Ptolemeaus, every object is
moon
around the earth. Hence the traditional system homogeneous on a large scale and therefore to be
advantageous.
(iv)
The laws of mechanics
Imagine
(being
a
stone
seem
falling
fixed to the
to contradict
from the top of
ground)
would
move
a
Copernicus' hypothesis. tower. Since the tower
together
with the
earth,
would not expect the freely falling stone to hit the ground at the foot of the tower. However, exactly this is everyday experience. one
(Cf. Fig. 1.3.2).
Fig.
1.3.1. Parallaxis effect
Fig.
1.3.2. Tower
example
Problem
(i) has been addressed by Copernicus himself. He just assumed sphere of fixed stars is so large that the parallaxis effect cannot be measured. Ironically, the true radius is orders of magnitudes larger than the radius he proposed. (He was just concerned with making the effect unobservable). The other three objections have been answered by that the
Galileo Galilei Calilei
was
(1564-1642)
church authorities effect that
some
60 years later.
least successful with Problem
(for instance,
(ii).
While he could quote
Aurelius
Augustinus (354-430)) to the bible literally when it comes to
one should not interpret the questions of physics, the establishment remained unconvinced. One of the reasons has been the fear to set a precedence. If people started to doubt any part of the writing they could as well start to be sceptical about other parts which are closer to the main doctrine. Hence there was a major threat to the whole building of Christian belief. The theory of Copernicus was put on the index and Galilei after having written -
1. Local
24
a
theory
of space and time
brilliant but rather defiant
matter
mild
-
semi-popular
book
(Galilei 1632)9
sentenced to house arrest. He obtained this
was
punishment
after
a
public but
on
the
comparatively
insincere abdication of his scientific
assertions.
Galilei solved Problem
(iii) by careful observation (Galilei 1610).
telescope had just been invented and Galilei was as a scientific tool. He observed that the planet
one
of the first to
The
use
it
Jupiter also has moons show that the to and used this observation cosmological system of Ptolemeaus of the universe was not more homogeneous than the system of Copernicus. On the other hand, since it was believed that beyond the moon the world was filled with a medium very different from air, many philosophers doubted the accuracy of the telescope. They claimed therefore that it was doubtful that the telescope which was acknowledged to work well on earth could be trusted when applied to the position of planets. Galilei argued that the telescope was accurate with respect to all the known phenomena in the sky and that it was therefore justified to
use
it
as a
scientific tool.
of inertia which asserts that a constant movement had no influence on physical processes. Galilei supplemented this law with the important physical assertion that complicated velocities can be decomposed into simpler ones. According to this law the stone would keep its initial tangential velocity while falling down and therefore come to rest at the foot of the tower regardless of the velocity of the earth. It can be argued that this solution of the problem was the most revolutionary act in natural sciences and started physics as a scientific discipline in the modern sense. Recall that everyday experience seems to point against Galilei's law of inertia: If we set a wagon into motion it will certainly come to a stop after some while. Moreover, there was a generally accepted physical theory by Aristotle (384 b.C.-322 b.C.) which explained this experimental fact. (The wagon has an initial impetus which is responsible for the movement and which is used up during the motion.) Galilei gave many examples to make his law of inertia plausible and to show that it is a law for a limiting case without friction. For instance, he claimed that a stone falling from the mast top of a smoothly sailing ship would also reach the ground at the foot of the mast-10 Galilei solved Problem
(iv) by asserting
a
law
-
9
10
This book is
a
literary and physical
master
piece.
Even
today
it is well worth
reading! As compelling this example may appear to us, at the time there were some good reasons to doubt it. Since the velocities involved are rather small it would be difficult to verify Galilei's claim experimentally. Also, while the wind is blowing into the same direction. it is conceivable that the just blown to the right position. (To value the merit of such counter arguments one has to be aware that at this time, good, quantitative physics has not yet been available). Some of these arguments have already been answered by Galilei, who, for instance, circumvented the wind argument by
ship
moves
stone is
1.3 Galilei's
theory
Galilei realised that his law of inertia is not notion of absolute rest. Instead he
postulated
a
of
relativity
25
compatible with the principle of
fundamental
relativity. Postulate 1.3.1 move
the
are
(Galileian relativity).
relative to each other with constant
For any two observers which all physical processes"
velocity
same.
It follows that the vector t in the definition of Newton
defines absolute rest does not have of the a
originality
of Galilei that
spacetime which another sign
physical meaning. (It is Newton thought he had to a
re-introduce
concept which had already been shown to be superfluous). While we have lost the notion of absolute space we can still retain
absolute time. obtain the
Spacetime is then fibred by hyperplanes following simpler structure of sPacetime.
Definition 1.3.1. A Galilei spacetime is
(An, where
Rn
7-:
product
on
R is
--
a
T,
non-zero
the vector space
(., .)
_
,
a
(0)
const and
t
we
triple
)
(1.3.8)
,
linear map and
is
a
scalar
T-1(0).
.40.00.."M
Fig.
1.3.3. Relative space, abso-
lute time
The linear map
-r
defines
a
world clock
time difference between to events
model
by defining 7-(y
-
x)
to be the
and y exactly as in the Newtonian above. In contrast to Newton's spacetime we do not have
presented
x
-
the vector field t at
our disposal and therefore there is no absolute rest replaced "absolute space" by a distinguished family of systems" or "inertial observers". The notion of "Rest" can only
space. We have
"inertial
be defined relative to
an
Definition 1.3.2. Let
(i) 'T
"inertial observer":
(An,,F, (., .),-,(0))
A non-relativistic observer is
( M)
=
I
be
a curve
a
Galilei spacetime.
7: t
->
-y(t)
E An
such that
-
claiming that the physics in a cabin of a smoothly sailing ship would exactly the same as on earth. Strictly speakingl'he only considered mechanical processes.
be
26
1.
Local
(ii) A -/(t) (iii) A
theory
of space and time
non-relativistic inertial observer -y is
==
+
x
tt, where
R', -r(t)
t E
non- relativistic
a
curve
of the form
1.
=
observer y is at rest with respect to
relativistic inertial observer
-y(t)
=
x
if A(t)
+ tt
a non-
t.
Hence
given a non-relativistic inertial observer 7(t) x + tt we obtain splitting of spacetime into space and time relative to -Y. Physically, this amounts to regarding the observer -y as being at rest. We can also a
interpret
t
as a
observer -y
relative time axis. Relative to the
non-
have thus recovered the structure of
we
Notice, however, that this one non- relativistic
is
a
relativistic inertial
Newton spacetime.
only possible by arbitrarily distinguishing following defi-
inertial observer. This motivates the
nition.
Definition 1.3.3. Let t is called
R' be
E
a
vector with
T(t)
1. The
=
For any via the
given reference frame (t, -F) we obtain a map Y + A where -F(V) unique decomposition v =
1.3.1. A
Proposition time
A:
(An',r'
T-1(0)
pair (t, -F)
non-relativistic reference frame.
a
(0),
T
>
---
a
(i) O(x) A(x-o) 1, and (ii) T(v) (iii) (Au, Au),-,(O) =
An
0:
map
--+
invariant
if
vector
Rn,
+
v
R'
=
G
-r(x-o)v
+
and
A'
leaves the there
only if
and points o, b
are
cz
F-+
Rn- I
0.
Galilei spacea linear map
An such that
b,
=
(u, u),-,(O) for
=
all
u
Cz
T-1(0).
Proof. It is straightforward to check that maps of this form are isomorphisms of Galilei spacetimes. Conversely, observe that any affine map 0 which maps each affine hyperplane E,, into some other affine hyperplane E,,, is necessarily of the form
O(x) where A is Since
V)
A'. Hence
,T
a
=
linear map of
preserves we
(A (X
which in turn
r we
A(x-o) -
1
T
(0)
have
+
into
T(x-o)v itself,
v
-r(O(x)-O(o))
+ b
and o, b c An. T(x-o) for all x,
E R' =
o
E
obtain
-
0)
+
T(X-O)V
implies -F(v)
=
+
b-b)
=
-r(x-o)-F(v)
=
7-(x-o)
1.
The third property follows since A must preserve the Euclidean scalar I product The Galilei group 9 is the group of maps which leaves the Galilei spacetime invariant.
1.4 Einstein's
special theory
It should be noted that the Galilei
Newton's
of
theory
spacetime
of
is
relativity
compatible
27
with
described at the end of Sect. 1.2. The
particles well accepted as the correct model of space and time for more than 200 years. However, in the 19th century a theory of electro-magnetism emerged which, together with this spacetime model, was incompatible with Postulate 1. 3. 1. Still, scientists continued to think that the postulate would hold for mechanical processes. Galilei spacetime
1.4 Einstein's
as
was
special theory
of
relativity
We start with a discussion of the fundamental Michelson-Morley Experiment which indicates that the velocity of light has an absolute value c. These findings indicate that the set of all possible light rays form a further invariant of nature. We will see that this leads to the structure of a Minkowski spacetime (Theorem 1.4. 1), or, equivalently, to Einstein's special theory of relativity. We use the results from the two preceding sections to show that there is no need for additional structures in spacetime. In Sect. 1.4.2 we give a short discussion of some consequences of special relativity such as the "Twin paradox" (which, of course, is not paradoxical at all). The proof of the fundamental Theorem 1.4. 1 requires section 1. 1. 2
to
ti
t2
Fig.
1. 4. 1. A flash of
light
at times to
=
0,
t17 t2
In the 17th century two
In 1676
Olaf
important properties of light emerged.
R&mer discovered that the
by noticing that the there of the moons of Jupiter. did this
was a
velocity of light is finite. He yearly oscillation in the periods
physicist Christian Huygens (1629-1695) developed a theory of light (Huygens 1690). In a very superficial way, we may light as an analogon to water waves.12
The Dutch wave
view
paragraphs should not be taken too seriously by the overly simplified version of the wave theory of light just enough in order to understand the Michelson-Morley experiment presented below. Moreover, today the theory of quantum electro dynamics provides a much deeper understanding.
The
following
two
reader. We give -
an
28
1.
Local
In water cle
theory
waves
of space and time
each
thereby inducing
a
"drop
of water"
individually
similar movement
(with
a
moves
in
small time
a
cir-
delay)
neighbouring drops. All these moving drops together form a wave (cf. Fig. 1.4.2) Since each drop is influenced by the neighbouring drops these time delays accumulate and the whole wave seems to move. If two different waves meet then (in a very rough approximation) they simply linearly superpose each other.13 This will result in a characteristic (and often complicated) pattern, the "interference pattern". In particular, this superposition will result in a much larger wave if both waves are synchronised and in the other extreme they may cancel, Diffraction experiments indicate that this crude picture qualitatively also applies to light for which, however, matters are mathematically simpler. Again using a very rough model, one may think of the electrical field E at each point as oscillating up and down with respect of a fixed direction. The influence of neighbouring points gives rise of to a wave as described above. The wave length \ is the distance between two consecutive maxima and very small. It specifies the colour of light. If two waves are superimposed then the result may be brighter if they are synchronised. In the other extreme, the waves may even cancel altogether if the setup is arranged such that maxima and minima (of the same size) are superimposed. In this case the result is darkness. (cf. Fig. 1.4.3). In order to explain the wave nature of light one used to believe that space is filled with a substance called "ether" which plays the same r6le as the water for the water waves. An important problem would then be of
to determine the movement of the earth with
Fig.
1.4.2. A
respect
to the ether.
consisting of linked
os-
directionally dependent,
in
wave
cillations
Since the
velocity
of
light should
not be
the non-relativistic reference frame connected with the ether
a
flash of
light should propagate in concentric spheres (cf. Fig. 1.4.1). The corresponding picture in spacetime would be a cone. To be more precise, consider the non-relativistic reference frame of the ether, given by the pair (t, -r). Let o be the event at which the flash of light is emitted and For water waves, this linear superposition is in fact mation.
a
rather bad approxi-
1.4 Einstein's
1.4.3.
Fig.
Superposition of
special theory of relativity
29
waves
Future
light
Galileian
rela-
Fig.
1.4.4.
cone
and
tivity. The observer moving with spatial velocity V measures a
different centre
OV of the flash of light and therefore different radii di, d2 for its wave front
E,,
Ix
=
have
a
can
v
v
An
:
be
-r(x-o) 01 be the instant of time defined by o. We of A with spatial hyperspaces. Each vector f Eo+ttjtGR =
n
uniquely decomposed
into
0. A
light
-r(v)t+V where
-r(,U)
spatial velocity
6 describes the
==
the
G
foliation
respect to the
CO+
=
to
a
C-
spatial and temporal components, ray which is sent out at
curve o,
+
reference frame fixed to the ether
future light
y
=
An
:
R(t + cl a
in A n
x .
flash of light
C
EO with
Hence with
corresponds
cone
IICI12_1(0)(,T(Y_O))2
=
Y_01 ))7-1(0) Y-0
I
7-
(Y X) -
-> 0 -
I
C + X-0 of fuspacetime. The fact that the field x --> C: with Galilei to transformations invariant not are cones respect light the would enable reference frame of the to one measure (cf. Fig. 1.4.4) earth with the of the the ether. This was movement to ether, Le, respect in
=
ture
the aim of Albert Abraham Michelson
(1852-1931) and Edward Williams Morley (1838-1923) (Michelson 1881), (Michelson and Morley 1887) in their famous interference experiment (cf. Figs. 1.4.5, 1.4.6). A light ray is
partially reflected
at
a
half silvered mirror H. The part of the
light
ray which is not reflected at H is reflected at a mirror M and then partially reflected at H before reaching the observer 0. The part of the ray
which is
immediately reflected at H is reflected by a mirror M' and then passes through H to arrive at the observer 0. The distance
(partially)
between M and H is 1 whereas the distance between H and M' is 1'. Both
light rays have the same intensity when they arrive at 0. Here they produce an interference pattern which allows to measure the difference
30
Local
1.
theory
of space and time
0
H
lamp
1
H,," H 11 H H11
lamp
M
H MMU
0
M/
M, MIMI
Fig. 1.4.5. Michelson-Morley experiment, at rest relative to the
Fig. Michelson-Morley experiment, moving relative to the
ether
ether
1.4.6.
of the distances which each of the fact that the
wave
light ray has travelled (Here one makes use lengths of visible light are extremely small
and that the superposition effect allows to measure the distance which a light ray has travelled at an accuracy of half a wave length). Since the
laboratory is relativity the
at rest with
HM and the
velocity of the earth. Let c (E R+ be the modulus of the light in the ether, Y be the velocity of the earth relative to
velocity
of the
the ether and
the
ct,
1 +
jjVjj,-1(0)tj.
the distance Ct2
:--=
1
-
7--
1
The other part of the same
Y
(cf. Fig. 1.4.6).
in time tj and
_
1/2
2
21/c 11,U112,r_,(O)IC2
+
cover
(1.4.9)
'
to M' and in
Jjyjj2_j(0)tj2.
gives
21'1c
2t'
(1.4.10) C2
(0)
We
The first part of the distance
cover
light ray travels in time t' from H H, thereby covering the distance
time back to
2ct' This
11
M,
earth, according to Galileian depend on the angle between
If it travels from M to H in time t2, it will jj'Ujj-r-1(O)t2- These equations imply
tl + t2
the
to the
first that HM
assume
ray will travel from H to
light ==
respect
interference pattern should
are
interested in the time difference zAt
Since the number
I I 61 I
=
2t-
(tl + t2)
for both
paths.
is very small we only need to calculate the , (0) Ic time difference to second order in I JUj Jr 1 (0) 1c. -
1
-
2
At
=
tl + t2
-
2t'
---
I
-
11
1
-
JjUjj2_1(0)1C2 Ir
-
C
1
11,6112
Ir
_I
(0) /C2
1.4 Einstein's
.j(jj,yjj2_j(0)/C2)j/ I 11,611 2(0) /C2
21+
2
C
of
special theory
relativity
31
j/
_
7'
-
-r
2
(_1
jj
j(jjyjj2_j(0)/C2 + jjjyjj2_j(0)/C2j Jjqj2_j(0)/C2
+
2
C
2
7'
2
1
1
(jj (_1 (jjUjj2_j(0)1C2) +
C
2
+
jjUjj2_j(0)1C2j
T
1
11,U112 -1(0)
2JI
C
C2
C
1 + J1. This
I
11,UI12_1(0)
2
C2
+
gives
a
displacement
C, At
zAZI1
2
(I jjUjj2_j(0)/C2)
X
where 1'
+
2JI
(0)
(
::Z
=
A
A
C
2
per
C
wave
length
JJV112 I + 2
21(o)
C
A of
)
-
depends crucially on the length difference accurately enough. In order to overcome this difficulty Michelson and Morley turned the whole setup by 7r/2 and It follows that the interference
51 which cannot be measured
measured the interference difference. For the rotated setup we must set 2t' ,At tj t2 and interchange 1, 1'. An analogous (but in the details =
-
-
calculation
slightly different)
C
zAZ_L
The relative
1
jjqJ2_1 (0)
2(-61)
A
C2
C
At -Z
gives 7'
+
-
,
A
displacement depends
on
61
21
ljqJ2
A
C
C2
only
up to second order and is
given by AZ
If
one
assumes
spectral
;:
-zz
-
AZ_L
sun
5.461.
1(0)
1
-
2
rests relative to the
10-'Om and has I
experiments had
61) 1
ether,
a negative possible explanation of this negative outcome
uses
21m then
0.4 which is well in the range which
all such
However,
AZI,
that the
line with A
obtain AZ
A
=
can
the
one
Hg
would
be observed.
outcome.
is that the earth rests
with respect to the ether. But the earth circles around the sun which itself rotates in our galaxy. Since in the course of the year the earth changes its
velocity
direction relative to these other
that all year round the
neglected. light
it is inconceivable
can be velocity explanation would be that light moves like particles Galileian relativity would apply to light as well. Since
Another
and that therefore the
velocities,
of the earth relative to the ether
was
from earth bound
sources
and the observations have been
32
Local
1.
theory
of space and time
made
on earth, this assumption would explain the negative outcome of experiment. The Michelson-Morley experiment has therefore been
the
repeated using
light
again with negative results. Now one could starlight is reflected at mirrors fixed to the earth, the light reflected light should be viewed as being produced on the earth. This would explain even negative results for star light in the framework of Calileian relativity (Hasse 1995). While this explanation is conceivable, it would demand a new theory of reflection. It is much simpler to assume that the velocity of light is independent of the movement of its source. This is the traditional interpretation which we will adopt in this book. It has been given further support by many consequences of the resulting theory (for instance, the possibility of obtaining huge argue that
star
as soon
as
-
the
amounts of energy from nuclear fission and nuclear
Since
fusion).
interpretation of the experiment of Michelson and Morley is in contradiction to Galilei's theory of relativity we have to reconsider the foundations of spacetime. In order to do so we start with our new
insight
our
about the nature of
future
light cones is an following we will chose c
=
2.99792458
light propagation,
invariant structure of units such that
c
i.e. that the set of
possible spacetime. Here and in the
(In
1.
=
the
SI-system,
one
has
108M/S.)
-
Postulate 1.4.1
(Invariance of the future light cones). Spacetime can be identified with An together with an field of future light cones C C + x-o, X E An.
invariant
=
We start the
investigation of this postulate, by first determining all maps light cone structure invariant. To simplify the dischoose again a non-relativistic reference frame (-r, t) and de-
which leave this future cussion note
we
by (-) the induced projection
of Rn to
-r-
1
(0). Defining
the bilinear
form 770: Rn
we can
C.,
=
x
Rn
R,
(u,v) i-4,q(u,v)
--F(u)-r(v)
=
ly
E
An
:
77(y-x, y-x)
light
cone
and
=
0 and
C1
Cx-
the past
Cx
y E
X
light
cone
transformation which leaves the field of the future
must also leave the field of
form qo is
a
light
Minkowski metric
as
cones x
-*
This
means
that there is
an
Cx
at
:
-r(y-x)
01.
It is clear that
x.
light
?7(0(x)-O(y), O(x)-O(y)) exist an L E O(n, 1), an a E R \ 101, an 0 E A', and a b E Rn such that TP--TT]1 5 aL(x-o) + b for all x E An. O(x)
Theorem 1.4.1. Let that
0 and 0
map
n
> 3
lightlike
and
0:
Rn
vectors into
-
-
=
=
=
-
1
There
are
several
proofs of
this result. For
instance, Benz (1992)
that the theorem follows from the fundamental theorem of
His
proof rests
on
Laguerre
ge-
who gave a particularly elegant affine results in geometry which have been given
ometry. We will follow Alexandrov
proof.
shows
(1975)
in Sect. 1.1.1.
0 and by Proof of Theoreml-4-1. Let y E Q,. Then 77(y-x,y-x) 0. The last equality implies assumption q(O(y)-O(x),O(y)-O(x)) 0(y) G Ck(x) and therefore that 0 maps generators of the light cone Cx into the light cone C,6(,,). Now assume that P is a two-plane which intersects C., in two genera1., tors lx lX. We will show that O(P) is also a two-plane. Since for any y 1,, and Py which is parallel to the cone C. intersects P in generators 1. 1',, (and similarly for y' E 1.,), P is ruled by two different families of parallel generators. Since 0 maps generators into generators O(P) must also be a surface with two different rulings and, by Theorem 1.1.2, be affinely equivalent to either a plane, a rotational hyperboloid, or a hyperbolic paraboloid. Each generator in P of one family intersects all generators of the other family. Since in a rotational hyperboloid the generators at opposite points of the circle c(s) lie in parallel planes, O(P) cannot be affinely equivalent to a rotational hyperboloid. To see that O(P) cannot be affinely equivalent to a hyperbolic paraboloid note that in a hyperbolic paraboloid all generators of a given family are parallel to a single 2-plane and that this property is affinely invariant. Consider 0(y) c 10(x) =
=
=
15
The
proof of this theorem requires the material presented proof is essential for the following.
The theorem but not its
in Sect. 1.1.2.
P. 34
34
1
Local
-
and fix any C,6(y) which
along lo(,,)
theory
of space and time
2-plane Q. Then there are exactly two generators 11, 12 Of are parallel to Q. If we now (parallely) translate the cone
we see
that the generators of the translated
cone
which
are
parallel to Q must also be parallel to 11 and 12. This implies that O(P) is generated by a family of parallel lines. Hence the hyperbolic paraboloid degenerates to a plane. We can now show that 0 is an affine map. Let 1 be any line and x E I and consider two different planes P1, P2 which contain 1 and intersect Q, \ fxJ. The intersections of these planes with Cx consist of two generators each. Then O(Pi) and O(P2) are also planes and their intersection 0(l) a line. It follows from Theorem 1.1.1 that 0 is affine. Since the property
?7(y-x,y-x)
0 is translation invariant is linear. Let t be
vector with
a
77(e,e) Then 0
0
=
be
0
77(t, t)
=:>,q(O(y)-O(x),O(y)-O(x)) =
generality
-1 and let
n(e, t)
1,
=
=,q(ct, et) implies 0(e)) -77(0(t), 0(t))
=
without loss of
we can
c
be
that
vector with
0.
=
(1.4.11) andtherefore
=
=:
assume
a
a,
n(o(e), 0(t))
=
Any vector v can Equations (1.4.11)
0.
decomposed into v v,c + vtt, where e satisfies and v, vt E R. We obtain q(o(v), 0(v)) aTI(v, v) which implies that 1 0 leaves the quadratic form associated with q invariant. But then it must also leave Tj invariant. I =
=
[I
p.
401 Definition 1.4.2. Let 77 be called Minkowski spacetime.
Minkowski metric.
a
The pair
(An, n)
is
Using orthonormal bases it is easy to see that all Minkowski spacetimes are isomorphic, i.e., we can speak of "the" Minkowski spacetime. In a Minkowski spacetime there is no designation of future and past.
(Observe that we needed the 1-form -F in order to define the future direction.) Observe that the set C,\ I ol consists of two connected components, say C0+ \ f ol and Co- \ f ol. We may now choose C (respectively, CO-) as the set of events in Q, to the future (respectively, past) of o (including o). Hence C0+ is the set of all events which can be reached by a light ray with
in
source
o.
By continuity, this
at any other event
x
E
An where
definition coincides with v
E Rn
(i) (ii)
a
w)
q (v,
> 0
or
we
vector
v
w)
can
E
< 0
w
alternatively Rn with
v)
for all
for all
==
(x-o)
=
w
G
E
C
< 0.
C
Then
we
and q (v,
and 77 (v,
w)
C0+-
+
previous definition
our
vector with q (v,
either 71 (v,
Hence a
be
also determines the future direction
C.+
Observe that this
in the
w) < 0
> 0
for all
for all
define the future direction
q(v, v)
01. E
=
:
be the invariance group of the
light
future light cone at light cone
and the past
structure and P be
cone
the group of Poincar6 transformations, P fx --> L(x-o) + b : b c orientation Given 0 L an E [v], we call P+ P+ n P (m, 1) 1. Rn, 0 E An, =
=
the group of time orientation preserving Poincare transformations. The discussion above suggests to reduce the group )5 to a subgroup 73+ by asserting that the elements of 13+ map future light cones into future light cones.
(An, 77) JbE
Lemma 1.4.1. Let
J5+
entation. Then
Minkowski
a
b(C,+,)
=
=
spacetime and [v] a time CO+v)(0) I is a subgroup of
ori-
Proof. Clear.
The
Lemma 1.4.2.
there is
only if there
is
R
transformation 0 is an element of P if and only if and a 0 E P such that 0 ao; 0 E )5+ if and E R+ \ 101 and a 0 E P+ such that 0 ao.
\ 101
=
=
an a
This follows
Proof. A
G
an a
directly
form Theorem 1.4.1.
it is conceivable that there exist other fundamental invariants
priori
of space and time which would restrict the group even further. We will now show that because of the validity of Euclidean geometry and the
principle
of Galileian
relativity
1.4.1. Fix
Proposition
1.3.1 this is not the
a non-
relativistic, inertial observer
and consider its associated Newton spacetime each
x
E A let
E
=
yE
An
:
case.
F(y-x)
=
(An, t,,r,
t
o
+ tt
(0)).
For
-->
01.
metricq which generates the light cones -* o + tt. as Further, this Minkowski by metric is unique up to a multiple. (ii) Let x cz An The map 0 E )5 restricted to the Euclidean space E.,, (., .),-,(0) is an isometry if and only if 0 E P.
(i)
There exists
a
Minkowski
the observer t
measured
.
(iii) Let P' be a subgroup of the Poincar6 group P such that E., there exists a 0 (a) for each Euclidean isometry 0: E with
(b) for
01
All
0
E
a
0
(E
P'
V),
each non-relativistic observer t
there is
(c)
=
E
P' with
O(o
+
R+t)
=
o' + tt' with
o' +
P' preserve the time orientation.
R+V,
77(t', V)
< 0
36
1. Local
Then P'
theory of
space and time
'P+.
=
Proof. (i):
The adapted Minkowski metric q is just given by q (u, v) Observe that T(w) -7-(u)-r(v) + -77(,T, w) for all W E Rn. (0) (ii): This follows directly form Lemma 1.4.2. (iii): We will first show that for any two different vectors eo, eo with -1 and 77(eo, e0 ) < 0 there is a Lorentz transfor71(eo, eo) q(eO, eO) mation L which leaves spanf eo, e'01 and its q-orthogonal complement spanf eo, e'01 Jf w E A : q(w, eo) q(w, eo) 01 invariant. There is a vector v c R' with 0. Since q (eo, v) we have eo I I eo + v and q (eo, v) 77(v, v) > 0 and the vector el vlV'rq(v, v) is well defined. By definition it satisfies 71(eo, el) 0 andq(el, el) 1. We complete the set of linearly vectors to an 77-orthonormal basis f eo, independent f eo, el I el, e,,,
=
=
-
=:
=
=
=
=
=
=
=
.
of R'. Let L be the Lorentz transformation defined
Leo
(eo
+
Lei
(el
+
Lei=ei
v)1 1 JJvJJR_ JJVJJR`-180)1V1
.
.
7
by
2
-
eo)
IIVIIRI-
2 I
,and
ViEf2,...,nJ.
It maps eo into eO and leaves the subspaces spanf t, 61 1, spanf eil (i E 2, ., n 11) invariant. This transformation is called a Lorentz boost. -
..
We will
now show that the group G generated by all Newton transformations with respect to the inertial frame (eo,,q(eo, .)) and all Poincar6 transformations 0 of the form O(x) o + L(x-o) where L is =
Lorentz boost coincides with the group P+ of time orientation preserving Poincar6 maps. It is clear that G is a subgroup of P+. To show the converse Let o, o' G An and f eo, two oren 1 1, f eo.... e'_lJ n thonormal bases with respect to q. We have to show that the Poincare a
.
transformation V) which maps i is
a
actly
into o' and
(respectively, e ) with 77 (6k 6k) -1) and 77(ek,t) -- 0 (respectively, 77(el,t)
vector ek
=
-
i
renumber the basis vectors such that ek let 01 be the map x -4 x + (o'-o). If eo tor
this
v
E
+ ei into o' + ei for all
o
of maps in JV and G. In each basis there is
composition
one
o
-
Rn with
let
77(v, eo) be
=:
0, 0
o
+ tt is also
a
relativistic observer and for this observer the non-relativistic and
specialspecial-
relativistic definitions for rest space with associated Euclidean scalar product and time difference coincide. The relativity Postulate 1.3.1 im-
plies that for every other special-relativistic inertial observer o' + tt' and time differences lengths and angle should be measured by by Att, (-, -) In conclusion, a Minkowski spacetime together with a time orientation contain all the geometric information of spacetime. This geometric structure is mathematically simpler and more elegant than the structure -
of
a
Galilei
spacetime.
1.4 Einstein's
Remark
1.4-1. The geometric
Minkowski
1905)
(1909).
special theory of relativity
structure of
spacetime
But before him Albert Einstein
39
discovered
was
by
(1879-1955) (Einstein
had realised that absolute time does not exist and
came
to
an
equivalent but less elegant description of spacetime. His work contains the main physical discovery which justifies to speak of Einstein's special theory of relativity. An important precursor of Einstein was Hendrik Antoon Lorentz (1853-1928) whose explanation of the Michelson-Morley experiment anticipated the length contraction17. 1.4.1
Causality
We start with
in
special relativity
terminology
some
Definition 1.4.6. Let
(A',,q)
which will be
justified below.
be the Minkowski spacetime.
A vector w is called spacelike, if j7 (w, w) > 0, timelike if 77 (w, w) < 0. A vector w is called causal 0, and lightlike (or null) if 77(w, w) if it is timelike or lightlike.
(i)
==
(ii)
Let
[t]
directed
A
(iii)
be
a
time orientation. A causal vector
(past directed) if q (u, t)
If
open set U C An.
are
Z2
connected via
J+ (Zl)
a
-
will prove the theorem. If x-y is lightlike then A (x, y) A(x, y) is formulated entirely in terms of causal relation-
ships, A(O(x), 0(y))
p. 40
a
y the
4
A(x,y)
=
equivalent =
to
0.
for
1
451
philosophically more appealing to demand causality than inlight cones but it should be noted that our original version to the actual experiments motivating special relativity.
It may be
variance of the is closer
1.4.2 Length contraction and time dilatation Since there
are no
that
absolute time
a
or
absolute space it should not come are observer-
in space and time- differences
lengths surprise dependent. To simplify notation at
let
JJwJJ
==
-\/ (_ww)
for any
spacelike
1.4 Einstein's
special theory
of
relativity
41
Fix an event x E A' and an infinitesimal observer t and conrod which rests in the affine rest space x + t 1 of t. If with respect to the observer its endpoints are given by x and x + f at a time to, then it will sweep out the subset vector
sider
w.
a
S
C:
z
A'
z
X
=
+ M +
/-tt, A
E
[0, 1],
E
RI
Lt
t'
Lt
K
O
Lei
Fig.
in
1.4.7.
Length
contraction
Fig.
1.4.8. Time dilatation
spacetime (cf. Fig. 1.4.7). We complete It, ei 111f Ii to an orthonorf t, ei, e,,- I I of A'. Now consider a second infinitesimal ob=
mal basis
.
t' who
.
.
,
(relative to t) with the velocity v II UII -ei. Let L be a Lorentz transformation which maps t into t' and leaves spanf t, el I invariant. We have V Lt _JIVI, Lei (ei + I I VII t) / _IVI (t + v) / I 12, server
moves
=
=
and
VIT
=
V-1
=
-
-
Vi E f 2,.. n Lei ei 11. The rest spaces relative to t' are all parallel to Ltj-. In order to determine the length of the rod with respect =
-
.,
to t'
we
must calculate the
S n
(x + Lt-L )
length
of
11 yllt..
ei + =
X
+
01
F_
a .
,
G
[01 IV III
V
It follows that
observer t' is
one
11 'Il
=
of the
reasons
V/1
-
11,61
-
Iltll
=
50
2.
Analysis
on
manifolds
A subset U C V is called dense
where t! is the closure if V of U, i.e., the smallest closed set containing U. (vi) A collection of open subsets fUa'jae_-A is a sub-basis if all finite intersections of sets U,,, form a basis of the topology -r.
(p)
Lemma 2.1.1. Let M be
a set and jUaja(EA be any collection of subM. Then there is uniquely defined of M which satisfies UaEA Ua topology -r of M such that jUafaEA together with the empty set 0 are a sub-basis of T.
sets
=
I
Proof.
This follows
from the definition of
immediately
a
I
topology.
We
can now describe those topological spaces which cannot be locally distinguished from A'. Let (M, T) be a topological space which is Hausdorff, i.e has the property that for any two different points x, y E M there are neighbourhoods U of x, V of y which satisfy U n V 0. The topological space (M,,r) is locally indistinguishable from A' (considered as a topological space) if each x E M has an open neighbourhood U such that there is a homeomorphism W: U -* V c R'. 2 The pair (U, o) is called a topological chart of M. Since for each open subset 1 C U the restriction of o to 1 is also a homeomorphism onto its image, we have =
the
following compatibility property. Let
(U, W)
and
0. Then the
(1 , 3)
map
be
topological charts
o
p-:
p(U)
n
( (a)
of
(M, -r) with UnO p(U) n ( (a) is
--+
a
homeomorphism. We would like to have not a
structure which allows
there is not
an
is
only
a
topological
structure
on
M but also
the tools of
analysis. Unfortunately, independent definition of a "differentiable space" which the definition of a topological space. To get an idea how us
to
use
analogous to this difficulty can be overcome we can view the charts (U, W) as a way to induce the local topological structure of R' on M. To be more precise, let M be any set, WiliEl be a set of subsets of M with Uj, Ui c-I M, and pj: Uj > Vi bijective maps onto open subsets of R'. We can now attempt to define a topology on M by using the sets f oj-'(Wj) : Wj C Uj is openj as a sub-basis for the topology of M. In order to get a topology consistent with a local description we have now also to demand the compatibility property above. Still, the resulting topological space could fail to be Hausdorff (cf. Fig. 2. 1. 1). Since this is a local property, we will demand it in addition. We have now defined a topological structure on M which is locally indistinguishable from the topological structure of R1. This definition
2
can
be carried
Recall from the definition of the x --+ x
-
o
is
a
homeomorphism.
over
to the differentiable structure.
topology
of A' that the map
V),:
X,
--+
R,
Manifolds
2.1
Fig.
identify
51
A topological space locally homeomorphic to
2.1-1.
which is
R but fails to be Hausdorff
Definition 2.1.3. Let I be
(i
c
index set and
an
A Ck-atlas of M is
which is
a Hausdorff. 1) of local homeornorphisms such that
(M,,r)
be
collection
(i) each Uj is open and connected,' (ii) each x E M is is contained in some Uj, (iii) for each i, j with Uj n Uj V - 0 the map oj (Uj n Uj) is a Ck -diffeomorphism.
oj
o
a
topological Uj
oj:
M
oj (Uj n
:
oj
C
space
-4
Kn
Uj)
pairs (Uj, pj) are called charts of M. A chart (Uj, oj) is centered at 0. Two charts A Oa) (a if x E Uj and Wi (x) 1, 2) are called compatible if they satisfy the compatibility condition (iii). Two atlasses are compatible if each pair of charts in their union is compatible. A Ck_ atlas A is called maximal if any Ck -atlas containing A coincides with The
x
E M
=
=
7
A. Remark 2. 1. 1. In the
erty that Wi
(Wj)-1
case
K
=
Ck -atlas (k
C every
>
1)
has the prop-
analytic, i.e., is locally given by its Taylor series. This follows immediately from the fact that C-differentiable functions are analytic. o
For technical
reasons
ogy of M has
a
(cf.
Sect.
2.1.2)
we
countable basis. This
will also demand that the means
that there
are
topolcountably
such that any open set VV is the union of sets Vi.
fVjjjeN
many sets
is
(M, -r) be a connected topological space which is Hausdorff and which has a countable basis. (M, -r) together with a maximal Ck -atlas is called a Ck -manifold. A C'-manifold is also called We will often refer to smooth manifolds simply as a smooth manifold.
Definition 2.1.4. Let
manifolds. A subset N C M is x
E
N there is
W(U)
n
ly
K'
E
submanifold
a
is
Observe that
a
manifold of M
chart :
an
m-dimensional submanifold
(U, W) of M
y'+'
often called
=
a
centered at
x
y' 01. An (n hypersurface. ...
subset N C M
=
can
=
be
a
of M if for
such that -
each
W(N n U) l)-dimensional
manifold without
=
being
a
sub-
(cf. Fig. 2.1.2).
The
following lemma guarantees that a manifold is determined by any (not necessarily maximal) Ck -atlas (k > 1) which is compatible with the given maximal Ck -atlas. In practice, it is therefore sufficient to work with any
given atlas. It
can
be shown
(Hirsch 1976)
that any maximal
2.
52
Analysis
Ck -atlas
contains
manifolds
on
Fig.
2.1.2. A manifold M c R 2which is
not
submanifold of R2
a
subatlas which is C'. Hence for most questions it
a
loss of
generality to consider only smooth (i.e., C"O) manifolds.' On the other hand, it should be noted that Co-manifolds are really more general. We will not consider such manifolds in this book. is
no
Lemma 2.1.2. Let A be
Ck -atlas. Then there
a
is
a
unique, maximal
Ck -atlas containing A.
Proof. Let
B be the set of
are
Clearly, compatible. To see
with set
x
E
01 (U
Ck -charts which
A C B.
chart in A.
Any
this let
U. Then the maps o o n V, n V2). It follows
Ck. That 01
0
-1
(02)
is
Ck
are
compatible which each
two charts
(Vi, 01) and (V2, b2) in B x G V, n V2 and (U, o) be a chart in A (0j)-1 and 02 (p-1 are Ck in the open by composition that 02 (01) -1 is also o
0
be shown in the
can
to prove that B is maximal and
same
way. It remains
unique. The first assertion follows from
the definition of B. Assume that B' is another atlas every chart of B' is
B
by the definition
as
well.
compatible with each
chart of
containing A. Since A, it must belong to
of B. Hence B' C B and the second assertion follows
I
Example 2.1.1. Consider the cylinder which can be obtained by identifying opposite sides of the rectangle [a,, bi] x (a2, b2)
f (a,, y) As charts
we can
can
a
:
y c
x
-
-
(b2
(a2, b2)
good
al x
2
-
a2), Y)
-4R
[a2, b211
for
x
for
x
M
maps 9
0
are
Ck -differentiable if
(Wa)- I:
K'
--
K' and
Ck -differentiable. The set of all Ck -differentiable and from K' to M are denoted by Ck(M, KI) and
are
to K'
K' and h: K'
54
Analysis
2.
on
manifolds
It is easy to
see
to any
atlas. For N
given
that this definition is satisfied if =
K',
M
=
f
Ck with respect
is
K', the definition coincides with
the usual definition of
differentiability in elementary analysis. Recall from analysis that a continuously differentiable map F: K' Km has rank r at x E K' if the subspace DF(x)K' C K' has the dimension Let
r.
Nbea Ck -differentiable map and x G M. If (Ul ol), x and (Viol), (V2) 02) charts centered at
f:M--
(421 W2) f(x), then the rankof the mapsolof oWl-1 at Vl(x) andV)20fo 02_ I at (P2(X) coincide. We can therefore speak of the rank of f at x and the charts centered at
are
following definition
is
independent
Definition 2.1.6. Let M. Let
be
(U, W)
(any)
of N centered
chart
(i) (ii)
has rank is
W(x),
(iii)
is
is
an
local
if 0
f: a
Proof
Since
if D(o
map and
(V, 0)
be
x
E
(any)
f
f o W-' has rank r if D(o o f o W-1) is x
and
x
o
f
o
p(x),
at an
o-')
injective linear
is
a
map
surjective linear
local
at
x
if D(o
o
f
o
o-')
is
a
bijective
-+
r
at y.
immersion (respectively, a submersion, x) then it is also an immersion (respectively, diffeomorphism) at any y E W.
diffeomorphism a
o
x
Ck -differentiable
N be a Ck -differentiable map of rank r at neighbourhood W of x such that for each y E W
M
the map f has rank ry > In particular, if f is
submersion,
The map
diffeomorphism
M. Then there is
local
of
a
M centered at
o(x).
Lemma 2.1.3. Let E
x
N be
W(x),
linear map at
x
f (x).
submersion at
an
map at at
(iv)
at
---*
chart
immersion at
an
at
r
at
M
f:
of the chosen charts.
an
at
D(oof oW-1)
is continous the existence of r
a a
linearly indepen-
dent vectors in D (o o f o o- 1) (,p(x))Kn implies that for y close enough to x there are also r linearly independent vectors in (DO o f o r 1) (,P(y))Kn. Hence the rank of f cannot fall in
a sufficiently small neighbourhood of given point. For the second statement observe that immersions, suband local diffeomorphisms all have maximal rank. p. 47 11 _ZT-11 mersions, F a
I
p. 61
2.1.1 Construction of manifolds
plays a fundamental r6le from infinitesimal assumptions. In this section we show that the inverse function theorem and also similar theorems carry over to manifolds. A special case (Proposition 2.1.1) allows a construction of submanifolds without specifying
In
analysis,
the inverse
function
theorem
because it allows to draw local conclusions
an
Atlas.
2.1
Manifolds
55
We will
who is not
skip this The
occasionally use the results of this section. But the reader primarily interested in analysis on manifolds is advised to
section and to return to it when needed.
following lemma
is
a
consequence of the inverse function theorem.
Lemma 2.1.4. Let U C a
Kn, V C K' be oPen, x E U, and f : continuous, differentiable map with constant rank r in U.
U
--
V be
Then there exist
(i) an open neigbourhood 0 C U of x, (ii) a homeomorphism 0: 0 f y E Kn : I y I < 11, (iii) an open neighbourhood V D f (U) of f (x), (iv) and a homeomorphism 0: Iz c KI : IzI < 11 -->
such that
f
=
Oop, oo where
MY',-
ly n)
-4
f)
(yi.... Y" 0'...' 0).
=
I
Proof. Let E be the (n r)-dimensional vector space E Iv Cz Kn and F Kn be an r-dimensional vector space with Kn C Df (x)v 01 =
-
=
f e,+,,. enj be a basis of E and f el,. ..' e,j be a basis of F. 4 For each y C Kn let A'(y) be the ith component of y with respect to the e basis f el, The vectors r 1) are a basis fi Df (x) ej (i E f 1, , en 1. E(DF. Let
=
.
.
.
of the r-dimensional vector space Df (x)Kn C K'. We choose linearly is a basis of K". independent vectors f,+l.... f, such that f fl, , f,,, I .
For
z
E K'
let
y'f (z)
.
.
be the ith component of
z
with respect to this
basis. We define
A'f (Z)
0
1-tf (Z)
0
A (Y)
+I(Y)
E
Kn,
,(Z)
f0
=
An (Y)
E
Kn.
0
e
The map g: U
-4
IK',
continuously differentiable
is
by
y
and
-4
[t
o
Dg(x):
the inverse function theorem there is
f (y) Kn a
+ A (y) -4
Kn is invertible. Hence
neighbourhood 0
such that g is a diffeomorphism from 1 onto the open set is an c > 0 such that B,(x) := Iz E Kn : Iz g(x)l < Ej C define g -'(B, (x)) and the bijective map
c U
g(Z ).
-
U
---
BI (0)
C
Kn'
Y
(g (Y)
This numeration of basis vectors of E 8 F may more practical later on in the proof.
prove to be
-
of
x
There
g(U).
We
g (x)).
seem
slightly
odd but will
Analysis
2.
56
The rank of for all y
for all
are
both
implies that dim(Df (y)Kn) bijective for all y E U and Dg(y)(v) the maps Df (y): F -+ Df (y)F and --4
K' (D
f01
C
v
o
=
=
Df (y)F be
show that the map h
y'+',..., yn. We write K' Kn. Since f (y) hQ it f (y)
E
K' E) K`
101
r
f
o
0-1
Kn
the inverse to the
does not
Kr E) K` and
variables any
=
is
Let vy: Kr E)
bijective. now
F
E
v
U which
on
Dg(y)
Df (y)F
/-t:
latter map. We will
manifolds
is constant
f
U. Since
E
I-t(Df (y)v)
on
ED
-!A(y)
-
Iu
o
depend
v
=
f (x)
the
on
V1 (D V2 for
!A(x))
(D
we
have
eDf(y)v
Djhj-!,Aof(Y)ED-,!A(Y)--! PO f (x) E) + D2h, ittof (y)(3)-i cx(y) -1
vy
o
p
o
Df (y)
into this
D2h,.I,,Of (Y)E)i,\(Y)- i./,Of(x)E)iL,\(x) where ay is the linear map
ay: Kr ED
Df (y)v
Av.
o
equation implies Av
=
ay
o
/-t
o
Df (y)v,
evy
-
Djhj-,!,UOf(Y)(D-,1A(Y)--! tto f (x) E)
is invertible and A maps K' onto =
0 in order to prove
D2h
=
-1 A (x) ,
f 0 j Kn-r
0. For
E
v
F
0. The map 0 and and therefore ay o ft o Df (y)v coincided on F with Dg(y) and is therefore bijective. In particular,
A(v)
have
pol)f /-t o Df F
==
Since can
--+
y
to show that ay
only
o
given by
,,pof (y)E)!A(y)
Since y have
we we
101 -4K',
o
P
-i.\(x)
c
Inserting Df (y)
0
A (x)
write
We
=
=
101
Kr
we
which is the domain of 7y. Hence ay vanishes. h(y) does not depend on yr+1.... yn
have proven that
h(yi)
identify
instead of
h(yi
(D
Y2)-
K' with Kr ED K'
r
and write
w
W1 E) W3.
=
be the canonical basis of K` and let
jbj,...,b,-rj spanjfr+j,...,f,,j We define the map
V): B, (0) This
C
be the linear map which is defined
-r:
K`
by -r(bi)
=
0 by
K'
--4
K'
ZI ED Z3
-'
h(ZI
+ A (f (X))) +
'T
(Z3)
-
implies 0
Pr
0
O(Y)
-=
0
0
=
V)
0
=
-0 E
Pr( g(Y)
g(x))
Pr(1P
-IP
0
Pr(P
we
0
f (Y)
0
f (Y)
-
A
0
0
fW
f (X))
+
1A (Y) -1AW)
Let --,
fi.
=
-
h(/,t
o
f (y)
-
p
o
f (x)
+
A(f (x)))
57
Manifolds
2.1
=
-
h(I-c
o
f (y)).
f
6
-
projection of tt to the span of the first r canonical basis 0 h o tt. Equation D2h by A. Then we have h o 0 0 0 (Y) h in which turn f IP (Y) gives Pr ( 1 A (f (y))) implies f (y) We still have to show that V) is a homeomorphism. But this follows I since both h and T are homeomorphisms.
Denote the
vectors of K'
=
=
Corollary 2.1.1. Let Mbe an n-dimensional, N be an m-dimensional N be a Ck -differentiable map which has conCk -manifold, and f : M stant rank in a neighbourhood of x E M. Then there exist charts (U, v), (V, 0) centered at x, f (x) such that W(x) 0, O(f (x)) 0, and --
=
=
0
0
f
0-
-
1 :
Kn
(xi
Km'
_
I ....
Xn)
F
---
(XII
...
I
xr, 01
...
10).
Proposition 2.1.1. Let Mbe an n-dimensional, N bean m-dimensional N be a Ck -differentiable map which has conCk -manifold, and f : M stant rank in a neighbourhood of x E M. Let y E f (M). Then the set f -'(y) is a closed (n r) -dimensional submanifold of M. -4
-
Proof.
Let
E
x
f
(y)
and take charts
as
given by Corollary
2. 1. 1. Then
have
we
p(U
n
f -1(y))
=
=
The assertion follows
p(U) o(U)
directly
n n
(0 fz
o
f
o
c- K
V-')-'(O)
n
:
ZI
=
...
=
Z'
from the Definition 2.1.4 of
=
a
01.
submanifold. I
Proposition
2.1.1 is
a
powerful
tool which is used to construct manifolds.
R, x i--> IX12. Example 2.1.3. Consider the submersion f: R' \ 101 M which O,in radius of > the that sphere p Proposition 2.1.1 implies -I of submanifold is coincides with the set f (p2) an (n I)-dimensional of atlas down an write himself herto for reader or R'. The directly try -+
-
the
sphere
of radius p. It is much
2.1.2 Partition of
more
work.
unity
we provide a method of localisation using functions practical if one wants to define a global object using charts. The prime example is the definition of integration in Sect. 2.5.4. This R. method works only for real manifolds: K This subsection is somewhat technical. The reader may therefore want to skim (or skip) this section on first reading and to return to its proofs when the results of this section are needed.
In this section
which is
=
58
Analysis
2.
manifolels,
on
The aim of this section is to construct
an
atlas and for each chart
a
function with support in this chart such that
(i) (ii)
point there
at any
the
sum
only finitely
are
of all functions
We start with two
)lVi
sequence
)1Vj+j
C
Proof. all
a
and
of
a
manifold. If
f)/Vilic-N
open sets
UiEN)/Vi
jUjjjErj
Let
by (i)) equals
I.
topological lemmas.
Lemma 2.1.5. Let M be exists
(which
many charts which intersect
is well defined
be
a
"::
M is not compact then there
with compact closure such that
M'
countable basis of the
topology of
M such that
Rj- are
compact. Let W1 := U1. Since the closure of this set is compact there is a k, E N with Wj- C Uk, Uj =:: VV2. We proceed now by ini= 1 duction.
Assume,
we
Ui. Then there
have is
already constructed )/Vj, kp+l > kp such that WP C-
a
.
_
Lemma 2.1.6. Let M be cover
all
of M.
.
a
manifold. And lualaEA
Then there exists
a
.
,
)IV,,
Uk,,+l j=1
where
Ui
)/Vp )/VP
be open sets which
countable collection
jVjjjEN of open
sets such that
(i) each Vj lies in some Ua and has compact closure, (ii) each Vj intersects only finitely many Vi. ("') M Uj E N Vj' ::-::
Proof. We will first show that countable. Let 0
=
I Oi IiErq
be
a
we
atlas of M. For each Vc there
an
that
Vc
U Zj Oi,,.
=
Let Oi be
index
c(j)
can
restrict to the
countable basis and let are
countably
Since 0 is countable
so
many sets
is the set
that A is
case
f (V,, Oc) I cc- c Oi,,
E
be
0 such
j0j,cjjEN,cEc
C 0.
of these sets and choose for each j C N an E C with Oi c V,(j). Then the collection j(0j, wc(j))IjEN is a a
re-numbering
countable atlas of M. Since Oi is
homeomorphic
to Kn there is for each
dense sequence fXi,j IiEN of points in 0j. For each (i, j) C N x N let a(i, j (E A) be an index with xi,j E Uqjj). Then the countable subset fUa(ij)jijEN Of fualaEA covers all of M. It follows that we can assume
j
a
without loss of Let Wo
=
generality
in Lemma 2.1.5.
by finitely
Ual, (k) N
x
n
many
O/ViliEri be the sequence of sets constructed The set )lVk+l \ Wk is compact and can be covered Ual(k), Ua.,,,(,,) (k) for every k c N. We set Vk'j =
...
(Wk+2 \ Wk-1)
N is countable.
property
(iii)
which defines
Property (i)
a
countable
family
of sets since
is clear from the definition of
follows from the fact that the sets
Finally, property (ii) by finitely many Vk,jM.
that A is countable.
0 and let
follows because each set
)/Vk+l \ Wk Wk is only
Vk,j
cover
and
all of
intersected
I
Manifolds
2.1
Lemma 2.1.7. Let M be
Then there is
(i) h(x) (ii) h(x)
r,
Fig.
real
a
manifold
C' -function h: M
a
1 0
--+
and U, V open sets with with
59
R
C
V
-
[0, 1]
for all x E a, for all x E M \ V.
r2
rl
2.1.4. The
proof of Lemma 2.1.7
Proof. We consider first the special
r2
case
but different radii: Let 0 < r, < r2 and Ix (E Rn: jxj < r2j. The map Br,(0)
of two balls with
B,,(0)
=
Ix
(E
origin
Rn
jxj
:
0 E Rn
rij,
0 for all x E M, and h(x) > 1 for all x E a. Hence in order to prove the lemma we x c M : h(x) < 1/21. only need to normalise h appropriately. Let U This set is open and its closure is contained in M U. Hence by the same construction there exists another smooth function h with h(x) > 0 for but
all
X
M,
E
and
h
h(x)
> I for
all
x
E
1,
and
do not vanish both at any
h(x)
then x
E
h(x)
M
\
> I
V then
h(x)
=
by
each
h(x) +
h(x)
=
h(x) e
[0, 1]
for all
0 which in turn
0 and therefore
h(x)
=
E M has a neighbourhood of finitely many fa, I for all x. fa(X)
point
x
Observe that
~
x
E
M. If
implies h(x)
x =
E
U
1. If
1
0.
Definition 2.1.7. A smooth partition of unity is f fa: M - [0, 11 IaC-A such that
(i)
\ a.
=
Further, h(x)
and therefore
C M
given point. Hence
h(x) is well defined and smooth.
supp(h)
a
set
which is
of functions
only
intersected
the support
(") EaEA
=
A partition of unity is subordinate to an open covering every a (E A there is a b (E B with SUPP(fa) C Ub-
fUbjbEB if for
2.2 Vector bundles and the
tangent bundle
61
(Existence of a partition of unity). If jUbjbEB is covering of a real manifold M then there exists a countable partition of unity ffjjjEN which is subordinate to jUbjbEBTheorem 2.1.1 an
open
For each
Proof.
neighbourhood exists
a
of
sequence
M let
E
x
be
b(x)
with
Ub(x) 1Xj1jEN and x
an
index with
x
G
and
Ub(x)
Lemma 2.1.6
ab(x)
be
a
implies that there
C
Ub(x).
a
countable collection of open sets
1Vjj
such that
(i) each Vj lies in some Ub(x.,) and therefore Vj (ii) each Vj intersects only finitely many Vj; ("') M UjEN Vj*
C
Ub(xj);
=
We
apply
can now
to find
a
the
same
argument
to the collection of open sets
countable collection of open sets
(i) each Wk lies in some Vj(k); (ii) each )/Vk intersects only finitely ("') M UkEN )/Vk
many
tWkIkEN
Vj
such that
Wj;
=
*
[0, 1] with By Lemma 2.1.7 there is for each k a function hk: M Since each Vj M for all 0 x E 1 for all x G Wk, hk (X) \ Vj(k) hk (X) 0 for have M each for we x chk (X) intersects at most finitely many Vi well is that the This k. x sum hk (X) all but finitely many implies have in each lies we Since smooth. and some x hk (X) defined Wk ---*
=
=
-
=
I for all
x
G
M. Thus
hk (X)
E' 1= 1 hi (x) Property (i) of Definition 2.1.7 is satisfied only finitely many Vi and supp(fk) c Vj(k). directly from the definition of fk. Hence ffkjkEN is
is well defined and smooth.
because each
Vj
intersects
Property (ii) follows a partition of unity. That follows from
supp(fk)
C
it is subordinate to the open
Vj(k)
C
I
Ub(.,.1(,)).
Remark 2.1.2. In Lemma 2.1.7 and Theorem 2.1.1 it restrict to the
eral. This is
so
case
K
=
covering UbjbEB
R. If K
=
C, both results
was
are
necessary to
wrong in gen-
complex- differentiable maps are automatically locally be written as a power series. This would be
because
analytic, i.e., can impossible for the function h
in Lemma 2.1.7.
2.2 Vector bundles and the
tangent bundle
ordinary calculus, the derivative Df of a map f : K' -4 K' is the linear approximation of f, i.e., it is defined by f (x) f (a) + Dfla(x-a) +
In
=
F-p. 5 j p.
62]
62
Analysis
2.
o(Jx
al)
where
manifolds
on
o(Jx
al)/Ix al --> 0 (x -+ a). To study the linear approximation of a map rather than the map itself is certainly one of the most powerful approaches in mathematics and physics. Because of the -
limit
-
-
0 this approach is often referred to as working infinitesa imally. Until the middle of this century people spoke of infinitesimal (or infinitely small) displacements (meaning the vector x-a if it was 'small'). This terminology can lead to misunderstandings but stresses the main idea of analysis. While we will give a modern presentation, it is a good idea to keep the 'infinitesimal way of thinking' in mind. x
--
-
The definition of a linear approximation of a function f rests on the linear structure of Kn. In the general case of a manifold, such a structure is not at hand. But it is possible to define a linear approximation of a map in two to the the
[P.
steps. First, we linearise the manifold itself. This gives rise tangent space TaM at a point a of a manifold M. Then we
linearise the map thereby obtaining between (linear) tangent spaces.
linear map
a
Taf : TaM
--+
Tf (,,)N
We will linearise the manifold M
by attaching an n-dimensional vecpoint x C M. At first one may think that it is sufficient to consider the product M X Kn and to define TxM JxJ x Kn. However, this would introduce a global structure via the (global) product x. In order to keep within the spirit of localisation, we can only demand that such a product exists locally. This motivates the following tor space to M at each
definition. Definition 2.2.1. A k-vector bundle
manifold
M is
a
and submersion
(i)
For each
triple consisting of
7rE: E x
c-
a
(E, 7rE; M) over an n-dimensional (k + n) -dimensional manifold E,
M such that
--
M is
(7rE)_1(x)
a
k-dimensional vector space
over
K,
(ii) for
each
x c
M there is
for each
a
linear
over
-
(7rE)
1
(U)
a
diffeomorphism
7
__4
(7rE) -I(Y)
isomorphism.
M is called the base
the fibre
k
-
and
y E U the restricted map
V)y: Kk is
neighbourhood U
UxK
0: where
a
manifold,
7rE
the
y, E the total space, and
projection, Ey := (7rE)-I(Y) 0 the bundle chart or local
trivialisation. The set of tangent spaces of a Manifold forms again a manifold of a special a vector bundle. While we will construct many special vector bundles
type, and
general
vector bundles
definition is not central to
are
our
of
importance
discussion.
in gauge
theory, their general
2.2 Vector bundles and the tangent bundle
We will often
just speak of the
vector bundle E instead Of
63
(E, 7E, M)
-
Kk. We call (E, 7TE) M) trivial if there exists a local trivialisation of the form 0: M X Kk -- E. In this case, E can be identified with M X Kk. An example of a vector bundle which is not trivial is given by the M6bius band. Notice that this is just
(M&bius band,
2.2.1
Example
band M is also
(Wi)-' (x', 0)
7rm(x)
(XI, X2)
localisation of M
a
a
continued
from
X
o
and i
The
projection
R2 is Wi(x) where pl: R2 E 11, 21 is an index with x E Ui.
o
pi
53)).
page
vector bundle. The bundle
--4
is
M,5bius
given by
the projection It is clear that
this vector bundle is not trivial. Definition 2.2.2. Let E be which
satisfies IrE(U(X)
==
x
a
for
vector bundle. A map
all
x
E
U is called
a
a:
U C M
--->
E
section of the vector
bundle E. A collection
ful,
...
7
9k
I of sections
spanf a, (x) for all
x
E
U is called
,...
following definition
nius
2.5.3).
will
The
play
a
(E, 7TE, M)
be
(F, 7rF, N) of the vector bundle E and submanifold N of M such that
(7E) IF:
IrF
(X) I
role later
dle
(i)
Uk
Ex
=::::
frame of E.
a
Definition 2.2.3. Let
7
such that
F
--4
N
defines
a
on
Theorem of Frobe-
vector bundle. A vector subbun-
consists
a
(cf.
of
a
submanifold
F
of E
vector bundle structure with base
manifold N,
(ii) Fy
is
a
vector
subspace of Ey for
2.2.1 Construction of the
all y
G
N.
tangent bundle
In affine geometry we had distinguished between points x E A' and translations (or vectors) in the associated vector space Kn. A translation a global concept. Notice that a translation is originally moving the point x along the curve yv: [0, 1] -* An, t -+ x + tv to its endpoint. The velocity vector of the curve is v which may be regarded as the infinitesimal (but in this case exact) approximation , An, t --> -y(t), we of -yv. Given an arbitrary smooth curve y: (a, b] take its derviative as its infinitesimal approximation at a given point x - (to). Taking all these velocity vectors at x we obtain a vector space TxAn which is attached to A' at x. This vector space is in a natural way isomorphic to the associated vector space given by the translations. In the following we will transfer these ideas to manifolds. The main
v: x
F-
x
+
thought of
=:
v
as
is
F
p. 62
p.
65]
64
2.
difficulty which
Analysis
we
manifolds
on
have to solve is the lack of take the derivative of
we can
Let M be
associated vector space in
an
a curve.
smooth
manifold, x c M and -y: (-E, E) --+ M be a through x at parameter value 0. Another smooth curve (-e, e) with x (O) is called x-equivalent to -y if d o -y) o This definition is independent of the chosen (W (W Tt 0. dt 10 chart and puts all curves through x with the same velocity into one equivalence class [-y,,] smooth
a
in M which passes
curve
=
=
-
Definition 2.2.4. Let M be
a smooth manifold and x cz M. The space of all equivalence classes [-y,,] is called the tangent space of M at x and denoted by T,,M. Its elements are called tangent vectors.
We must show that TxM has indeed the structure of
Choose any chart
(U, W)
centered at
vector space.
a
This chart induces
x.
a
bijective
map
d (9 x': TM which Let
a
we can use
E
[-y,,]
dt
(W
to induce
[-yx], [px]
K and
K',
---
on T,,M the vector T.,M. Then we define
E
a[-yx]
:=
[AXI
:=
-
-y) 10
space structure of Kn.
(eP)-l (aRP([7x])) x
x
and
[-Yxl
+
(ex
((9x1 OXI)
+
(9x,([-Yxl))
-
This vector space structure is independent of the chosen chart. In let (V, 0) be another chart centered at x and denote (9 ' o ((9P) x
(9'P,w-'.
Then
we
x
have
x
((9x
(a(9xI ([,Y.])) =
('9")
-
1 0
e"
x
=
((9V))-1(9V) x
=
and
-
x
(en
-1
'W-1
o
o
x
x
(90 (I
0
x
(exI,)-,
-1 (aev (eOX) e"([-Yxl))
x
x
x
,
Yxi))
(01(9X, Q -Y-D)
analogously
(ex
(ex10([-Yxl)
+
(ex ) + (9w x
x
1QAxD)
ex (9x
0
0
((9x,P)
-
I
(e'O
0
x
(,g")-l .,e"([-txl) x
x
(e b)-l e b([Mxj) ,
x
x
x
x
x
x
fact, by
tangent bundle
2.2 Vector bundles and the
((gqp,w-I)-I
+
(ex
We call the set TM
vector bundle
TxM for all
on
=
Tfa: (7rTM) The
are
infinitesimal models
[p. I
a
smooth
manifold.
There is
associated with M such that
a
1]
63
p. 65
natural
(7rTM)
(X)
UXEM TxM
-I
and define 7rTM
x.
of M and define the structure of
f ((7rTM) -I (Ua)) Tfa) I
(Ua)
Kn
-4
'/x I
K n'
We choose
an
smooth manifold
a
where Oa (X) 0) 19'p' x
-4
(/ [XD
-
bijections bijections Oa : Ua
are
as
M.
through the atlas
M
+
be understood
can
(TM, 7rTm, M)
x E
I A, (Pa)IaEA
atlas
[_YX D
2.2.1. Let M be
Let TM
Proof.
.
UxcmTxM the tangent bundle of the manifold
:=
M. The vector spaces TxM of M at x.
Proposition
(1YXD) ex0 QYXD)
191,
,
x
(ex
65
X
Kn
-4
(7rTM)
diffeomorphisms linear isomorphisms.
then
-I
(Ua)
such that
(X, V)
i
(Oa.)x:
Kn
-4
-->
(ef"') _1(v)
-
x
TxM,
v
1-4
(ew-)-'(v) x
p. 65
always practical to work with equivalence classes. We will therefore also give an equivalent definition which is better suited for calculations at the cost of being less intuitive. The key observation is that each tangent vector 1-yx] E TxM induces a map
It is not
If
D
E
Coo (U,
K)
:
U is
an
open
neighbourhood
of
x
K,
d
f
-4
at- f
This map has the
0
-Y(O).
following properties.
(i) D[,yl.,,, is K-linear, (ii) for any smooth functions f
D[,yl., (fg)
=
.;
,
g: M
---
K the derivation property
D [,yl. (f) g (x) +
f (x) D [,yl. (g)
holds,
(iii)
for any open neighbourhood U of which coincide in U we have , g
f
This motivates the
following
x
and any two smooth functions
(f
(g).
[I
p.
1
75]
66
2.
Analysis
manifolds
on
Definition 2.2.5. A map
If
v :
C' (U,
c-
K)
U is
:
an
neighbourhood Of xj
open
---->
K
satisfies properties (i)-(iii) above is called a derivation. The vector space of derivations at x with addition and multiplication being defined pointwise, (avx + bw,)(f) a(v.,(f)) + b(wx(f)), is denoted by D ,. which
=
Remark 2.2. 1. The reader may wonder why in our definitions derivations U is an open neighbourhood of xj instead of on If E C' (U, K) :
act
the
simpler
C' (M,
K),
set C'
(M, K)
but in the
For K
-
K
case
R
=:
could indeed have chosen
we
C there exist
=
only
very few
globally
defined differentiable maps in general. In fact, in the extreme case of the complex torus only the constant functions are smooth. However, locally there is
always
Lemma 2.2. 1.
v.,(f)
=
0
for
Let
I +
lvx (1)
a
-
If f :
M
K is constant in
--+
=
a.
=
vx
(f)
2avx (1)
Theorem 2.2.1. Dx is
=
(f /a) 2vx (f ). avx
==
is
a
Let
=
--+
avx
=
(1)
=
(I 1)
av.,
-
=
avx
(1)
-
I
n-dimensional vector space.
an
Proof. We know already that Dx n. only to show that dirn(Dx) and let x': U
neighbourhood of x, then
a
all derivations vx E Dx-
f (x)
Proof.
abundance of smooth functions.
an
vector space and have therefore
(U, W)
be
a
chart centered at
W-'.
K the ith coordinate component of
x
Observe that
K the identity h(y) W(U) h(O) + Enj , hi(y)y', 1 ah(t') dt and y is the ith coordinate where hi is defined by hi(y) := L --5 1 K we in Kn. Applying this identity to the function f W- : W(U) we
have for any h:
--+
=
o
obtain n
f(Y)
=
f (X)
Xi (Y)
+
Hence for any derivation vx E Dx to
(f W-') i W-' (Y)
f
-
-
get
we
n
vX(f)
=
VX
(f (X))
+ X, (X)vX
The first summand vanishes
x'(x)
vx is
uniquely determined by
0. From vx (f)
((f
by
since
=
E (vX (Xi) (f
+
=
-
0-')
-
i
0
0))
W
I
WW
-
i
-
Lemma 2.2.1 and the last term vanishes
Ei'- I (f
the
n
o
p
-'),
o
(x')
W (x) vx
numbers vx (x 1),
.
.
.
,
VX
we see
(Xn).
that I
2.2 Vector bundles and the
Definition 2.2.6. Let M be
be
manifold, (U, p)
a
tangent bundle
a
67
chart, and
vi
p":
Kn
--,
vs.
K, vn
Then x': U
-->
K,
-4
y
p'
is the ith coordinate function with
(p(y)
o
(xl,...,x') a coordinate system. defined by axi(xi) Jj3 is called the
and the collection
(U,W) respect The pointwise basis to
=
(U, W)
CauBian basis associated with GauBian vector fields. We will
Corollary
2.2.1. Let M be
centered at
x
C'(M)
we
write
often simply smooth
a
fields axi are called 0i instead of
and the vector
manifold
and
(U, W)
M. For any derivation v., E D., and any have E
n
V-M
=
n
Evi'9xif
.'9f
=
ST
a
chart E
W-1
axi
i=1
i=1
where D7 is the usual
0
E v%
be
function f
partial
derivative in Kn.
Proposition 2.2.2. The tangent space T ,M is canonically isomorphic to Dx. The isomorphism is given by the map i: TM --+ D,,, iQ-1] ,)(f) A o dtf y(to) and well defined. Proof. Clear by
construction.
dispend with the symbol Dx and always use TxM instead. Our first definition using curves has the advantage to work also in infinite dimensional settings. However, we are only concerned with finite dimensional manifolds and derivations are more practical to work with than equivalence classes of curves. Hence
we can
Definition 2.2.7. Let M be
vx
*
f
:
=
The map
a
manifold,
x
E
M and U be
a
neigh-
For any v., E Tx M and f E C" (U, K) the number df (vx) : vx (f ) is called the derivative of f in direction vx,
bourhood of
x.
=
df :
TM
-->
K,
-->
vx
df (vx)
is the differential
of f
2.2.2 The derivative of maps between manifolds
In the
preceding
section
we
linearise differentiable maps
taining linear
maps
Tx f :
Definition 2.2.8. Let a
differentiable
map.
f: M
Tx M
--->
--+
N between manifolds
Tf (x)
can now
thereby
ob-
N.
manifolds, x E M, and 0: M -4 N be Txo: TxM --> TxN, Txo(v)(f) v(f o V)) is (Or simply the derivative) of V). We will often
M,
Then
called the tangent map denote T,,o by 0.,
have linearised the manifold. We
N be
=
68
2.
Analysis
manifolds
on
Observe that in terms of
given by [-(.,] of the curve 0 o -y ,0
is
equivalence classes of curves, the derivative of which is clearly the linear approximation
[V) -yp(,,)] at O(x).
-4
o
It is instructive to calculate the
.
systems. In these we
tangent
map with
respect
to
a coor-
o) and (a, 3) be charts of M and N respectively, Xn) and (5 1......; k) the associated coordinate charts 0 has the representation T1 o-' and p 7P
dinate system. Let (U, and denote by (x 1, .
.
,
o
==
o
calculate n
,O.V(f)
E vi'9xi- Y
=
-
0)
V'
=
C W
-
V)
0
-
Oxi
i=1
vi'9f
0
(V, aP 0
a
,9T13 =
a.V Hence with respect to
Y
0
V
'9xi
W
a;,7 -if .
coordinate system, the tangent map 0" is just Again we see the tangent map Txo is the
the derivative of the map T1. linearisation of 0 at x. The
following is
an
immediate
corollary of Definition
2.1.6 and Lemma
2.1.3.
Corollary 2.2-2. Let M be a manifold and x E M. A continuously differentiable map 0: M ---+ N is a submersion (respectively, immersion, local diffeomorphism) near x if and only if TxV): T M T b (x) N is surJective (respectively, injective, bijective).
2.3 Tensors and tensor fields Tensor
fields play a central r6le in geometry and physics. Differential forms are not absolutely necessary for the theory of space and time. However, their usage has many advantages and they also provide a very natural way to define integration. In particular, the integral theorems of Stokes and Gaufl have a very simple, common form when stated in terms of differential forms (cf. Theorem 2.5.5). Unfortunately, the introduction of differential forms requires some
technical preparation.
Some readers may
differential forms
on
The tangent bundle of mations
JTMJ.
linearisation
we
therefore wish first reading.
a
to
skip the
sections
dealing
with
manifold is the collection of its linear approxiuse of the simplifications arising from
In order to make
need to express
physical
and geometrical
objects
in terms
of maps which are adapted to the linear structure of T,,M for all x E M. We will see later in the book that the notion of a tensor field provides a
good
section
framework for this we
simply
(here
still rather
introduce tensor fields
as
vague)
idea.'In the present
mathematical concepts.
2.3 Tensors and tensor fields
2.3.1
Algebraic preliminaries:
In linear
algebra,
the concept of
69
tensors
tensor unifies
a
vectors, linear forms,
bilinear forms, linear maps, determinants etc. Let V be an n-dimensional vector space over K. Then its dual space V* is the vector space of all linear maps V ---> K. It is easy to see that V* is isomorphic to V. In fact, if jej,... , enj is a basis of V then the set 10 ...... O'l C V*, defined by
Oi (ej)
=
6 ,,
is
a
basis of V*. It is
defined
uniquely
by f el....
7
e,,, I and
called the dual basis. While the
isomorphism
the choice of basis
of V and V* defined
lei,..., enl,
there is
by
iOi depends on isomorphism tv of
ei
canonical
a
V and V**.
V*,
tv: V
tv(v): f
V
-->
f (V)
E
following we will freely make use of this canonical identification given by tv and write v instead of tv(v). Using this identification we not only can view a vector v as a linear
In the
V**
V
Tap V*
--->
A: V*
V
x
generalised
K but also
and
linear map A: V
a
K, (f, v)
--->
A(f v)
i,
(")-tensor
Definition 2.3.1. An
is
a
V
X
which is linear in each
space
s
VXV*
X
and an
a
linear map V
V is
--4
bilinear map can be
V*
--+
K
copies
or a
0
tensor
is
an
times
r
of order
co-
The
by Tsr(V).
is denoted
s
=
For
x
We say that
its entries.
of
...
r
The most important special cases T,O(V) V*. A bilinear form such
T20(V)
X
times contravariant tensor
(r)-tensors
of all
...
copies
s
variant and
a
as
map
8
0:
V
f (Av). This reasoning following unifying concept. ,
let to the
we are
-->
=
are
Too (V)
as a
scalar
K, To' (V) V, and product is an element of =
element of
an
T11(V).
explanation of the terminology "covariant/contravariant"
see
will need to define the components of a tensor. This in turn requires the introduction of the (natural) "tensor product" & of tensors which generalises the usual product of numbers. Remark 2.3.1 below. First
Definition 2.3.2. Let sor
0
product 0 0
0 (Vil
&
...
)
0
E
LS+p
...
O(Vi
Tsl(V), 0 (V) by
E
,+q
V87 1011 :=
0
we
)
wp)
....
7
W11 Vs,
...
Wi
I
G
I
Tpq(V).
Then
we
define the
Wr, 77,.... 77 q)
...
IWr) O(wj,
wp,
n1,
ten-
70
2.
Analysis
manifolds
on
Observe that the tensor Lemma 2.3.1.
The tensor
Proof. Let 0, 02 03
is not commutative.
product
is associative.
(ri), (r2), (r3)
be tensors of order
7
we
product
81
82
respectively.
83
Then
have
(01
0
02)
(D
Wil (01
03 (V1
...
I
X
and
...
,
.
.
.
1,
....
V"'
I
*
'
'
W1
03(VSI+S2+11
-
V81+821 V81+82+11
rl+'r2+1, Wrl+r2+1,
,
vs, VS1 +1
I
VS1 +32 +S3
I
rl I
...
VS1 +,92
7
7
W
.
.
.
...
I
V81+82+831 ,
W
rl
+r2 +r3
,)rl,
I
)
,
03 (Vs I +S2 +11
01 (vi
+
rl+r2
,,rl+l, X
r, ,
b2) (V1
0
VS1 I VS1 +I I-
I
ri
...
)02 (Vs, +1,
I
VS1 +82 +,93
7
)rl +r2 + 1,
I
-
'
*
ri
I
VS1 +82
,rl+r2+1,.
+r2 +r3
r1+1 1
rl I
...
+r2
I
,rl+r2+r3)
analogously
01(9(V)2
(D
7P3)(V1
Wil (9
...
rl, W +11
I
....
VS1,
I
( )II
?P3)(Vsj+17-
,,ri +
ri ...
I
-
vs 1
I
,rj+1....
.
*
I
,
W
I
1
IVSI+S2+S3)
+r2 +1
rl+r2+r3)
,
rl)
.,
VS1+S2 7 VS1+82+11' rl
I ....
rl+r2
rl
1W
...
+r2'
V)3 (VS1 +S2 +11
Lemma 2.3.2.
.
)VS1+827VS1+S2+11' +r2 +1
ri
i
01 (vi,
X
IVSIIVSI+ll ri
01 (vi
(02
....
+r2 +
1,
.
.
.
,
I
VSI+S2+S3'
Wrl+r2+r3)
U)rl)'02(Vsl+l
....
I
VSI+S2'
) rl
VS 1 +82 +S3
If jej.... enj, 101,
+r2 +1
on I is
I
rl+r2+r3
a
pair of dual bases then
the set
f 0" forms have
a
(9
basis
...
(9 0'. (2) ej, (9
of the
dim(T,,r(V))
=
space
...
(9
T,(V) of
all
(") -tensors. S
in
particular,
Proof. The set of tensors f 0" independent. In fact, let
(9 0i. (9 ej, (9 & ej,, I is be numbers such that ...
n
V)
E V)ij'---j"'0i1
=
j3
.....
we
nrn'.
j'-1
(9
...
(9
0i.
(2) ej, (9
...
(9 ej,
=
0.
linearly
2.3 Tensors and tensor fields
Then 0
:::::::
O(ek,
I
....
e-k,,
011 ,
...
,
1017,)
71
hence the tensors 01'-*1',,, k,
=
...
k
linearly independent. Conversely, we see that for and any v,.... v, E V, 77' E V we have
0
any tensor
are
all
Tr(V)
E
*
,
n
O(Vi
....
Vs, 17,
......r
O(eii
&oil
I
...
I
ei,
0 ej,
oilI... Oil Oil
I
1
ej,,
(VI
....
VS,
nil
7r).
....
T,,(V) is nrn' since there are exactly n' choices of t-tuples (with possible duplication) from a set of n elements. I
The dimension of
ordered
Definition 2.3.3. Let
01,
on
V)
c
Tr (V) and
f ei,
I
e,,
be a. basis
be the associated dual basis. The numbers
0,"
of V and defined by
n
ii
are
the components
In the
physical (and
...
i,' 0',
(9
...
of 0 with respect old
(9 0'. & ej, (9
(9 ej,,
f ei,
to the basis
mathematical)
...
en
literature it is the standard to
use
for contravariant entries upper and for covariant entries lower indices. This provides a checking mechanism for the syntactical correctness of tensor formulas and also
ing
effective notation in the
simplifies
the
(Einstein's
of formulas involv-
interpretation
components. In Remark 2.3.4 below
tensor
summation
will introduce
we
which is
convention)
a
very
prevalent
physics literature and has at its core the difference between Unfortunately, many modern mathematicians indices for all entries on grounds of "aesthetics".
per and lower indices.
lower
Remark 2.3.1. The
terminology "covariant/contravariant"
arose
upuse
in the
19th century and refers to the transformation of tensor components under transformation of a given pair of dual bases f ell Oil Onj. , e,, 1, f .
Let
v
==
T11 (V) fined
I:ni=1 Viei
be
we can
=
write
V,
W
=
I:ni=1 wiO'
E
v
Eni=1 f),Ei, n
W
=
rn _i=l CoiO'. n
W(W)
w
(z-v%)
.
.
V*, and A
invertible linear map. Then f jI, 0' o (A-') are also Aei and
an
by ji
(E
.
.
a
.
.
=
Ei,j Aiei 0 Oj
,
.
For any
w
G
V
.
.
we
n
Cv'A'i-w (ej)
n
Cv'AiWk Ok (e j)
E
n I den 1, f ', pair of dual bases and
ib'w (Aei)
n
.
Cv'A3j wj
,
have
72
Analysis
2.
manifolds
_Tj'=, Aj' wj.
Hence cDi
i.e.,
on
in the
The components of
Similarly,
n
(A)
v
(j
( j 0' A-')
j 0'(A- v)
-
n
j(A-1)jVkoiej k
j(A-)'k Vk.
i,j,k=l
i,j,k==l
I:nk=l(A -1)'vk, k
=
covariantly,
n
v
n
Hence f)'
transform
as
way
n
v
w
the basis vectors ej. for any A C- V* we have
same
antly with respect
the components of
to the transformation
v
transform contravari-
A, i.e., opposite
to the basis
vectors ej.
Another natural operation which is defined for tensors is their "contraction". Definition 2.3.4. Let
of dual bases. The
slot and the 9th covariant
C,rO(vl,
T,' (V) and f el, en; 0 ...... of 0 with respect to the th slot of 0 is defined by
contraction
0' 1 be
a
pair
contravariant
W1....
Vr-1,
dth slot
,Pth slot
n
6-
Vr
-
1, W
....
WS-1)
02,
1
We have to show that this definition is In
fact,
1 1,
if
there exists
and Ok
=
a
.
-
-
I
linear
=
.
-
6k
qro(vi,
-
independent of the choice of basis. pair of dual bases then nj, A: V V --+ with AEj isomorphism ej En 1 Aj j is another
A-'
o
.
.
,
Enj= (A-1) 3 03
=
W1
Vr_ 1,
I
...
.
We calculate
I
n
O(Vl,..., ej,..., Vr-1, W1,
-
-
-,
0i'.
-
-,
W,
-,
n
A1j*(A-')k'O(vj,---, j.... 7Vr-17W17
-
i,j,k=l n
O(Vl)
-
-
-
jji
Lemma 2.3.3. Let V be
T,r (V).
-
-
a
-,Vr-1,W1,
-
-
vector space
-'0 over
K and
G
TqP(V),
Th e n and
(C9
Z (0 Cq+sr
E
2.3 Tensors and tensor fields
This follows
Proof.
immediately
73
I
from the definitions.
Another class of natural operations on tensors are symmetry operations. We introduce below the two most important symmetry operators, symmetrisation sym and anti- symmetrisatio n alt of entries. First some
technical
we
need
a
bijec-
preparation.
Definition 2.3.5. A
permutation of the numbers (1,...,p)
is
tion
f(il'. .,ip) : fil'. .,ipl
up:
=
-
-
11,
1(il'. .,ip)
-
-
-Al
fil'...'W
11'.. 'Al.
=
If orP is a permutation we write a(ij, ip) (i,(j), i,(,)). The set of all permutations of the p integers j1,...,pj is denoted by Sp. A transposition is a permutation which permutes only two consecutive elements and leaves all other elements fixed. =
-
-
Lemma 2.3.4. mutation
The set
group)
and is
of all permutations Sp forms generated by transpositions.
-
.
a
,
group
(the
per-
Proof. That Sp forms a group is clear since the collection of all bijections of a given set forms a group where the composition of maps is the group operation. be any permutation. Starting Let a(ij,...'ip) with the p-tuple (i 1, ip) we can use successive transpositions in order to
move
the index
to the last
i,(p)
position. Assume
now
that
i,,(k)
i
...
I
Since ia(kfio-(k) ,... I ic(p) I it must be , p. 1) k It that 1. follows we can Move i.(k, 1) to
positions k, at one of the positions 1, k 1 by successive transpositions which all leave the last p position k have there finite shown that is induction invariant. we a By positions sequence of transpositions which is equivalent to a.
i,(p)
are
at
.
.
.
-
.
.
.
-
-
Lemma 2.3.5.
There is
a
natural
sign: Sp
of
the
mined
permutation
by sign(-rp)
--+
homomorphism
(1-1, 11, .)
group into the group
=
-1
for
of
two elements which is deter-
all transpositions -rp.
prove this lemma by showing that every permutation a is product of an even number of transpositions (sign(O-) 1) or the product of an odd number of transpositions (sign(a) -1). First we show that the identity permutation id is not the product of an odd number of transpositions. Assume that id is the product of finitely many transpositions and denote by n1k the number of all those
Proof. We
either the
=
=
74
Analysis
2.
manifolds
on
transpositions which interchange the numbers I and k. The number must be
the
since at the end 1 must be
even
and since there
beginning
change
k and 1. If
p
E 1=1 (Epl=k n1k) Let
now a
Since id is the
(-ri
=
=
o
set n1l
o
-ri
...
0
0
...
Irk)
-
of k + 1
product
Hence k and I
we
=
which is the
are
Irk 1
0
0
0
...
of
even
even or
A permutation up acts in
0
tl
=
(Tk)
0
set up 0 (vi,
...
k + 1 must be'even.
tensor
on a
and any 0 E TO(V) p (v,,, (1), V,,, (p)).
vp)
0
...
even.
transpositions. 0 ;F_J (-Fl)-' 0 ;F, 0
are
1
-
Definition 2.3.6. For any we
at
both odd.
natural way
a
n1k
as
numbers and therefore
o, i where Ti,,Tj
...
transpositions the number
both
side of k
other transpositions which inter0 then the number of all transpositions is
-71
11
same
are no
sum
=
the
on
.
.
.
0
TO (V).
c
p
permutation up
E
Sp
,
Lemma 2.3.6. For any permutations -rp, up E T'(V) we have (up o Tp)V) -rp(apV)).
and any tensor
SP
0
=
p
Proof. Let
vi,
vp
V. We calculate
E
-r,) 0 (vi,
(UP
Vp)
(V,P.-,, (1), (V,, (", (1)), 07po (VTT, (1))
V-P .-,, W) V-,, (-,, W) -
(p))
*
Tp (CPO) (V, (1), which
implies
the first
Lemma 2.3.7.
sym:
equality.
The maps 0
To M
T M,
p
1
P!
alt:
E
0
To P, M
Yp M, I
I: "
are
W
up
V)
V)
P!
S7,
1:
sign (up) up 0
ESI,
linear projections.
Proof. We only
prove the lemma for the
operator alt since the proof
for sym is completely analogous. That alt is linear is clear.from the definitions. For given V) E To (V), vectors v1, and any permutation , vp p we have -Fp .
alt,0 (vl,
.
.
.
,
vp)
P!
Y_ E
P!
.
.
sign (up) 0 (v,,,, (1).... Va" (p)) I
Sp
E ,pEST,
sign(up) sign (,Tp) V)(v,,,, -T, (1),
V-P -', W)
2.3 Tensors and tensor fields
=
where
sign(,rp)
alt
have used that R,:
we
-
-
Sp
Sp,
-
,
75
v_gp)),
up
apTp is
1-4
bijection.
a
It
follows that alt
alt
o
0((vj,..., vp)
=
E
-
P!
sign (-rp) alt V)(v -" (1),
-
-
-
1'r', W
,
7-7, E Sp
1 =
P!
o
alt
=
alt
O(vj,..., vp)
-7, E Sp
alt?P(vj,..., vp)
=
and therefore alt
Y
-
alt.
I
Definition 2.3.7. A covariant tensor V) E T ,(V) is called symmetric (respectively, anti-symmetric) if for all s-tuples of vectors (vl,...,v,) and all permutations a,
b (vi,
V')
of (1,
0 (v,' (1),
=
-
-
.
Symmetric analogously.
and
s)
,
the
equality
-V"' (,))
(respectively, 0 (vi, holds.
.
.
.
.
v,)
,
anti-symmetric
=
sign(o-,) 0 (v,,, (1)
7
...
contravariant tensors
7
va" (S)))
defined
are
p. 65
I
The set of all
r
times contravariant and
TXM, where
E
M, form
x
a
s
times covariant tensors
vector bundle which
generalises
on
the tangent
bundle. 2.3.1. Let M be
Proposition set
T ,M
an
U. CMTr(TxM) of
=
n-dimensional, smooth manifold. The
all
(r) -tensors
carries
natural vector
a
.
bundle structure.
Proof.
Let
associate
0,:
a
(U,, o,,,)
be
an
atlas of M. Then with each chart U,,
we can
map
U
T,,r(TxM)
-
0.(U)
x
K
n"n'
xEU,,.
W. W,
where
0", (a) ii
are J.
...
GauBian basis lies in at least
the components of the tensor
Oxi ......9xr,,. one
....
TrU,,,
:=
that for each
0,
(TrU,, 0,) forms an atlas pair of indices (a, 0) the map o
V), 1: V)O(TrU,
n
TrUe)
Oy
UxEu,,, Tr(TxM) (a
that the collection
--
....
nj
Ox with respect
It is clear that each tensor
of the sets
isEfl
of TrM
0,(TrU,,
n
we
E c
to the
T,"(TyM) A).
To
see
have to show
TrU,8)
p. 84
76
is
Analysis
2.
We denote the coordinate system associated with y") and let x E U, n UO. For any vector v E Tx M we
diffeomorphism.
a
(UO, oo) by (y', then write
can
manifolds
on
.
.
.
Eni=1 V i a.xi.
v
Eni= , v'0 0.,i
a
(WC, ( 00)-')j
=n
V3.
where
-
v
Oyi In other
words, the column
nents of the vector
DW,,e (va), row
where
=
=
of the components of
the charts must then be related
components
By the
v. I
by w(v)
defined
are
by (w')
with respect to since these
(w") (v,,)
=
w
=
=
-
(wO) (vo) -
argument it follows that the components
same
for all vectors
and
..........
(a) i
I
T,
Ozi ()3) ji ...
related
are
by
,
oki
...
k,
i, oil(0) ilj ...
I"
p!q!
sign (0-,+q) '7p+q(W
(2)
77)
,p+qESI,+q
L
p!q!
sign (Up+ q) sign (Tp+q) '7p+q Tp+q(W
0
77)
,p+qESp+,l 1
(_I)pq
=
p! q!
'PE
sign(Cp+q)9p+q(?7
0
W)
+qESp+q
(_l)pqT,
=
Lemma 2.3.9. Let 1
(i) w A (ii) wl A (iii) wl A
*
*
wl
A wP
...
A wP
...
7
=
A WP
*
A
....
wP E V* and irp E
Sp.
A w", (P) sign(7rp)w"P(1) A F,.,ES,, sign(up)wIz,(1) 0 ...
=
=
W.
0
...
if and only if the 1 -forms
w
0 WIP
1,
-
(P),
wP
are
linearly
dependent.
Proof (i):
Since
to prove this
transpositions generate all permutations, for a transposition
7rP (1 7.... But in this
it is sufficient
equality
case
i'j'...'n)
=
the assertion follows
immediately from Lemma
2.3.8.
2.3 Tensors and tensor fields
The assertion is
(ii):
true for p
clearly
79
2. Assume that it is true
=
for collections of up to q 1-forms wl.... I Wq and denote for any Uq E Sq the permutation (ij, .... iq+1) -4 (Uq(il), '07q(iq) iq+1) by tq+l(gq)Then
we
have
w1A
A
...
w
q
A
I
q!
Wq+1 (V,,.
E
.
.
Vq+l)
,
sign(,Tq+l)(wl
A
A
...
Wq
", +jESq+j W
q+1
(1)
1:
q!
E
sign (-rq+ 1) sign(gq)W
T,I+iESq+l o-qESl
Wq (g
q!
'Y'
Y-
qESq
0
W
q
Wq+1 (Va q-Or,,+1(1)1
1:
W
Sign(tq+l (Uq)
VUqO'rq+1(q),
0
Tq+l)W
1
VT,1+1(q+l))
(9
...
ESq+j
-,,+I
0
I
q+1
(Vlq+
I
(Uq) 07',1+ 1 (1)
Sign(Tq+l)W
1
0
1
'
*
'
7
Vtq+ 1 (Uq) 0r,,+ I (q+ 1) )
...
,T,I+iESq+l (9
(iii): a
If
W
U)q(9
1,
W
q+1
wP
are
linear combinations of
w ).
(q))
(1)
linearly dependent 2
(otherwise we 2,...,p) with wl
wP
w
There exist numbers ai of
we can assume
(i
that
1 w
is
could renumber the i=2
aiw'. Hence
right hand side
the
P
W1
A
...
Aw p
E Wi A U)2 A
...
A wP
i=2
products each of them containing some factor w3 twice. by (i). If the forms w 1, wP are linearly independent then we can complete w' I of V*. Denote the dual basis by f e 1, them to a basis f w 1, 6n I If w' A A wP would vanish then so would wl A A w'. However, we is the
sum
of
Hence all summands vanish
.
,
...
...
have
W1
A
...
A Wn
(el,
e,) all, E
S",
sign(a,,)J1,,,(j) a
a,, E
S"'
jn o,,,,
(n)
.
-
,
-
80
2.
Analysis
manifolds
on
If lei,
Lemma 2.3.10.
en
1, 101'...' Onj
is
a
pair of dual bases then
the set
fO" forms
a
basis
Proof. First
of
we
A
...
A 0"
AP(V)
the space
In
1 I
A
77(Wl
(P
7...
7
we
set
v
J
w
=
0.
v
=
wo. For any vectors wl,...'
sign(Up+q-l)W(WO,W,P+,,-,(l),
ESl,+q-1 X
and
(wo J 71) (Wi,
set
E
1)!q!
7WO'j,+q-1(P-l))
A
we
wp-l)
Wp+q-1)
+q- I
(- 1)Pw
0
W(V' Wi,
The interior product w E AP(V) and 0 E Aq(V). (v -i w) Aq + (- 1)P w A (v Jq) for all p-forms W and
For notational purposes G V we have
(WO J W)
=
-->
-
.
.
i
Wp+q- 1)
77(WO'T,+q-I(P)
W,,,+,,-I(p+q-1))
......
2
"
-11tHisors
11.'.F
I)P 1)!
p!(q X
Sign(Tp+q- 1)LO (W7-T,+,j-j (1)1
....
81
W7-j, +,, -I (P)
-
'rp+q-1ESp+q-1
77(W0IW-rp+q-I(P+1),'-I Ip+q-1(P+'1-1))* W
We consider
all
now
We
Wp+q-l)-
....
E
and tensor fields
of the ordered set
possible permutations divide them into two
can
(wo, wl,
the group SP p+q
subgroups,
where wo is in one of the first p positions and the subgroup Sq where p+q it is in one of the last q positions. Using this notation we can write
wA?7(wo, wl,..., Wp+q-1) 1 =
p!q!
1:
(
sign(6'p+q)U) (W&,,+,, (0)
ID&,+,, (1),
7
-
-
-
,
W&,+,, (p- 1))
&, +, E S,','+ q X
W&p+q(p+q-1))
77(W&I,+q (P)
E
+
sign( p+q)U) (W. ,,+, (0))
10, p+"(P-1))
....
Sq
P+q
X
?7(W- p+q(P)7W' p+q(P+I)I
....
Wr,,+q (p+q-
Consider the first summand. For each the first entry of is
a
by executing unique permutationCp+q-1 w
( p+q
(&p+q)-1(0) E
=
shift wo to
Sp+q-lwith
(- 1) (&p+q)-1(0) W (WO Wlp+ I
X
&p+q
can
IW&p+q(P-1))?7(W&p+q(P)I
W(W&P+q(0)1*
Observe that
Spp+qwe
E
transpositions. Hence there
is the
7(W,,+q (p),
*
*
"
I
WO'p+q (p+q- 1))
Composition Of
WO'Z)+q (P-
7
q
-
1 (&p+q)- (0) transpositions
and
the permutation i
This
each
implies sign(&p+q) of the p possible
one
Up+ q (i)
for i > 0
0
for i
=
0.
(-1)(4+q)-I(0)sign(9p+q)entries of
w we
Since wo
can
be in
obtain
I
E
p!q!
sign(&p+q) W (W&I,+
q
(0) 7 W&I,+,l (1)
1....
W&7,+,,(P-I))
E S"
p+q
X
P -
p!q!
'7(W&p+q(P)l
....
W&p+q(p+q-1))
sign(9p+q-I)W(W0, Wor,,+,I-j (1), X
q(W,P+ll-
I
(P) I
I
-
-
-
WUp+q- I (p+q-
,
W-p+q- I (P- 1))
82
Analysis
2.
manifolds
on
(Wo J w) Aq(wj,...' Wp+q-1).
=
Consider
Sq
p+q
an
now
allows
the second summand. An
us
to
additional factor
transpositions,
a
wo to the first
move
(-l)P
since
E q
E
- p+q
entry of 17. In this case we obtain is the Composition Of (' p+q)_1(0)
'rp+q (i)
for i > 0
0
for i
and the p transpositions which first entry of 17. This and we obtain .1.
for
permutation
i
p!q!
' p+q
analogous argument
wo from
move
implies Sign(' p+q)
sign(' p+q)U)(W_ p+q(0)
7
0,
=
the first entry of -, P+')
(-I)P(-I)(
=
W' p+q(l
I...
1
to the
w
(O)T(07p+q)
M 7'+'J(P_ I))
'q '
Sp+q >(
77
(W' p
(-')P- L
+q
p!q!
W- ,,+,(p+q-1))
(P)
E
Sigll(Tp+q- 1)W (WTI,+q-
7'
1
Wr,,+q- 1 (P))
'rp+q-jESp+q-1 77 (WO,
X
=
W-r,)+q
I
-
(-1)PW
(p + 1) 1...
A
7
W-r,,+,,_j(p+q-1))
(Wo J 77)(Wi,..., Wp+q-1). I
Finally,
we
relate the
theory of p4orms to
the determinant of linear maps. following from linear al-
To motivate the definition below recall the
gebra. Assume that orthonormal basis
have
we
Euclidean scalar
an
f ej,...' e"j.
If
fol'...' onj
product (, -) and
is the dual basis then
an
one
the n-form 01 A A on in order to measure the volume of parallel b, G V one defines the volume of the epipeds. For any vectors bl, A 0 n(bi, parallel epiped spanned by these vectors to be 01 A bn)This number depends on the chosen scalar product but not on the orcan use
...
.
.
.
,
...
.
thonormal basis. The determinant of
defined
as
det(B)
:=
01
This definition of
.
.
,
linear map B: V -* V is often B ej. where bi , b,,) determinant obscures the fact that the deter-
A
a
...
A on
(b 1,
a
=
.
.
.
independent of the choice of scalar product. The following equivalent definition is probably the most natural way to introduce the minant is
concept of
a
determinant.
Definition 2.3.12. Let V be k-dimensional vector space
n-dimensional vector space, W be K, and A: V -4 W a linear map.
an
over
(i)
The pull-back of 0 under A is the defined by A* 0 (vi, (Avi, vp) .
.
.
,
.
.
map A*: .
,
vp).
TO (W) P
--->
a
T' (V) P
2.3 Tensors and tensor fields
(ii)
W. An(V) \ 10} and assume that V det(A) of A is the number defined by A*/-t
Let tz G
minant
83
Then the deter-
=
=
det(A)M.
We have to show that the map det is well defined. First observe that A* maps A,(V) into AP(V) and recall that An is 1-dimensional by Lemma 2.3. 10. Hence A* y must be a multiple of M. If A is any other non-vanishing n-form then there exists a number a 7-1 0 with A ap. Hence we have =
A*A(vl,.
Vn)
=
A*(a/-t)(vl,..., vn)
det(A)M(vj,..., vn)
=a
which
implies that
definition for
our
=
a/-t(Avl,..., Avn) det(A)A(vj,..., vn)
=
det(A)
does not
depend
on
the
chosen n-form. Lemma 2.3.12. Let V be
an
n-dimensional vector space A: V
be
a
linear map and
Let the components
A* (Ol
A
...
A O'P
)
f el....
A3
be
particular
we
n1, fO'7
E ( E
==
V
--
....
defined by Aej
=:
onj be a. pair of dual bases. rnj= 1 A3j ej. Then we have A"'
sign(up)A"0,7, Ul
( P)) Oil A
17P
...
AOiP.
U7,ESP
jlf(t,x,y)
f:j'xUxV-+Kn,
j
in standard
mathematics for
U
x a
x
V there exists
unique
map -y:
(t, y)
neighbourhood
a
F-->
-y(t, x)
E
]Kn
with
d-y (t, y)
f (t, -Y (t, Y), Y)
dt
for all (t, y)
E
j
x
V and
This theorem translates
-y(to, yo)
=
xo.
Further,
straightforwardly
Theorem 2.4.2. Let M be
a
smooth
7 is C'.
to manifolds.
manifold and
V be
a
vector
field
M. For every X E M there exists a subset j C K and a curve x and (t) Vy(t) for all t E Jr. -yx: J --+ M with -yx(O) If VX : 0 there exists a neighbourhood U of jxj x 101 C M x R on
=
=
such that the map F: U -- M, (t, y) 1--4 Ft (y) -yy (t) is well defined. Moreover, the map y -4 Fi(y) is a local diffeomorphism for each i, and =
its inverse is
For t, The
s
curve
given by F -i.
small
enough
-y is called
an
we
have Ft
integral
o
curve
F,
=
Ft+,.
and the map
Ft the flow of V.
Proof Let (U, o) be a chart centered at x. Then the fundamental theorem ordinary differential equations (Theorem 2.4.1) implies that there
for
exists
a
solution
0(t)
of the differential equation
d dt
NO
=
( O-V)'3(t)
88
Analysis
2.
with
13(0)
-/ ' (0)
=
o(x).
=
manifolds
on
Hence 7,,
o-'
=:
o
0
is
integral
an
curve
of V with
X.
That F is smooth and well defined follows from the fact that
integral depend smoothly on parameters (cf. Theorem 2.4.1). The map (Fi)-' is given by (Fi)-'(z) -y_(i) where -y- is the locally unique of -V with curve integral -y-(O) z. Hence Fi is differentiable and has curves
=
=
a
differentiable inverse. The equation
of differential
Ft o F, follows from local uniqueness 'of solutions Ft+, and the fact that both t equations Ft+, (y) and Ft o F, (y) =
-->
are
integral
curves
An
integral
curve
j
-y:
-4
M of
same
a
initial value.
I
vector field V is called maximal if
integral : j --+ M with j C j C K implies J. By the lemma of Zorn and local existence of integral curves
the existence of
j
of V with the
=
an
curve
each
integral curve is contained in some maximal integral curve. The R maximal -integral following Proposition shows that in the case K =
unique.
curves are
K
R then there is
Proposition
2.4.1.
j
unique solution -y
C
K and
a
Let -y,
Proof.
If
=
of - (t)
=
unique maximal subset Vy(t) defined on j. a
be
integral curves of V with -y(O) integral curves coincide on the
=
(O)
=
x.
We must
j of j. In order to do so we will prove that the set IC It E j n j : -y(t) (t)j is both open and closed. Since in the case K R the set jnj is the intersection of two open intervals and therefore show that these
intersection j n
their domains J,
=
=
=
connected, this
set must then coincide with 1C. It is clear that IC is closed.
(U, o)
Let t E K and
be
a
the
chart centered at
=
y(t)
=
(t).
Given local
coordinates, problem finding integral curve solving a system of ordinary differential equations. An application of the fundamentat theorem for ordinary differential equation (Theorem 2.4. 1) proves that there is a unique local integral curve 0 through 0(: ). Hence there is a neighbourhood of t such that -/ on this neighbourhood. I of
of V reduces to
an
=
The
theorem
following
fields
implies that, locally,
Theorem 2.4.3. Let M be vector x
with
field
V(x) 7
such that
(P-
1
(t, X2,
...'
Proof. Let be
There is
a
0.
a
smooth manifold,
Then there is
and the
non-vanishing
vector
integral
N be
an
(n
curves
x
chart
of
V
G
M, and
(U, W) are
V be
a
centered at
given by
t
I)-dimensional submanifold of M through x (i.e., RV(y) (D TyN TyM for all y c N). Let
-
chart of N centered at
an E
a
xn).
which is transverse to V
(V, 0)
all
alike.
are
> 0
such that the map
=
x
and denote the flow of V
by Ft.
2.4 Vector fields and
f (- 6, 0
X
(X 1, X 2'..
0M
M,
n)
FX,
X
.'
is well defined. The differential of
FO
=
ordinary differential equations
df (xl,.
id and
(2.4.1)
(0 -1(X2.... X'))
(2.4.2)
1
at 0 is
f
89
X')
an
)(01
dx
isomorphism
V(X) =A
since
0.
0
a neighbourhood VV of 0 where this map is an diffeomorphism. The pair (U, W) (f (W), (f 1) ju) is therefore a chart centered at x. For any y E U denote by (yo, prN M) E R x N C R x M the unique pair defined by Fy,) (prN M) y. Then we have o o Ft (y) -' 1 o Ft (y) f o Ft+yo (prN M) (t + yo, 0 (prN (y))). Hence the intef gral curves of V are indeed the curves t F--+ o`(tX2'...'Xn). It follows I Ox immediately that o. V
Hence there exists
-
=
=
=
-
=
=
=
Given in
a
vector field
V,
1
-
one can
define the derivative of
a
tensor fiels
direction V.
Definition 2.4.1. Let and
E
x
M, V) be
tensor
a
field,
U be
a
vector
field,
Ft the flow of U. Then
((d) Ft*,O) ,
XU'O(X) is the Lie derivative
(-!!-) dt jt=O
Here sion x
Ft*V)
dt
(x)
It=0
of 0.
is the usual derivative in vector spaces. In
denotes
a
tensor field which is
if t is fixed and small
evaluated at
:=
x.
As
a
enough.
In
defined
particular,
function of t this gives
on a
fact, the
this tensor field
a curve
expres-
neighbourhood can
of
be
in the vector space
T'r (TX M) The Lie derivative
measures
Lemma 2.4.1. Let V be rection V is
a
a
the
vector
derivation, i.e., for
.Cv( O 0 0) -COW + 0)
=
=
change
field.
of
0 along
V.
Then the Lie derivative in di-
any tensor
fields p,,O
the
formulas
CVW 0'0 + W 0 -PVO, -Coo + 40.
hold.
Proof. These formulas follow immediately from the properties of derivatives.
I
90
2.
Analysis
on
manifolds
Theorem 2.4.4. Let
V be vector
U,
fields
and
f
smooth
a
function.
Then
XUf =Uof and .CUV*f =U*Vof -V*Uof. Proof.
The first
equation follows immediately from
d
.Cuf Let
X
d
=
at-It=0
Ft*f (x)
(U, p)
E M and
V C U of
x
and
d
Tt I t=0 f oFt(x)
=
be
a
chart centered at
number
a
of
a
x.
> 0 such that
e
satisfies Ft (y) E U for all for all the
df
==
(t, y)
E
(dt
Ft (x)
-
df (Ux).
It=o
There is
neighbourhood
a
Ft (y) is well defined and
(- 6, E)
x
V. An
application
of
formula to the map t i--+ f o F-t o o-' implies the existence smooth map j: (-,E, c) x o(V) -- R with f o F-t o W- 1 (z) f (z) +
Taylor
=
tj(t, z) for all (t, z) E f (Ft) (y) f (y) + tgt (y) o
=
V. The map gt(y) := j(t, for all y E V and we obtain x
w(y))
satisfies
d
-dt I t=o (Ft*V).,
f
d
d
"
((Ft V)
dt lt=o
f)
==
dt It=o
(((F-t),,V),,
d
d 0
*
dt It=o
e
d dt I t=o 0
V
0
where in the last step
f)X we
(f
*
(V
-
+
0
t9t))
U
gO)X
==
V
t))
0
(dt
0
=
-
dt It=o
(V
*
f)Ft(x)
+
Vx
90
f),
have used
d
0
F-
o
d
(VFt, (x)
-
(V
f)
(((F- t) VFt (x)) f) =dt (VFt (x) (f I t=o
=::
(U
9
f
o
(Ft)
-
1
(-)
It=o
)
=
Vx
9
df
X
(
d
dt lt=o
(Ft)
d =
Vx
9
df
it-lt=o
(F- t) (-)
(V
0
U
0
f),
Theorem 2.4.4 shows that the Lie derivative of V in direction U is the commutator of U and V. This motivates the
Definition 2.4.2. the Lie bracket
if
or
If U,
V
are
their Lie bracket vanishes.
fields
vector
the commutator
of
following
then
U and V.
we
definition.
call
Vector
[U, V] fields
=
CUV
commute
2.4 Vector fields and
Commuting fields
ordinary differential equations
91
are of particular interest since GauBian vector necessarily commuting. The following lemma gives the
vector fields
49,,k, 19.,1
are
to this observation.
converse
(Geometri'c interpretation of the Lie bracket). fields U, V have vanishing Lie derivative, CUV 0, if and only if their flows commute.
Lemma 2.4.2
Two vector
=
Proof Denote the flows of U and V by Ft and G,. The equation Ft o G, G, is the flow of V. Hence we have G, o Ft implies that F-t o G, o Ft o G, o Ft) V Ft*V and therefore T(Ft)-l (jd,,G,) o Ft jd-,,(F-t ds ds =
=
=
d
_pUV
( dt ) t=0
=
Conversely, *
Since Fo V that
s 1--4
integral
F-t
o
curves
Ft V
=
=
(-.4-) dt t=0
V
that X UV
assume
=
0.
0 which is
=
equivalent
* to - !- Ft V
dt
=
0.
*
V for all t. This implies integration yields (Ft) V G, o Ft is an integral curve of V. From the uniqueness of I we get F-t o Gs o Ft G, for all t, s.
V
=
*
an
=
=
2.4.1. Let M be
a n-dimensional manifold and f U1,...' U"T pointwise linearly independent, pairwise commuting vector fields defined on an open neighbourhood of x E M. Then there exists a coordinate chart (V, p) centred at x whose Gauflian basis vector fields Ui. satisfy Oxi
Corollary be
a
collection
=
Proof.
Denote the flow of
k Uk by Ft and let
O(X',...,x')
=
F.1,
o
...
o
F.1, (x)
sufficiently small (xl,...,x') E K'. Since the vector fields Uj are pairwise commuting so are their flows F j (cf. Lemma 2.4-2). Hence we for
have for every i E
O(X 11
...
fl,...'nj
Ix n)
=
This
implies V).(Ei)
basis
JE1,...' Ej
Fx i Fx1, o
d =
(Xi
-1
o
,
...
0
_P i_l
0
P+1 Xi_1
p(X1'...'xn))
of K'. Since the vector fields
=
0
...
o
F; , (x).
Uj for the standard
f U1,...' U,,j
are
linear
independent the map 0 has maximal rank and is therefore a local diffeomorphism. Let VV C K' be an open neighbourhood of 0 such that O(z) is well defined for all z E W and one-to-one on W. We can now define
A W)
=
(V)()/V)l 0-1).
1
Corollary 2.4.2. Let M be a 2-dimensional manifold and U, V be vector fields which are at each point linearly independent. Then M admits local coordinates (XI, X2) such that a,,i 11 U anda,,2 11 V.
92
2.
Analysis
on
2.4.1
Proof. By Corollary f, h with [f U, hV] =
[f U, hV]
we.
have to show that there exist functions
only
0. We calculate
=
0
manifolds
Vf U(hV)
=
=
f h17UV
=
f h[U, V]
=
f h QU, V]
-
+ -
'7hVf U
f dh(U)
-
hdf (V)U
+ d
f h17VU
hdf (V)U
-
f dh(U)V
+
ln(h)(U)V
d
-
ln(f)(U)V).
wu, wV be the 1-forms which are dual to U, V. Then any solution (f, h) of the uncoupled system of linear ordinary differential equations
Let
0
=
d
ln(h) (U)
-
wV QU, V]),
0
=
d ln(f) (V) +
wu ([U, V])
[f U, hV] 0. This system of differential equations can be solved by the fundamental theorem for ordinary differential equations 2.4. 1. 1
satisfies
In the
=
following
sections
we
will encounter various kinds of tensor deriva-
tiv,es. It is therefore practical Definition 2.4.3. Let D be it
fields. If
to formalise their a
common
map which maps tensor
properties.
fields
into tensor
satisfies
(i) D(T,(M)) C T,'(M), (ii) D(W 0 0) D(O) 0 W + V) 0 D(W) ("product rule"), (iii) D commutes with contractions, =
then D is called
derivation.
a
2.4.3. Let M be
Corollary
The Lie-derivative
Proof.
Ft*Cr^o
Xv
We have to =
S
CrFt*o
is
a
manifold and
d
dt
be
can
a
Proof. Writing
an
field
on
M.
Ft*o
CfFt*0
-4
is linear
(so
the
interchanged with this operation).
Proposition 2.4.2. Two derivations coincide if they fields and junctions.
we
vector
the third property. This follows from
verify only
and the fact that
8
derivative
V be
derivation.
a
arbitrary
tensor field
V)
in
coincide
on
vector
a
coordinate representation
(9
dxj,
obtain
Do
=
D(0'1`,9j, ii j.,
0
...
(9
aj,
0
dxjl
(9
...
...
r
+
E'O"ii
-
-
-3,",9j,
+ 3,
t=1
aj,
0... &
D(ai,)
&
aj,
...
&
(2)
0... 0
dxjl
(3
aj,
...
0
0
dxjl
(9
D(dx3")
...
(9
0
...
dxjl
(9
dxjl.
2.4 Vector fields and
Hence
we
only have
differential equations
ordinary
to show that D is
uniquely determined
93
for tensor
But this follows from D(w(V)) & V)) EO(M). D(C'(w I I Cll(D(w&V)) Cj1(DwOV+wODV)) Dw(V)+w(DV) for arbitrary I vector fields V and tensor fields w E 710 (M)
fields
E
w
=
=
=
-
Recall that vector fields tions. We show
alised to
now
can
be considered
b,
D,
can
on
func-
be gener-
be derivations. Then the commutator
[D, D] a
acting
arbitrary derivations.
Lemma 2.4.3. Let
is also
derivations
as
that the commutator of vector fields
derivation.
:=
D
D
o
-
D
o
D
Moreover, the Jacobi identity
[D, [D, b]]
+
[D, [D, D]]
[D, [D, D]]
+
=
0
holds.
Proof. For the first
assertion
we
to check that the
only need
product
rule
is satisfied. This follows from
D(DWOV)+W&DO)
DoD(WOO)
.6 o
and the fact that the term
respect to D and
The second assertion is
+
Do
D o
+
0
special incident
a
commutators of the form AB
[D, [D, DI I
0
J5V)
symmetric with
is
A -
of
a
general property of
BA: The summands in
[D, [D, Dfl, [D, [D, DI I D
o
b
o
b
-
D
ob
+
b
o
D
o
f)
-
b
o
J5
o
b
b
-(f) b o
b
o
D
-
o
D
o
D
-
b
o
D
-(D
o
b
o
o
5
o
D
-(D
o
D
2.4.4. For any vector
IU, IV, W]
+
fields U, V,
1W, [U, V11
+
W
we
IV, [W, U11
o
D)
D
o
b)
o
D)
2
o
D
-
pairwise.
Corollary
b
6
0
cancel
o
1
6
3
+
o
4
5
4
3
2
1
have
=
0.
D
o
D
94
2.
Analysis
Proof. Clear on
on
manifolds
since vector fields
be considered
can
as
functions.
Proposition
2.4.3. For
both X
Proof. Clearly, tion 2.4.2
we
only
vectorfields U, V,
[U, V]
and
need to show that
vector fields. For any function
-PV.CU*f
UoVof
=
[.C U,
-
Vo
formula holds for functions.
have
we
X V]
are
they
coincide
=
=
2.4.4
X[U,V]
=
[XU, _PV1
derivations. By
f we have [.CU, XV] Uof [U,V] of
Corollary
derivations acting I
on o
Proposi-
functions and
f
=
CUXV
C[U,Vjf.
implies for
9
f
on -
Hence the
any vector field
W
[.CU,.CV]W
XU(.CVW) [[U, V], W]
CV(XUW)
-
=
X
[U, V]
if
one
to
diffeomorphism but there
considers smooth maps which
[U, [V, W]],- [VI A W11
W.
The Lie bracket of vector fields does not
respect
=
only
is also are
not
transform naturally with especially simple relation necessarily diffeomorphisms. an
N be a smooth Proposition 2.4.4. Let f : M tor fields on M. If f', T7V are vector fields on N -+
and
T'f (W ,)
[V, W] f (x) Proof.
=
fVf ( )
for
all
c
x
M,
then the
map and
with
V, W be Tf (V,)
formula Txf QV, W])
holds.
Let
V (W (V
o: N o
f
R be
a
smooth function. The assertion follows from
V ((Tf (W) (
(V(W(w)))
o))
0
o
f
(Tx f (V) (Tf (W) ( o)))
o
f
f
2.5 Differential forms While it is
possible to avoid the usage of differential forms, they are important tool in analysis and mathematical physics that I have chosen to include them in this book. Differential forms will be used occasionally in the book, for instance in the treatment of electromagnetism. The reader can skip this section on first reading but she or he is such
vec-
an
advised to read the motivation below. This section builds on the theory of anti-symmetric tensors which is presented in Sect. 2.3.1 starting at page 77.
2.5 Differential forms
Differential forms areas
are
totally anti-symmetric covariant tensors. physics where this anti-symmetry
95
There
in mathematics and
are
proves to
be of great importance.
(i) Systems of partial differential equations: derivatives of
of Schwarz the
Recall that
by the
lemma
C' function commute. If
one a higher solution satmust differential has a equations, any partial isfy this "integrability condition". For the existence of a solution it is often sufficient to ensure that this integrability condition holds. Since anything symmetric applied to something anti-symmetric vanishes, such conditions can be naturally expressed by the requirement
system of
that certain differential forms vanish.
(ii) Integration: Recall from linear algebra that the volume spanned by n vectors Jbi,..., b,j in K' is given by the determinant I det(B)l where B is the linear map given by Bei bi and enj is the =
standard basis of K'. As the determinant is differential forms car6
are
totally anti-symmetric, generalisation. The lemma of Poin-
its natural
(Theorem 2.5.5)
and the theorem of Stokes
(Theorem 2.5.2)
integral theorems of Gau.B and Stokes are the for superiority of,using differential forms. good examples (iii) Physical applications: There are also direct physical applications of differential forms. They are a prerequisitive for understanding gauge theories (cf. (Bleecker 1981)) of elementary particles and in particular the theory of electromagnetism (cf. Sect. 5.2.3). which unifies the classical
Recall from Definition 2.3.8 that the set APM
p-forms
is
a
vector subbundle of
Definition 2.5.1. p
by
OP (M)
If M is
(cf.
a
To (M)
over
of all differential forms of degree w I (cf. Definition 2.3.13).
=
will sometimes denote OP (M)
we
of all
by
S?P (M,
R)
below)
The definitions and to
o w
P
real manifold
Remark 2.5.2
alt
:
U,,CM AP(T,,M)
P
We denote the set E
=
T'M.
properties of p-forms given a pointwise manner.
in Sect. 2.3.1 carry
differential forms in
Lemma 2.5.1. Let w,,q be
differential forms
and V be
a
vector
field.
N the exterior product satisfies (i) For any smooth map 0: M A O*W O*n. O*(W Aq) N is a local diffeomorphism then 0* (v I w) (0* v) _j (ii) If 0: M holds. (O*w) (iii) The differential form w can uniquely written as --->
=
=
--+
U)
(X)
1:
=:
1
coincides with
differential. The operator d
Definition 2.5.2.
of Theorem
2.5. 1 is called the exterior
derivative. Observe that for the definition of the exterior derivative
we
any additional structure. This fact indicates that in many
it will
play
fundamental r6le. In comparison, the Lie derivative only defined with respect to a given vector field.
a
tensor field is
2.5. 1. First
Proof of Theorem is
an
(dw)jU
h: M
--+
show that d is
a
local operator, i.e., if U
a
compact closure and w,,q c QP(M) satisfy WjU =,qlu (dq)ju. To see this let V be an open set with f) C U and
=
R be
h(w -,q) d(w
This
we
of
open set with
then
Since
do not need
applications
-
a =
smooth open function with -,q we obtain from (iii)
77)
implies (d(w
=
-
d(h(w -,q))
77))1V
the arbitrariness of V Since d is
a
hjV
=
0 and
hlm\a
=
1.
w
we
=
=
dh A
(w -,q)
+
hd(w
-
0 since both dh and h vanish
have therefore
local operator
we can
proved
(dw),U
==
restrict to chart
77)).
on
this set.
By
(d?7)lu.
neighbourhoods.
We will prove the theorem by showing that for each chart (U, W) there is a unique operator d which satisfies properties (i)-(iv) above. Let W (E
QP(M) and write (i)-(iv) imply
dw
w
=
El
0
(W ' )R3
are
will defined since the scalar
product
to the standard orthonormal basis this
column and
row
The scalar
::::::::
an
isomorphism
(V7')R3.
The inverse
These
w. isomorphisms non-degenerate. With respect isomorphism just interchanges =
is
vectors.
product also induces an isomorphism between O-forms an isomorphism between 1-forms and 2-forms. Let
and 3-forms and 1 (X ,X2,X3)
be the standard coordinate system of R3 and set
This defines linear
*1
dxl
A
dX2
*dxl
dX2
A
dX3'
*dx
2
dX3
A
dx',
*dx
3
dxl
A
dX2.
isomorphisms
*:
A dx
A'(R3)
--,
3,
A3(R3)
A 2(R 3) They can be extended to 2- and 3-forms by for all forms in R 3 .
-
and
*:
A'(R 3)
demanding
**w
=
w
2.5 Differential forms
df 0, rot(F) grad(f) above relating F (ii)
Observe that Hence the
=
(i)
(*dP) , div(V)
=
and
and
=
103
*(d * 0).
V and W translate into
f,
the assertions
P satisfies dP P f with df (ii) If the 2-form *0 satisfies d *0. I-form W with dW If the I-form
(i)
0, then locally there
exists
0-form
a
=
I
0
=
0, then locally there exists
a
=
The lemma of Poincar6
generalises these
(Lemma
Theorem 2.5.2
QP(M) with p > 1. If dw bourhood U of x and a (p The
proof
facts.
Poincar4). Let M be a manifold and w 0, then for every x E M there is a neighdO. I) -form 0 E S?P-'(U) such that wjU
of
=
-
=
of the theorem will be
Lemma 2.5.4. Let M be
a
corollary
a
to the
smooth manifold and
following lemma.
define for
any t E
[0, 1]
(q
with
the map
it: M There exists d
K + K
o
Proof.
For
o
a
d
w
G
(il)*
-
Qq+ 1 ([0,
Vl,..., Vq+j be
Let
M,
X
([0, 1]
X
X
linear map K: Qq+ =:
1
--+
(t, X).
M)
_+
Qq (M)
>
0)
(io)*. 1]
Kw,,(vl,...,Vq)
we
[0, 1]
--+
M)
x
=
and V1,
J0 i(it* (at
vector fields
on
J
...
i
Vq
E
TxM
w(t,x))) (V1
M.
I
...
we
define
Vq) dt.
I
Then, using Proposition 2.5.1,
obtain
dKw(Vj,..., Vq+l) q+1
1
E(_l)a-IV
t
a
J
Z
W(t, X)) (VI,
.....
Vq+l) dt
0
a=1
1
(_I)(a-1)+(b-1)
+
fo
it* (at
J
W(t,x)) ([Va Vb] V1.... 7
,
I 2 there is
notation
we
renumber these
j < k with Vj n Vk =A 0. We set (Ul, Wj) (Vi, 01) thereby trivially defining an atlas for V1 which satisfies the positive determinant condition This atlas can be extended by an induction argument. Assume that we have defined an atlas a
=
f (Ul (PI) for the set V1 U
and let
j
A
...
U
Vk which satisfies the positive determinant condition be an index with Vk+j n Vj =h 0.
a
A
nowhere
dxn and
satisfying (Ok+l)*l/ vanish
on
fk+,(y)
0 0 for all y G pj(Uj) n Ok+I(Vk+l). In the first case we -
(Uk+1 (Pk+l)
=
7
whereas in in the second
(Uk+li Ok+l)
=
(Vk+l; Ok+l)
case we
(Vk+l
7
1
2
,Y ,Y
3
fk+l
=
fk+l
set
X1-2
where Xl-->2 is the reflection defined
XI-2(Y
and
Yn)
0
V)k+l)
and
fk+l
=
-fk+l
by =
(Y2' Y1, Y3'...' Yn).
--
R
fk+l
or
set
fj
2.5 Differential forms
In either 0. This
case we
have then
(Wk+l)*I'
fk+ldx'A
=
Adx"' and
...
ill
fj-fk+l
>
implies
det(D
(Wj (Wk+l)-') dx1A o
...
A
dx"
(Wk+l)-l)* dx1
0
A
fj fk+1
det(D (Wj det(D (Wi
and therefore
0
0
(Wk+l)-l) (Wk+l)-l)
(CPj
A dx"
...
dx1
A
...
A
dXn
(Wk+l)-')
0
> 0 in
either
case.
We still have
0 for any i E let fj be defined
kj with by (Wl)*v f1,...'k+lj A dXn (1 E fp, qj). Since all fj (1 E 1, fldx' A kj have the same sign and also fk+l and fj have the same sign it follows that sign(fi) sign(fj) sign(fk+,). Hence det(D (Wi (Wk+l)-l) > 0 Let jgk: M [0) 1] JkEN be a partition of unity subordinate to fUkjkEN and let
to show that
>
0. For 1 (=-
4
UinUk+l
o
=
...
-
-
-,
0
=
n
V
T_gk(Wk)*(dxl
...
dXn).
k=1
This is
gk(x) x (E
a
smooth, well defined n-form
since at each
x
only finitely
many
and the support of each A is contained in Uk. Let all indices with gi., (x) -4 0. Then
are non-zero
Uj and il.... iP
P
( oj)
*
v.,
=
E gi., (x) D (Wi.,
o
( oj)
-
dxIA
1)
...
A
dXn
(x)
j=1
does not vanish since all gi., D
have the
same
are
( oj,.,
strictly positive o
( oj)
-
1) 1
wi,
and all
(x)
I
sign.
Definition 2.5.8. Let M be
E on (M) an oriented manifold and v representative of the orientation. An oriented atlas is an atlas I Pa (Pa) JaEA such that for each a E A there exists a strictly positive function Vaj n: Ua -- R+ with
be
a
i
...
( Oa)*V where
(xl
....
ented chart is
I
xn) a
are
::::::
Vaj
...
ndX1
...
dx',
the standard coordinates
chart which
belongs
to
an
of Rn. A positively
oriented atlas.
ori-
112
Analysis
2.
2.5. 1
Example
on
manifolds
(M&bius band,
continued
from page 53). 2.1.2 is not orientable.
band M defined in notation
VV+
Example definition,
in its
as
7r-1 ((a, 2a)
=:
map D
(WI
Using
==
has positive determinant for all
.
the
the set U, intersects U2 in two and and W_ 7r-'((O, a) x (-b,
(-b, b))
x
(W2) 1) -
0
The M6bius
E
x
same
subsets',
b)).
The
W+
and
negative determinant for all x E W_ Assume now that M is orientable, i.e.-, that there is a nowhere vanishing 2-form v on M. Then there are .
nowhere
vanishing functions fl,
:
V1
-4
R,
f2: V2
R
-4
with
fldxl
v
((W2) 1)
*
A
f2dx'
dX2 and dX2
(( P2)-1)*
P
==
f2dxl
A
dX2.
(((Pl ((P2)-l)* (fldx' A dX2) we f2 (x) (W1 (W2) 1) h for all c )IV+ U W_ Since the determinant det (D (W 1 ((P2) 1) changes sign get contradiction. From
-
V
obtain that
det (D
A
0
=
0
x
-
0
we
a
Thus the M6bius band is not orientable. It is
a
simple but good exercise to actually build a M6bius band from verify using this model that there are closed curves along
paper and to
which there does not exist any continuous frame.
2.5.4
Integration
on
real manifolds
In this section, we restrict to K R. This is necessary since we need to employ partitions of unity. See Remark 2.5.2 below for the integration of complex valued functions. =
One
usually introduces integration as a method to determine the volume open, bounded region B C R n- The main idea is as follows. We divide B into small parallel epipeds Bi. Each Bi carries a number f(Bi) representing the volume of Bi. Summing up all these numbers gives an approximation for the volume of B. Clearly, the function f which maps parallel epipeds into real numbers must satisfy certain properties. The most obvious property is that if we divide Bi into two disjunct parallel Ai U Ci, we have f(Bi) ,:z f(Ai) + ffi), epipeds Ai, Ci with Bi at least if Bi is sufficiently small. Then, choosing an infinite sequence of
an
=
fJBi,,,1iE1(a)1a (a E N) ofsuch divisionswe obtain a sequence of numbers lEiEI(a) f(Bi,a)JaEN which in most cases of interest has a well defined limit, the choice for
volume
vol(B)
of B. Linear
algebra indicates the following
Let
e,,J the standard basis of R', 101.... 0- 1 its dual basis, and bij,..., bi,, those vectors which span the parallel epiped 01 A Aon (bi, I,. Bi. The number f(Bi) bi,,,,) is then the Euclidian f.
)
=
...
-
-,
volume of Bi with respect to the standard Euclidean scalar product. This function clearly satisfies the additivity condition above. If one knows
2.5 Differential forms
how to determine volumes
0: B 01 A
...
(for
R
-->
113
also integrate continuous functions differential form
one can
densities) by replacing the differential form 001 A A on.
instance
A on with the
mass
...
turn to manifolds. The main
problem here is that we do not have a linear space in which to embed the cubes. However, by now we are familiar with the idea of translating concepts to their infinitesiLet
us now
mal counterparts. Since the tangent space was introduced as the linear approximation of the manifold, it is natural to place our parallelepipeds which
use
the linear structure of Rn into the tangent spaces rather than words, we divide M into small sets Vi
into the manifold itself. In other
such that each of these sets
corresponds
to
a
parallel epiped
in
T,,.M,
where xi is a point in Vi. We cannot define a canonical volume because E,, 1. This a general manifold does not have a preferred frame I El, , .
indicates that it is 'more natural to define the volume of
special To
(cf.
case
are
set first. We will later
of
of all
We
clearly have
recover
.
.
than to
directly
the volume
as
a
our
discussions
we
will define integration for
n-
necessarily smooth.
not
Definition 2.5.9. Let M be set
n-forms
4.2.1).
Definition
simplify part
forms which
a
integrate
continuous
a
real
manifold.
We denote
On (M) the
by
n-forms. Q (M). Let (U, o) be
f2n (M) c
a
chart and
w
cz
f2cn (M)
an n-form with compact supp(w) C U. Writing o (xl,..., x') there A A dxn. We is a unique smooth function wi n with w ndxl w,
be
=
...
...
...
define
W
Wi
...
ndxl
A
...
/\ dXn
where the last expression is the usual
U)i
integration
...
n
o
cp-ldxl
...
dXn,
in Rn. We still have to
show
(i) (ii)
fu
that
w
does not
' O
depend
on
the chosen
chart,
how to extend this local definition to manifolds which may not be covered by a single chart, how to extend this local definition to n-forms which do not have
(iii)
compact support.
(i):
Let
(U, 0)
be another
chart, denote by (y 1,
yn)
the
corresponding
coordinate functions and set axa 77
Then
we
=
sign
(det(I
ayb
1))
have gxa
W
:::=
Wi
...
ndxl
A
...
A dXn
=
Wi
...
n
det(f
ftb
1)
o
Odyl
A
...
A
dyn
114
Analysis
2.
on
manifolds
qxa 77wi
...
det(f
n
J)
o0dy'
A
...
dyn
A
and therefore
JOM
W1
oO-1dy'
n
...
10 =
dyn
...
Wi
...
0'0-1 det(I
n
(U)
77
=
Wi
W1
7
...
n
...
n
-0-1
O-ldxl
o
Hence the definition is indeed coordinate orientation in advance and restricts to In order to address
employ
a
(ii)
E
w
Qn(M)
be
a
...
(p
o
dyn
0)-ldxl
...
dXn
dXn.
independent
if
fixes
one
an
integration globally
we
will
C
an
oriented n-dimensional real
I (Ub) Ob)
closure, fa index with
GB
SUPP(fa)
Since each
is
an
partition of
a
E
x
C
b(a),IP6(a)
oriented atlas such that each Ub has compact unity subordinate to fUbjbEB and b(a) is an
Ub(a)
M has
SUPP(fa)
many
manifold
n-form with compact support. Then
aEA
finitely
o
...
oriented atlas.
and to define
IM where
1) dy'
qyb
partition of unity.
Definition 2.5.10. Let M be and
an
qXa
a
and
-
neighbourhood
supp(w)
which is intersected
is compact the
sum
by only
in the definition
above is finite. Let and
19c1cEc
6(c)
finite
be
an
we can
be another partition of unity subordinate to f(UbIbEB Since all sums involved are supp(g,) C
index with
Ub(c).
calculate
aEA
J(U
faw b(a) Wb(a)
aEA
9c
fa U)
9C
f" W
9c
fa W
cEC
I EEI I: E
aEA cEC
(Ub (a)
,
W b (a)
)
cEC aEA
E cEC
f(U6(WP6(c))
gC
1: aEA
f.
2.5 Differential forms
115
grCEC
implies that the definition is independent of the chosen partition of unity. Since by coordinate invariance it is also independent of the chosen oriented atlas our definition of integration over n-forms with compact This
support is well defined. We will now address point (iii). If w G Qr' (M) does not have compact support, its integral may not exist. This is completely analogous to the integration of functions f : R ---> R. For our purposes the following extension is sufficient.
Definition 2.5.11. Let M be and let
w
E
w(a) : Ua 1...n
modulus
oriented n-dimensional real
an
manifold
A, (Pa) I aE A be an oriented atlas and for each a A dx'. The R be defined by (Wa)*W W (a)ndxl A I... the continuous n-form jwj locally defined by ((Pa)*lwl
S?n Let C (M).
of
p(a)" jdx1 A
is
A
...
I...
Let M be
":
--+
w
...
dxn.
oriented n-dimensional real manifold and
an
(Ua, W,,)aEN
be
a countable oriented atlas such that each Ua has compact closure. As a preparation to the following definition we first need to give a meaning to
arbitrary w E S?n(M) which do not 11 E kj let Ij necessarily have compact support. For each j E f 1, fj+l,...,kl : UjnUj = _ 01. ifq E Qn(U,U UUk) has compact support
the
expression f fulu
...
JWjjkEN
uU,
for
C
=
.
.
.
,
...
G
then WE
we
fu,u
have
Qn(M)
is
...
UUk
general
a
C
Ekj=
1
(f(Z i,Wj)
differential form
we
f(i jnuwj)
if
define
k
Jul
U
...
UUk
jm'wj)
j=1
Definition 2.5.12. Let M be and
be
A, Wa)aEN
a
oriented n-dimensional real
w
E
Qn(M)
is
c
manifold com-
integrable if the (monotonically
f fak: U1 b(a) is an
...
UUk
U
and
index with
w
E
...
U
Q (M)
independent
aEAk
Ukj
where
any
I
jWjjkEN
UI U
nition is
n u, oj
sequence
11 Clearly,
J(41
countable oriented atlas such that each Ua has
pact closure. The n-form
increasing)
an
1E j
a
fakW7
Pb(a),Vb(a))
partition of unity subordinate
supp(fa)
C
Ub(a)
-
to
f(UbIbEB
is bounded.
with compact support is integrable. The defisee this let (Vb, V)b)bEN be
of the chosen atlas. To
second countable oriented atlas such that each Vb has compact closure. Then for each k E N there is a j(k) E N such that U, U Uk C a
...
116
VI
2.
U
Analysis
on
Vj(k). Hence JU)JJkEN is UUk
...
IfUJU
...
We
can now
manifolds
fu,u
...
I (A) I
UUk
bounded if
fV, U-I ,(,) 1wJ which implies ffv,u U-I j IWIIjEM is bounded.
2).
integrations:
00
fm
dw
pulled back
E
d(fjw) j=1
j=1
fmnL j d(fjw)
00
Y j=1
Corollary
2.5-4. Let M be I E S?n-
mani
Id and
Proof.
Since M is
fam
=
As
W
fo
Jamnul. fjW Ja
W
w
=
a
an
(M).
W.
M
n-dimensional, oriented, compact,
Then
fm
manifold without
dw
=
real
0.
boundary,
OM
0 and
fm dw
0.
1
application of Stokes' theorem we prove that for every evensphere every vector field must have a zero.' This theois also known as the "theorem of the hedgehog" since it shows that impossible to perfectly comb an "ideal" hedgehog.
an
dimensional unit rem
it is
Lemma 2.5.11. Let M be
manifold and f,
f:
M
--+
M
an n-dimensional, oriented, compact, homotopic maps. Then
f
f
m
for all 7
w
E
M
Qn(M).
Recall that
we
only consider smooth
vector fields
real
2.6 Connections and
on
-
lary
1
implies that there
Lemma 2.5.6
Proof.
(M)
with
f *w
-
f*w
=
exists
a
121
structures
differential form Co
1
2.5.4.
Sn.
on
Proof.
If
E
dCv. Hence the assertion follows from Corol-
sphere in R'+1 and V be a a point x0 E Sn with V(xo)
Theorem 2.5.6. Let Sn be the unit
field
projective
is
If n
even
then there is
0 for all
V(x) :
x
Sn
(E
we
can
vector =
0.
normalise V with respect
product -, -)R11+1 of R'+', i.e., we can assume 1. For each x E S' we without loss of generality that T) V)R11+1 can identify V(x) with a tangent vector of R'+1 and, since T.,Rn+l and R1+1 are canonically isomorphic further with a point in R'+'. Our S'. normalisation implies then that we have defined a map V: S' 1 Let 7 be an integral curve of V with -y(O) x. Then ('Y(t))'YW)R-+1 2 (X7 V(X))R",+I. Hence V(x) is implies 0 I ('Y(t)1'Y(t))R-+1)Jt=o Ut to the Euclidean scalar
=
--->
=
perpendicular
to
and the homotopy
x
[0, 1]
F:
x
is well defined. It satisfies
Since -id is
=
=
=
homotopic
orientation of Sn
(for
n
S'
--+
X
_4
S'
cos(7rt)x
F(O, x)
=
x
and
to id and the
even)
+
sin(7rt)V(X)
F(1, x)
-x
=
for all
x
E
Sn.
diffeomorphism -id changes the 2.5.11 implies that for every W E
Lemma
on (sn)
is holds. This in turn not true. Hence
implies
our
is",
is" fs,,. w
=
0 for all n-forms
initial assumption
which is
w
V(x) 7
0 for all
x
E
certainly
S' must be
wrong.
2.6 Connections and There
is
one
projective
feature of A' which
we
have
structures
ignored
far in
so
our
efforts
p. 89
spacetime. Given any two different points in An there is a [I unique line passing through them. This global structure has an infinitesimal counterpart given by the directional derivative of vector fields. In fact, these lines are exactly those curves -y: [a, b] --> A' which satisfy 0. (Observe that this expression is well defined, i.e. for any Dff- y)
to localise
=
vector field V with
Recall that in
V(-y(t))
Chap.
I
=
we
(t)
we
have
have relied
to introduce inertial observers. Here
we
on
DV( ) this
=
0.)
affine
will introduce
structure in order a
generalisation
of
p.
1
125]
122
2.
Analysis
manifolds
on
general manifolds. This will be done by generalising
it to
the directional
derivative D. In the affine space An the difference of a derivative of a map 0: An a vector field V: An --> K' is blurred. In fact, let e 1, en
A' and
-
be the standard basis of Kn. Since -
the derivative of
in direction
w
at
x
is
given by
-
the derivative of V in direction
w
at
x
is
given by
it is difficult to
0:
map
commonly both
are
M and
-->
x
+
2-24wie.
and
wi ei av" axi
the difference between both kinds of derivatives.
see
Accordingly, they M
0
a
vector field U
denoted
on
by
D. Consider
now
0 at x in analogous
direction wx is now given by Txo(w.,) E To(x)M whereas the derivative of V is given by TxV(wx) E Tv(x)TM. We do not obtain vector but
a
M. The derivative of
a
element in the tangent bundle of the tangent bundle. It follows that this derivative cannot be used for defining straight lines. an
In order to obtain in TM
ues
we
ing definition
an
will need is
an
same as
the usual derivative D.
Definition 2.6.1. A covariant derivative
Tol(M)
V:
-4
711(M),
V
--+
VV,
or
VV(W)
connection V is
VWV
:--:
such that the
with val-
additional structure, a connection. The followexpression of the idea that infinitesimally a connection
an
17 should be the
analogue of the directional derivative
for all vector fields U, V, W, all functions f, h, following holds.
'--:
a
map
Va VbWa OXb
and all a,
0
E
R
(i) '7f V + hWU f VVU + h'7WU, (ii) VW(au + OV) aVWU + 017WV, NO 17wf U (W f ) U + f VWU. =
=
-
=
The torsion
of V
is the tensor
(U, V)
-->
field
Tor(U, V)
=
VUV
VVU
-
A covariant derivative 17 is called torsion-free
if
-
[U, V].
in addition to
(i)-(iii)
the equation
(iv)
Tor
=
0.
holds. This definition is
justified by
the
following
Theorem 2.6.1. A map V: Tol(M) ant derivative if and only if for each
(xO
I
.
.
.
,
xn- 1) centered at
x
such that
theorem.
T11(M) x
E
is
a
torsion-free
covari-
M there exist coordinates
2.6 Connections and
projective
123
structures
,9Va
( ) '9Xb
holds at
x
for all
Proof. Assume
first that there exist coordinates such that the
properties (i)-(iv)
2.6.5 is satisfied. Then it is clear that covariant derivative
Let V be
centered at
x.
of
Equation
torsion-free
a
satisfied.
are
(U,
covariant derivative and
a
3)
be two charts
We denote the coordinates with respect to these two charts
W (y), -P(y) (D( o W- 1))ab igXb
by x (y)
(2.6.5)
V.
fields
vector
(7b Va
=
Ix
3'(y).
=
=
Observe that
(D(W
,-1))ab
0
ax' =
ayr
and
o
Setting
,VWV
Va
V,95 19: b
fVa
fVa -9.:i'Vb, .97
a
f'C 19X,::
-95c'Wb
=
have
we
lacblqX ,WaVbg
Wa,9X", Vc,9X; +
=
=
where
and
F'b,9x.
V,9X.19Xb
X
f;,cb,9, ,::fVafrbaxc,
+
a
' Xb ab. Hence
and
a
obtain
we
C
(VWV)
fVaa:i", frc a,,ta
Wd
axd
PcbWaVb
+
'a
aXe 9.,
a
ax
a;,-a CqXe a;,c
,ox d
9;-,a
&,-a +
aj b
Xd Xf
, 'c af
-
a: c
(_)c VWV
a., , ax,
0_ _c
_Vf
+
axf
P
a. a
ax
d
a, b
Vf
9xf
a2;i c
a, ,a CqXe +
Wd
c
ab
WdVf
_
Xd a.: a aXf aXe
.cb WdVf a
+
WdaxdVe
aXe
(
Wdax Vf
WdaXd, Vf
9xf
From
5X7
"
a.:; a 19: b
a2,, c
(
+
9Xf,9X d aXe
+
92j h
(5
-
a., h
(,VWV)e
=
cb)
WdVf
Pa c
19x d 9Xf
XfgX d
a;,-a +
aj b
9x d gXf
2LWd aXd Ve ax,
b)
fahI
WdVf
yWdVf
+ re d
get
we
therefore
(9Xe
Fdef Now
we can
show that
9j a
h
92,: h
9Xe 9. a
9., b
,h
-jX-f_9Xd +_;X h_ -Xd 5Xf ,
covariant derivative is torsion-free if and
if there exist coordinates such that the
equation
only
in the theorem holds.
Expressing the condition of being torsion-free in coordinates we see that it is equivalent to Fab Fbca in any coordinate system (xo.... Xn-1 If the covariant derivative is induced by coordinates in which the Facb =
I
vanish at x, then in any other any coordinate system have red f
a2: l which is _5 r D_X_ -Tax 'T aX.
clearly symmetric
in
(xo.... Xn-1) I
e
and
f.
we
For the
124
Analysis
2.
assume
converse,
generality)
that
manifolds
on
that the
z-a(x)
Facb
transformation of the form :F in b and
c.
At.;p
0
=
we
5-Xb
A ac ab
=
Pacb
=
(Pacb)
-
+
a, b and
consider
a
(without
loss of
quadratic coordinate
xa + .1 Aa XbXc, where A' is symmetric bc 2 bc
=:
2;,- c
Aacb
=
5X _b at JF
=
ja b
-
19Xa,9Xb
Acab,
0. Our assertion follows
by choosing I
Ix
For later reference
proof
now
aXa
ja b i
=
Facb
symmetric in
have then
a,, a
whence
are
0. We will
=
collect the coordinate expressions derived in the
we
of Theorem 2.6.1 in the
following corollary.
Corollary 2.6. 1. Let (xo, Xn-') be a local coordinate system on a manifold with connection (M, V). Then there exist functions FbI, ra (bc) such that for each vector field V the covariant derivative VV is given by .
.
.
V b Va and the
I
==
9b Va + _pa buc
function Fbac transform under
(x0 according
xn-1) aj a
92Xh
19X h
7Xbqj c
a C
Lemma 2.6.1. Let extension
unique
(M, V)
of V
to
transformation
(: o ....... n-l)
VV: T,,r(M)
--
19. a 19Xd
19Xeh d 7Xh Yb_ a. c Fre r
+
be a manifold with connection. There general tensor fields,
V: Tr (M) such that
coordinate
to
A a
_4
a
-
T'r+ , (M),
Tr(M)
is
a
0
-
exists
V V)
derivation for every tensor
field
V.
Proof. It is clear that we can extend VV to tensor fields Uniqueness follows directly from Proposition 2.4.2.
as a
derivation. 1
In
special relativity, timelike straight lines represent freely falling partilightlike straight lines represent light signals. In Euclidean space, straight lines are the shortest curves between any two points. One way to define a straight line -y: (a, b) F--> A' is to require that the acceleracles and
tion
=
Dffy)
connections
Let t
as
vanishes. This definition carries
over
to manifolds with
follows.
V(t)
be
vector field
along a curve t 1-4 -y(t). Then we can ' which is defined in a neighbourhood of the path of -y. It follows from Corollary 2.6.1 that the covariant derivative of '1 restricted to -y in direction is independent of the extension ' . Hence the following definition makes sense. 1--4
extend V to
a
a
vector field
2.6 Connections and
Definition 2.6.2. Let V be and t
-->
V(t)
vector
1 (t)
Then
(i)
a
covariant
a
field along
V (t)V(t)
=
covariant derivative of V
projective
derivative,
125
structures
t
-y(t)
-4
a
curve,
9a
is the
-y.
(AVa(t) +.pbayc(t) b(t)) dt
=
along
-y.
(ii) A pregeodesic is a curve -y satisfying V 0. (iii) A pregeodesic -y is a geodesic if V (iv) A geodesic is complete if it is defined for all t G K. (v) A projective structure is a maximal equivalence class of =
tions which all have the
The notation field with
V (t)V(t)
W(-y(t))
=
can
V(t).
(V (t)W)
0
same
be
justified
Then
we
as
follows. Let W be any vector
have
(( babWa
Y
connec-
pregeodesics.
rba,_ b Wc)Oa)
+
0
C
-Y
d dt
d dt Hence the notation
comes
from
(W 0,Y)a
+
Fba_ bWc
0
'Y) 19a
Va + ra bW Vc)Oa-
identifying
V and W which
are
different
maps but assign to each point -y(t) the same vector. We will now introduce the notion of parallel transport. The follow-
ing
motivation may
seem
to be
mathematically imprecise. Nevertheless,
mathematicians used such arguments (before the advent of the French Bourbaki school which introduced a new level of precision in mathemat-
ics)
in order to introduce connections. These
and
are
with
arguments capture very well
of the intuition which leads to the notion of
some
a
therefore worth
knowing,
even
of salt: One may view
grain
a
a
covariant derivative
though they
have to be taken
covariant derivative
as a
connec-
tion between
infinitesimally neighbouring tangent spaces. Let V be a x E M and E TxM. The idea that T,,M is the infinitesi-
field, approximation of the manifold M near x is sometimes expressed by stating that x + is a point infinitesimally close to x. (Strictly speaking, this "addition" of points and vectors in general manifolds does not make vector
mal
Assume that M is
submanifold of Kn+k
Then we can identify subspace of K k+m and x + does make sense. This point will in general not lie on M but still be close to M if is small.) Let v,, E TxM which we want to "parallelly" translate to the point x + . sense.
T,M with
a
a
.
linear
We will then have vx+C v., + Jv where Jv is small. Since this is an infinitesimal process Jv should depend linearly on v and the difference =
. This bilinear map mally neighbouring points.
vector
sider the
now
curve
connects the
In
tangent
spaces of
our
infinitesi-
17bcVb c. Co n coordinates, we have jva a curve -y between two distant points x, y E M. We can divide into infinitesimal segments and parallelly translate a vector a
=
_
[p. 1211] I
p. 132
126
2.
manifolds
on
TXM successively along these infinitesimal
E
vx
Analysis
curve
segments. The
concatenation of these infinitesimal map P
:
ly
TxM
which in
TyM
-+
parallel translations gives a linear general will depend on the interme-
diate tangent spaces and therefore on the curve -y. These ideas will be made precise in Definition 2.6.3 and Proposition 2.6.1 below.
Definition 2.6.3. Let
(M, 17)
be
-/:
be
a
manifold
(a, b)
-+
with connection and
M
A vector
field V along -y is called parallelly transported along -/ if 17 V(t) for all t E (a, b). A parallelly transported vector field is often simply called parallel.' curve.
a
=
0
In
particular, a geodesic is a curve whose tangent vector is parallelly transported. For this reason, geodesics are sometimes called auto-Parallel curves.
2.6. 1. Let
Proposition
[a, b]
-y:
--+
M be
a
curve
(M, V)
be
which
can
a
manifold with connection and smoothly extended into both
be
let di-
rections. The map
P^y: T-Y(a)M where V:
V(a)
=
Proof.
[a, b]
v, is
Since
a
-4
T-y(b)M7
TM is the unique
--*
linear
-y([a, b])
v
parallel
-4
V(b),
vector
field along
-Y with
isomorphism. compact, there exist finitely
is
many charts which
the
curve -y. Without loss of generality we can assume that -Y cover is contained in a single chart. (Otherwise we could divide -Y in segments which are contained in single charts. A successive application
of the
the a
proposition to these segments would imply Vt 0 general case.) The equation 17 V(t) =
the G
proposition
in
reduces to
(a, b)
system of first order differential equations for the coefficients Va(t):
1 a + rba . bV, 0. It follows from the fundamental theorem for ordinary differential equations (cf. Theorem 2.4.1) that the map P is well dely C
fined.
(t)
Letting
-4
-y (a + b
-
t)
and
f/ (t)
=
V (a + b
-
t)
we
clearly have -
17,:YV(t) and
a
=
0 if and
only
fortiori that P
is ^/
It is
possible
to
if 17
recover
an
V(t)
=
0. This
implies that
isomorphism.
the covariant derivative from
(P
,
P I
parallel transport.
The relation between these concepts is similar to the relation between the Lie-derivative and the Lie-transport. But note that there is no Lie-
transport along
a
single
curve.
2.6 Connections and
Proposition -y:
[a, b]
rections. For Then
for
2.6.2. Let
(M, 17)
be
which
can
curve
a
Proof. Let W(a)
=
S-0
(P t, (V(t
8
+
-
s))
FbacWb au ( t,,)c. Using
a
t
V(t))
-
.
Then 17W 0 implies -!LWa da 1419 expansion it is easy to see that there
P t (V(t + s)).
d
a
be
--
lim
V (t)V(t)
exists
a
-
any
127
structures
manifold with connection and let smoothly extended into both dilet each s E [0, b M, a -4 -y(t + s t) a). t,,: [0, s] vector field V(t) along -y we have
M be
-
projective
=
Taylor along t,, with
smooth vector field U
C
W07) Setting the
a
=
W(O)
s we
=
d
((au t") lc=O)
o'Fb(c Wb(0) a
-
obtain from W (0)
=
V (t + s) and
aa
+
U(9)9 2.
(jddo, , t,,)1,=0
=
- (t)
equation
p t'sV(t + 8) which
V(t)
_
implies the
Corollary
=
V(t
+
8)
_
V(t)
+
SrbacVb(t)
c(t)a
U( 8 )82
a
assertion.
2.6.2. A connection is
uniquely
determined
by
its
parallel
transport. Since inertial observers played an important r6le in Chap. 1, a thorough understanding of geodesics and projective classes should be important for the globalisation of the results in Chap. I (cf. Chap. 3). But they are also of independent geometric interest. In fact, the classical development of Euclidean geometry is built on the concept of straight lines (and therefore on the concept of projective classes). Theorem 2.6.1 implies that torsion-free connections are a straightforward globalisation of the usual derivative of vector fields in vector spaces. Proposition 2.6.3 shows that also from the viewpoint of geodesics it is sufficient to consider only torsion-free connections. Lemma 2.6.2. Let
difference
V
-
t
Proof. We have
is
V, t be a
tensor
connections
S(U, f V)
only
a
manifold
M. Then their
field.
to show that
function-linear. The
on
(U, V)
S(U, V)
,
=
VUV
-
7_ UV
is
non-trivial part of this assertion follows from
=
Vuy V)
=
(U
=
f VUV
0
f )V -
-7uy V)
-
+
f VUV
ft
uv
::--
-
(U
-
f )V
f S(U, V).
-
f tiuv
p. 159
[I
p.
1
129]
128
Analysis
2.
on
manifolds
Proposition 2.6.3. Let t be a connection. Then there exists torsion-free connection 17 which has the same geodesics as Proof
We define
.
VVW
-
VWV
17VW [V, W] is
=
-
t VW
1
Tor(V, W),
a
unique
where
Tor(V, W)
the torsion tensor of V. Since
Tor(W, W)
-
2
0 for all
W, both connections have the same geodesics. Further, V is torsion-free by construction. For uniqueness note that if we add any additional, non-vanishing, in the covariant entries skew symmetric tensor field S
T21 (M)
(E
for any
hand,
field S E
loose the property Tor 0. On the other in the covariant entries symmetric tensor there exists a vector v., with S(v.,, v,,) :h 0. This implies to V
we
=
non-vanishing,
T21 (M)
that the
geodesics with respect to the connections 17 and V have initial velocity v., do not coincide.
+ S which
I
However, there exists infinitely many torsion-free connections with the same pregeodesics. This means that each projective class contains infinitely many torsion-free connections. Lemma 2.6.3. Let V and 17 be torsion-free connections and 93 be the projective structure generated by V. Then 7 cz T if and only if there exists a one-form 0 such that t 0 (& id + id & 0 (or, in coordinates, V =
-
1'6'r
-
F1 be
2ja 0 C) ). (b
.
Proof. Let Zbac Pba,. 1-b`e. Then their pregeodesics can coincide o n ly d VbVc Za for all _Tdbe VbVcVa va vectors v'. This implies _Ta Vb v'V 11 be be =
-
=
-
for all vectors
symmetric Sbea C,d
in
which is
equivalent to 6e(d Zbae the lower indices b, c we get v
6e _Ta (d be)
:=
1 =
3
(d
_
cd
-
dc
6a_Tb d be
that Za
.
cd
For the
=
(n
a
6a c
-deb)
+
(6eZa d be
_
ja_Te be ) d
)
-
e
=
-
I
.
26a Od) (C converse
notice that same
2.6.3. Let
connection such that
VVV
=
VVV
+
O(V)V,
whence both
pregeodesics.
will be used in Sect. 3.2
(M, V)
be
a
manifold with M there is
connection. Let
be M "
for every pregeodesics with initial velocity v., (E itx with respect to V t coincide. Then V and t generate the same projective structure
such that the and
_
c
-
following corollary
Corollary
(6e_radb
b a a and b gives Sbbcad nZCd 'Ec'd + rdac jac Zd0b + 26a where It follows + 1)_ra Od (n + 1)'-IZbbd ed (C Od) ,
connections have the
The
0
0. Since Z is
be)
6a_re b cd ) +
Contractil-ig the indices Za
6a Zbee (d
ja Ze
( (je_ra b
-
if
x
(E
an
open set
it ,
C
2.6 Connections and projective structures
We
Proof.
the
use
in the proof of Lemma 2.6.3. It is Zd VbVeVa 0 for all vectors v E i1x be
notation
same
as
VbV,Vd sufficient to note that Z, be a (I
already implies
6(,dZbc)
-
129
6a Ze (d be)
0.
[p. I
Written down in reduces to it
natively,
0 coordinates, the geodesic equation a + Fbac b c system of second order differential equations on M. Alter-
a
be considered
can
tangent bundle TM. In is given by d
system of first order equations on the coordinates, this system of differential equations d a
_1Y
=
dt
The
corresponding
701(TM),
and
can
as a
vector field
be
Va
Va,
=
dt on
_Fa bcVb VC. ,
-
TM is called the
geodesic
spray F E
invariantly defined by r (V.)
TO '. (at)
where -y,.,,. is the unique maximal geodesic with 7,.,, (0) v,,. The following proposition justifies the definition.
=
x
and
(0)
Proposition 2.6.4. Let (M, V) be a manifold with connection and V be the geodesic spray. If A: (a, b) -- TM is an integral curve of V, then 7rTM o A is a geodesic. Conversely, for every geodesic -y there exists a unique integral curve A of r with ?rTM o A y. =
Proof
Let vx E TxM and -yv,, be
defines d
Av, (t)
=
translation at
a geodesic with (to) vx. This geodesic v. (t) in TM. Clearly, 7rTM o Avx -yvx and Tt V :(a) T0 11_ (-Yvx o -rt) (at), where Tt is the t dt
Avx (t) TtAvx (,9t)
a curve
t, Avx is
s
-4
Conversely, 7,\(t,,) with
t +
=
=
s.
-
A(-yv,,, oTt)(0) dt
curve
of _V.
let A be
(to)
I
Since
integral
an
=
=
an
integral
A(to). By
curve
is the
velocity
vector of -Yv.
of r and consider the
the construction
geodesic above, ,\(t(') is an
of V. Since its initial
point in TM is A(to), it must coinuniqueness part of the fundamental theorem for 1 ordinary differential equations (Theorem 2.4.1).
integral
curve
cide with
We have
A(to) by
that
the
manifold is
locally isomorphic to A' (considered as a It is not true, however, that a manifold with connection is isomorphic to A' with its affine structure. In fact, for a manifold with connection there do not generally exist charts which map geodesics into straight lines. For this to be the case a necessary seen
a
set with differentiable
structure).
condition would be that the connection is in the as
the canonical connection of A'. We will
identify
the
lines in A'
2.6.4).
now
same
projective class
show that
one can
still
geodesics which pass through a given point with all straight which, pass through a given intersection point (cf. Lemma
127
1]
p. 159
2.
130
As
a
Analysis
on
manifolds
consequence of
Proposition
2.6.4 and the fundamental theorem
for
ordinary differential equations, each x E M has a neighbourhood W of 0 in TxM and there is a J > 0 such that for all v c )/V the
M with (O) v is defined. By choosing W geodesic 7v: (-J, J) small enough we can normalise the interval [-J, J]. In fact, observe that ,,v (t) a , (at) for any a E K. Hence for every x E M the zero vector 0 E TxM has a neighbourhood U C TxM such that for all v E U the geodesics t 1-4 -yv(t) with initial velocity v is defined on the interval -
=
=
[-1, 1]. Definition 2.6.4. Let
exp:
jv
E
TM:
(M, V)
-y,(1)
be
a
manifold
definedf
is
--+
v 1-4
where
X
(V)
7rTM
=
is called the
,
M
exp(v)
exponential
It follows from the fundamental theorem for tions that the set of all
with
A(O)
=
v
v
TM,
E
is defined up to
with connection. The map
:=
map
expx(v)
:=
-yv(l),
of V.
ordinary differential equaintegral curve A of F
for which the
(including) parameter
value 1, is open.
Hence the domain of exp is open in TM. This also implies that for any x the intersection of this domain with T,,M is open in TxM. This set is also
all
a
star-shaped E
as a
consequence of the
equation ,,v(t)
a ,(at)
for
R.
Proposition 2.6.5. For each point x E M there exists a neighbourhood U of 0 E TM such that expx is a diffeomorphism from U onto a neighbourhood U of x E M.
Proof.
Let
V
E
TxM and
f)
d T expx (f))
dt
expx (tv)
)
dt
(tv)) It=0
E
ToTxM. Then
d
dt
It=0
-YtV (1)
)
d dt
It=0
-Y' M
)
V
lt=O
which
implies that T exp,, is an isomorphism. Now the assertion follows from the inverse function theorem.
Corollary
2.6.4.
There is
Exp: 1 is
a
diffeomorphism
--
a
M
onto its
neighbourhood x
M,
vy
F--->
U
of Ox
E TM
(y, exp(vy))
image
Proof. The corollary follows from the fact that TExp non-singular whenever T exp is non-singular.
is
such that
=
T7rTM T T exp I
2.6 Connections and
e,,J
be
a
V"E7,'1(va)2
sufficiently small
for any
r
> 0.
Hence
a
we can
is
a
diffeomor-
a
coordinate
These coordinates
coordinates and the
corresponding
chart is called
Lemma 2.6.4. Let
(M, 17)
(U, p)
be
be
of 0,, c
define
(expx-1 (y)) a.
a
M.
c
neighbourhood
:=
xa(y)
system by
lei,
Let
ei for each vector v,,
=
f3,(O.,)
Then
i
basis of T,,M and write v,,
v
131
structures
provides especially practical coordinates.
2.6.5
Proposition
projective
are
called normal
normal chart.
with connection, x E M, and Then W maps the x. lines through 0 E K'.
manifold
a
normal coordinate chart centered at
a
x onto the straight Furthermore, the Christoffel symbols with respect 0. satisfy _ra (b c) (X)
geodesics through
to the chart
(U,
=
Proof. The
first assertion follows
immediately from the
construction of
normal coordinates.
vb vc vanTo prove the second assertion we must only show that ra bc ishes for all vectors v E TxM. Let y be the geodesic through x with (O) v. Since its coordinate expression is a straight line the coordinate =
components satisfy a
rbac(,y(t)) b(t) c(t)
=
At t
In Euclidean space,
0 and therefore 0
=
b
a (t) +rba
C
=
0 this
a convex
equation reduces
to
U set is characterised
(t)
rba,(x)vbv
C
C
(t) 0.
by the requirement
that any two points x, y E U can be joined by a straight line which is contained in U. For a manifold with connection we call a set U convex if any two points x, y E U can be joined by a unique geodesic which is contained in U. We will now show that each point has a convex neigh-
bourhood. Theorem 2.6.2. For each
bourhoods U,, with
x
nn" , Un
=
E
M there is
a
sequence
of
convex
neigh-
JxJ.
Proof. Let (xl,...,Xn) be a normal coordinate system and (U,W) the corresponding chart. For y E Image(expx) n U we define the distance function
d(y)
:
=
[(Xa (y)
_
Let
B, (x)
=
ly
c
Image(expx) n U
:
d(y)
V(t)
(U(t), V(t)) =
0 which
=
be two
V (t) (U(t), V(t))
implies
sequently, parallel transport
is
an
i-->
- (t)
parallel vector fields along -Y.
that
(U, V)
isometry.
is
V (t)U, V(t)
independent
be
Then +
of t. Con-
2.7
Conversely, u, v,
vector fields
transported 0
parallel transport
U, V
a curve
with
(t) A V) ='('Nt) g) A V) + (t)g) (U, V). At t 0 this implies =
of connections
is
an
-y with
U(O)
V
=
(V
that
assume
be vectors. We choose
w
Examples
=
u,
(O) V(O)
=
=
isometry and let w and parallelly
v.
Then
we
V (t)u' V(t))+(U(t), V 0
=
(Vwg) (u, v)
135
obtain
(t)v
=
which proves the
assertion.
2.7.2 The
Let M be is g
Weyl
a
conformal
02j.
=
connection
real manifold and g be a metric on M. A second metric to g is there is a positive function Q: M ---> R+ \ f 0 1 with
A
conformal structure Q is an equivalence class of conformal chapter we will see that the Michelson Morley experdirectly leads to a conformal structure rather than a Lorentzian
metrics. In the next iment
metric.
Given This
a
conformal structure (t there is
generalises
a
class of
adapted
connections.
the Levi-Civita connection of the previous section.
Definition 2.7.3. A
triple (M, Q , V), where M is a n-dimensional manconformal structure on M, and V a torsion-free connection is called a Weyl structure if for every g C- (t there exists a one-form p such that Vg W 0 g. The connection V is called a Weyl connection
ifold,
C-'
a
,
=
In the w
E
following
TjO(M) (cf. the lemma
use
will
we
use
the exterior derivative dw of
Theorem
2.5.1). In Theorem of Poincar6 (Theorem 2.5.5).
2.7.2 below
we
a
p-form
will also
Readers who have omitted Sect. 2.5 can replace dWab by 2!a[aWb] Using thi equality the lemma of Poincar6 can be understood in our
special
case.
Lemma 2.7.2. Let
(M, Q , V)
AdW
is
Proof.
Let g E (t and
independent of
=
Q2g.
2S?dS2 0 g +
Vj Hence F
j
=
2
dW
=
-
-1 2
be
a
Weyl
structure. Then the
2-form
F
g E it.
d o
The 2-form F is called the
Then
Q2Vg
=
does not
length
we
(p
have +
2dlnf?)
depend
on
0
the choice of g
curvature of the
Weyl
structure. We
will motivate this term in Sect. 2.8.1 below. Theorem 2.7.2. Let x
E M.
structure
(i)
F
(M, (t, V)
be a manifold with Weyl structure and neighbourhood U such that for the induced Weyl (U, (t, V) the following statements are equivalent.
Then
=
0,
x
has
a
Aijalysis
2.
136
(ii)
c,.rL
There exists
masafolds
e
a
(t which has Levi-Civita connection V.
Proof. We first show that (ii) implies (i). Let g 0. Then we have Vg 02g such that Vj j Q3dQ (D g. Hence W QMS? and dW 2SMS? (& j
(t be any metric and (p (9 g
=
=
=
For
"(i)
=:
=
note that F
(R)"
=
ld o 2
0. Hence
=
(Theorem 2.5.2) implies x and of a function f : U ---> R -e-f df 0 g + e-f df (3 g 0.
lemma of Poincar6 hood U of
V(e-f g)
=
Corollary there exists
=
0.
an
application of the a neighbourW. Consequently,
the existence of
with
df
=
1
=
(M, E', V) be a Weyl parallel, non-vanishing n-form
2.7. 1. Let a
.
assume
y. Then there is
an
that
(up
to
(t such that V is the Levi- Civita connection
sign) unique metric g En) g and I S?(E,,. .
structure and
I
,
for
every
g-orthonormal basis f El,
.
.
..,
of En 1.
Proof. For any metric g (=- Q we define an n-form A as follows. We let f Ej,..., Enj be an orthonormal basis with dual basis fOl, O'l and 01 A denote g (Ei, Ej) E I 1, 11 by ej. Then A is defined by A A on. Since An(TM) is I-dimensional there is a unique g E Q such that -
=
-
p. For this metric and any vector
0
=
VV (y (El,
.
.
.
,
v we
-
-,
...
calculate
E,,)) n
=
(17vp) (E,,.
.
.,
En)
+
1: I_t(Ej,..., Ei-1, 17VEi, Ei+l,..., E,.)
n
=
1: I_t(Ej,..., Ei-1, g(17VEi,, Ej)EjEj, Ei+,,..., En) i=1 n
=
E g(VvEj, Ei)ei n
=
Y- -2 Ej (Vv (g(Ei, Ej))
-
(Vvg) (Ei, Ej))
i=1 n
=.=
-E 2 cj o(v)g(Ej, Ej)
n =:
-
2
P(V).
i=1
Definition 2.7.4. Let
timelike
(or spacelike)
(M, ( , V) curve.
be
a
V (t) (t) for
all t in the domain
Weyl structure and -y be a smooth affinely parameterised if
We call -y
I
of definition of -y.
(t)
137
2.8 Curvature
It is clear that timelike
only
if
they
pregeodesics
if and
affinely parameterised
are
geodesics.
are
(M, (t, V) be a Weyl structure and t -y be a smooth (t. and there exists a all Then all t 0 E g g( (t), (t)) :h for that such is affinely parameterised. 7(t(s)) reparameterisation %s) 7(t(s)) is affinely param-y(t) is affinely parameterised then s If t
Lemma 2.7.3. Let
F-->
with
curve
=
t-4
-4
eterised
and
if
only if
Proof. We denote -y (t (s)).
there exist a, b E R such that
- 4dt
by
(),
dot,
a
by
ds
a
=
at + b.
%s)
Let
prime,
Then
Xv (t) W' W) X'71Y(sm) 7
ly (s (M)
(0)
ds
g
(7Y(S) Is t
1(8)
) (g (dsTt )2 (dt_ds is
dt
ds
d
(s))
(s) Tt
(V '
dt
that
ds
( ' (S) Tt
2
ds
implies
d
and
s
ds
2Sg(
(s), ' (s))
dt 2
affinely parameterised
(s),
+Tt g if and
d's
ds
9('V (t) W,
dt2
dt
gMt)
only
(s))
if
W)
W)
I
immediately from the fundamental thealready affinely parameterised, the differ-
holds. The first assertion follows for
orem
ODES.
If t
-+
'y(t)
ential equation reduces to
is d2s ft_2
=
0 and the second assertion follows.
1 13
2.8 Curvature 179
In Sect. 2.6 of
we
have
seen
that the covariant derivative defines Given
parallel transport along -y(l) x, this parallel transport
-y(O)
curves.
=
=
R,y: TX'M
ly
=:
vx, in
TX'M'
small
defines
VX
-4
loop a
-y:
[0, 1]
i-->
notion
M with
map
P vX ly
spacetime and in Euclidean space we always general the vector P ly vx depends on the loop -y.
While in Minkowski P vx
-->
a
a
have The
p.
141]
138
Analysis
2.
manifolds
on
'Ya3, b
f (a, b)
V 2
4
la,b
f (a, 0) a,b
theorem shows that if
following
there exists
a
Theorem 2.8. 1. Let U, V,
w
E
(M, V)
TxM. Then there R: TM
with the
be
a
exists
a
TM
x
rectangular loops then
restricts to
one
well defined limit where -y
--->
jxj.
manifold with connection and well defined tensor field TM
-->
TM
(u, v, w)
t--4
R(u, v)w
x
x
E
M,
following property.
Let U C R 2 be
an open neighbourhood of (0, 0) and f : U -+ M be an v. For any a > 0, b > 0 2-surface such that f,,al u, f,,a2
immersed with
=
[0, a]
7a,,b:
x
[0, b]
[0, 2a
+
2b]
=
let N,b be the closed
C U
--+
f (U)
curve
C M
f (t, 0) f (a, t
t
f (2a
f (0,
-
a)
+ b
-
t, b)
2a + 2b
-
t)
fo r
0 < t
TM with IrTM 0 X(X) f (x) for all x G Z. We denote the space of vector fields along f by E' Q Any smooth vector field X on M induces a vector field k: x t--> X.,
Chap.
5
we
be described
--+
A vector field U
Xf (.,). f
via
on Z also induces a natural vector field. along f,,Ux. Important examples of this construction are given by fields along curves and by vector fields along canonical immersions
x
vector
-4
of submanifolds
(cf.
Sect.
Lemma 2.9. 1. Let
4.4).
Z
f:
M, U, V
-->
E
To'(Z),
and
X, Y
E
To' (f
f
Then V
UX
(U,60OXa
:=
Fbac(f*U)bXc),qa
+
is
a
well
defined
vector
field along f and satisfies the following properties. f
(i)
UX
V
is
function-linear
in U and R-linear in
f
(ii)
V U (WX)
=
f
(iii)
+ W
Vf UX for all functions W E C"O (Z);
f
V
Uf* V
Proof. We have
-
-
f* [U, V]
=
Tor (f* U,
f* V).
to show that the definition is invariant under coordinate
of M. Then f
Vf* U
V
transformations. Let
phism
dW (U) X
X;
0 be
we
a
diffeornorphism
of Z and
0 be
V)*U(x
V
'Y
+
O,3U-y(ajXaq0(0-1)5)00-1 (ayfba,3(0 -1)-Y)(0jO0UJ)XC 0-1
a
rb
0
C
((UOa Xa M
-->
M be
Of
UO*X
=
a
iC(OOfbUO)Xc) Oa) 00-1.
depend on the coordinates chosen diffeomeorphism. We obtain
( U,3 a
0
+
pdoa)
Fba (O0(0
0
e
0
0
for Z.
f Xd)
f)bUO) (ador
(U,3 p_adoa)
aa
+ ra b
Hence the formula does not
0:
diffeomor-
f-'r,
UX)
(,q
Let
a
obtain
fa
0
f xd
)aa
3feXd + (adoa)
0
f aoXd)
2.9 Variation of
b
rb,(ae (C -0 ) of 19Of 'U'O(Od0c) a
+
((adoa)
UO
0
Fba_('9e_0b)
+
f xd
e
0
f 19'8Xd
((,g-ad 0a)
+
f (adoc)
0
0
e
145
geodesics
)19.
f
f)aefeXd ) 0a
0
The GauBian basis vector field with
respect to the coordinates induced by 0 are given by 0.0a. Taking this into account we see that the Christoffel symbols transform as given in Corollary 2.6.1. This implies that our coordinate formula defines a well defined vector field along f. Equations I (i)-(iii) follow directly from our coordinate expression. If
f
is
lated
immersion then the covariant derivative
an
entirely
Lemma 2.9.2. Let
X, Y
E
along f
can
be calcu-
in M.
To' (f).
Let
f:
Z
0,
M be
--->
be vector
Tol (Z),
and
M which coincide with
f. U
immersion, U, V
an
fields
on
E
f
and X at all Points y X
f (x). Then
we
have V
UX
o
f
at all
E Z.
This lemma
justifies writing
this notation
extensively
instead of
Vf* UX
(VCj) of.
We will
use
in Sect. 4.4.
Proof of Lemma 2.9.2. Let x E Z. Since f is an immersion there exists n-dim(Z) of a neighbourhood of U of x, a neighbourhood V C K 0, and a local diffeomorphism F: U x V --4M with F(x, 0) f (x) J6r all x c U. X o F-'. We may extend U, X to U x V such that FU and X =
=
Then
we
obtain
('7 CJ .' )
-
,
o
F
=
-b
Ua,9a X
F
o
(1,b,&akc)
_
=OiF allia
a
(Xb F-1) o
o
i9iF alliajXbaa (F-1)i
Uja.Xb J Restricting the
for all
x
E
last
_
(_Vb
o
ac
o
-
F
(Tab, F) (F U)aXc (1-abc F)* (F U)aXc o
*
o
-
F) (F U)aXc.
expression
to U
(VCj9)
F(x, 0)
o
F
x
101 gives ==
(Vf UX)
x
I
U C Z.
Lemma 2.9.3. Let
the
F
o
'a
f:
Z
4
M, U, V
E
7-01 (Z),
and X
E
To' (f ).
equation
R(f*U, f*V)X holds
=Vf U Vf VX- Vf V Vf UX- Vf
[U, V]X
Then
146
2.
Analysis
The
Proof.
manifolds
on
equation follows directly from the definition of the Riemann
tensor.
I
Observe that
f
may not be
immersed submanifold. This is important for the
an
does not need to be
an
immersion and therefore
f (z) following
application. Definition 2.9. 1. A
geodesic
M, (s, t) geodesic.
M such that
f (s, t)
variation is
for
each
a
the
6, 6)
f
map
(a, b)
x
-4
f (s, t) is a We denote the velocity of the geodesics by ft : T(,,t) f (19t) and the deviation vector field along the geodesic f (s, -) by f, : T(,,t) f (a,) --+
E
s
curve
t
F-->
=:
=
.
Proposition
2.9. 1. Let
f : (s, t)
-4
f (s, t)
M be
a
geodesic
variation.
f
Then vector
ft satisfies the geodesic equation V field f, satisfies the equation
f v
f
at V 9t fs
+ R (f,
ft) ft
-
0 and the deviation
tft
f
(Vf at Tor) (ft f ,)
-
Tor (ft, V
at fs)
=
0.
Proof. The geodesic equation follows directly from the definition of geodesic variation. Observe that [f, ft] f. [,9s, at] [f. (0,), f. (at)] =
a
=
0 and that therefore =0
f
V
f
f
atV ath
=V
f
9, ft + V at [ft, f,] +
atV
=
R(ft, fs)ft+ f + Tor (ft, V
t7at (Tor(ft, f,))
0
Vf 0,Vf 9tft + (Vf ajor) (ft, f,)
at fs)
It is often sufficient to consider the infinitesimal variations. This
justifies
the
Definition 2.9.2. A Jacobi field is which
satisfies
the Jacobi
V V J + R(J,
)
analogue
of
geodesic
following definition. a
vector
J
along
Tor( ,
V J)
field
a
geodesic
-y
equation -
(V Tor)( , J)
-
=
0.
Proposition 2.9.2. Let -y: [a, b) ---> M be a geodesic. The Jacobi fields along -y span a 2n-dimensional linear space and any Jacobi field J along 7 is uniquely determined by J(a), V (a)J-
Proof. Without loss of generality we may consider a single chart which geodesic. The Jacobi equation reduces then to a system of
contains the
2.9 Variation of
n
second order differential
ordinary
equations,
or,
geodesics
equivalently,
147
to
a
sys-
first order differential equations. Hence the assertion follows from the fundamental theorem for ordinary differential equations
tem of 2n
(cf.
ordinary
2.4-1).
Theorem
1
Corollary 2.9. 1. Let -y: [a, b] --> M, t F-4 exp,, ((t a) u.,) be a geodesic. A vector field J along -y is a Jacobi field which vanishes at x -/(a) if and only if there is a vector vx E TM with J a) (ux + svx)).,9,. exp ((t -
=
=
It is clear
Proof.
that, given such a x. Proposition
vector vx, J is
-y which vanishes at
along
-y which vanish at
characterised
x
by their velocity vector
'7 ('a) exp ((t for all vx
TxM and TxM is
E
Definition 2.9.3.
geodesic
-y
joining
vanishes both at
Proposition there exists and
T,
2.9.2
points
and y and and y.
2.9.3.
an
expx:
x
Two
ux E
+
svx)). a,
=
vx
n-dimensional vector space.
an
x
along
the Jacobi fields
'7 (a)J.
a) (u.,
-
Jacobi field
a
implies that
n-dimensional vector space and are The assertion follows since
an
span
-
E
x, y
M
a non-zero
are
conjugate if there is a field J along -Y which
Jacobi
Two points x, y G M are conjugate if and only if TxM in the domain of exp such that exp(ux) y =
T,,.TxM
---
TyM fails
to have maximal rank.
M joining these Proof If x, y are conjugate, there is a geodesic -/: [0, 1] points and a Jacobi field J along -y which is non-zero but vanishes at x and y. Let ux be the uniquely determined vector which satisfies exp(tux) -y(t). By Corollary 2.9.1 there exists a vector vx E TxM \ 01 --
two
=
such that
J(t)
=
T(O,t) exp(t(ux
+
swx))(,9,).
The assumption
J(1)
0
=
that the linear map T(o,l) exp(t(ux + swx)): K2 --* TM does not have maximal rank which in turn implies that T". expx: T,,.TxM TyM does not have maximal rank.
implies
To prove the
by
the
converse
requirement
assertion,
that expx (ux)
0. The vector field
J(t)
expx (tux) is then
non-zero
a
=
=
we
just choose the
0 and
T.,,,,,
T(O,t) exp(t(ux
exp
+
((A
ds I ,=o
(ux
+
0
:
vectors ux, vx
svx))
swx))(i9,) along -y(t)
Jacobi field which vanishes at
x
and y. I
2.9.4. Let y: [a, bJ -- M be a geodesic without conjugate points. For every pair of vectors W-Y(a) E T-Y(a)M, 17v-y(b) IE T- (b)M there is a unique Jacobi field J along -y with J(a) ?-D-y(b) W-Y(a) and J(b)
Proposition
=
=
-
148
There
Proof. and
Analysis
2.
ii-y(b)
vectors
are
Ty(a)M with -y(t)
E
U-Y(a)
with
T-y(b)M
E
manifolds
on
-y(t)
exp.,(b)((b
=
-
=
expy(a) ((t
Oa-y(b))-
-
a)u_ (a))
Since -y does not
have conjugate points the linear maps
T(b-a),,,(, )
:
T(b-a)u,(,,.)T-y(a)M
---
T-y(b)M
expY(b):
T(b-a)i -,(b)T-y(b)M
-4
T-y(a)M
expY(a)
and
T(b-a)i,-,(,,) are
both
isomorphisms.
Hence there
vectors
are
and, -y(b)
V-y(a)
d
170-y(b)
T(b-a)u,(.)
exp_Y(a)
(b
W-y(a)
T(b-a)fi,(b)
eXPy(b)
(b
a)
-
ds ls=o
(U-y(a)
such that
SV-y(a)))
+
d
Let J,
Then
,
-
a)
ds I s=o
S' -y(b))
+
J2 be the Jacobi fields defined by J,
=
J2
=
JI vanishes
T(0,t) exp((t T(o, t) exp, ( (b -/(a)
at
-
-
a) (U-y(a)
+
8V7(a)))(a,9)7
t) Oi-y (b)
+
8 -y (b) ) ) (Os )
-
and has the value
has the value W-y(a) at -/(a) and vanishes at J, (t) + J2 (t) is well defined since 7rTm o J, (t)
'CV-y(b) at y(b) whereas J2 -y(b). Observe that the sum 7rTM 0 J2 (t) -y (t). By the
=
=
linearity of the a
(ft-y(b)
Jacobi equation the vector field J J, + J2 along -Y is also Jacobi field which has correct values at both 7(a) and -y(b). This proves
existence.
=
is clear since the space of solutions has dimension
Uniqueness
2n which is
just the dimension of the vector space
Jacobi fields
can
Proposition 2.9.5. field along t -4 -y(t)
=
Let =
E
x
M,
exp(tux)
Since
Let
f
is
(f,),,,=o
is
f: (s, t)
ft
=
f,
=
a
a
-4
expx (t (u +
T(,,t)f(,9t) T(,,t)f(a,)
geodesic
J(O)
=
=
TxM, and J be the Jacobi
=
0 and V
map in direction
expx(v[u])
Tu;,:
E
ux,vx
with
differential of the exponential
Proof.
I
also be used to calculate the differential of the exponenan n-dimensional "vertical" subspace of Tu.TM - !+ tvx) I t=0 E Tu. TM, where vx E Tx M. v[,,] dt (ux
tial map restricted to which consists of all
the
T-y(a)M (D T-y(b)M-
sv))
=
E
v[u)
J(O) is
==
v,
Then
given by
J(I).
M and
Tt(u+,,) expx((u + sv)[t(u+sv)])7 Tt(u+sv) expx((tv)[t(,,,+,,)I).
variation with
Jacobi vector field
f(s, 0)
along
=
x
Vs the vector field
-y which vanishes at 0. Rom
2.9 Variation of
f
[f,, ft] we
=
have
0
f,
we
obtain
V J
=
0 and
ft
Fas) Is=O To exp., ((u as
+
field J also satisfies
=V
=
geodesics
f
atf,
To exp. ((u
Tor(f,, ft). At t 0, sv) [o])), whence V 1 (0)
asft
=V +
+
v. The Jacobi To exp., ((v) [o]) sv) [o]) J(I) (,9,) T(0,1)f T(u) exp.,(vfu]). The ==
=
149
=
=
=
vector asser-
tion follows since initial value and derivative characterise Jacobi fields
uniquely.
I
Space and time from a global point of 3.
view
p. 137
The content
of this chapter is mainly physical. In Sect. 3. 1 we show experiment of Michelson and Morley indicates that spacetime admits a conformal structure. A conformal structure is not sufficient to describe spacetime adequately. In Sect. 3.2 we generalise the notion of inertial observers which leads to the existence of a projective structure. (One of Einstein's key observations was that this projective structure is closely linked to the phenomenon of gravity. This will be pursued in Chap. 5). In Sect. 3.3 we use our physical postulates in order to show that the conformal and the projective structures of spacetime form a Weyl structure. Here we closely follow (Ehlers, Pirani, and Schild 1972). The proofs in this section are technical and can be omitted without loss of continuity. In Sect. 2.8 we introduce a further physical postulate which restricts the Weyl structure to a Lorentzian manifold.
[I
that the
3.1
rays: the conformal structure
Light
In Sect. 1.4
field of
we
light
have
(cf.
cones
that spacetime is endowed with
seen
Postulate
1.4.1).
an
invariant
In
analogy to the discussion in light cones infinitesimally, i.e.,
the previous chapter we will define these in the tangent spaces rather than in spacetime itself. The discussion in
Chap.
I may
seem
to indicate that for each
x
E
M
the
tangential space TxM can be identified with (R', TI). Since for each non-degenerate bilinear form g., of signature (-, + +) there exist .
linear coordinates
(xo, x',
(i, j
11),
f 1,
E
.
.
.
,
Xn- 1) such that gx
may be
tempted
=
-
d
.
.
,
(x 0)2 + Jij dx'dxj
simply replace Minkowski spacetime (An, 71) by a general Lorentzian manifold (M, g) However, the Michelson-Morley experiment only determines the paths of light rays.1 In other words, from the Michelson-Morley experiment alone one can only infer the existence of a conformal structure (An', E,,)' where (t17 .
.
.
,
n
-
one
to
-
=
ff?277
:
0 E
COO(A
n ,
R+ \ 10 1) 1.
For the definition of space. This
wave
(In
length
to indicate that
Sect. 1. 4 we
we
used the affine structure
needed the Euclidean structure of
implicitly we used a Lorentzian metric rather than a conformal structure in order to interpret the Michelson-Morley experiment. However, the outcome of this experiment is a null-effect, i.e., , AZ ,z 0 which is independent of the Euclidean structure chosen. seems
M. Kriele: LNPm 59, pp. 151 - 169, 1999 © Springer-Verlag Berlin Heidelberg 1999
p.
1
156]
152
Space and
3.
time from
global point
a
of view
of A' to
single out a constant representative n E (1_1'77 This is not possible general manifolds.) Hence in a global setting the Michelson-Morley experiment leads to the following postulate. .
for
(Existence of a conformal structure). (Conformal) spacetime is a pair (M, Q ), where M is a n-dimensional manifo Id and (t a conformal structure of signature (-, + This
Postulate 3.1.1
We will call We will to
recover
to
it)
given by the paths of light
structure is
conformal
a
conformal structure of signature
rays.
(-,
+,
.
.
.
,
+)
Lorentzian.
that, given a Lorentzian conformal structure, it is possible light rays (cf. Postulate 3.1.2 below and the discussion leading see
-
(M, (t) be a manifold with Lorentzian conformal (or lightlike) hypersurface N is a hypersurface such (t) the induced metric on N is positive semi-definite
Definition 3.1.1. Let structure. A null
that
(for
any g E
but not positive Let
(M, ( )
be
definite. manifold with
a
a
Lorentzian conformal structure and
hypersurface. At each point x E N there exists a unique 1-dimensional subspace 1., C TM which is tangent to N and 0 for all v E lx, g E C For, if there where two such satisfies g(v, v) N C M be
null
a
=
vectors V1 i V2 E a
vector
E
w
impossible
TxN which
spanf vi V21 ,
Ux
E
TxN lx \ Ox I Vx I
of V and U in N
curves
Definition 3.1.2. Let
(M, (_")
structure and N C M
null
satisfies g( , geodesic
vector
null
not collinear then there would exist
g(w, w)
since g restricted to
vector fields with
integral
were
with
a
='O
are
the
be a conformal differential equation
a Proof.
First
respect
we
+
gad '9bgdc
null
geodesic
b c
C9dgbc 2
will show that the coordinate
representative
E
t But this is
and g c Q . Then -y
11 a.
expression
to coordinate transformations and with
g G
w
be a manifold with Lorentzian conformal hypersurface. A curve -/ in N whose velocity for every metric g E Q is called a conformal
Lemma 3.1.1. Let
satisfies
for all
< 0
positive semidefinite. If U, V are E lx \ jOx I for all x c N then the reparameterisations of each other. is
respect
is invariant with to
changes
C If V is the Levi-Civita connection of 9 then
7a
+
gad 19bgdc
'9dgbc 2
of the
we
have
3.1
whence
have
we
2f9ad
rays: the conformal structure
to show that for any other metric
only
Levi-Civita connection
e-
Light
t'
we
have
11
V
1Od(e 2f gbc)
Ob (e 2fgdc)
b c
2
e
2f
153
g E (E"with
But this follows from
==
149dgbc
gad abgdc
Y
2 0
+ 2abf
Let N be
dinates
a
null
a
ad
9
_
ad f gbcyb
hypersurface containing -y. We may chose for N such that 91 spans 1,, for each x.
c
coor-
(xl,...,x n-1 )
Since
the bilinear form gJspanJa2-.'a,,_11 is positive definite and 91TN does not have full rank, we have gii OVi E f 1, I I We may exn , =
-
.
.
.
-
tend the coordinate system to (xO.... I Xn- ') such that g (,9o, at N. In these coordinates we have (after normalising ) we
gad (abgdc
obtain
ordinates
we
tion follows from
proved
is
equivalently, adgll
or,
9igll
!gad (_adg1 1). 2
=
0, the lemma
9ad (_,g011) 11 (al)a
=
0 Vi G
f 1,
n
Since in
once we
a gad
have
-I
our co-
seen
-J1. This d
=
=
a, and that
equa-
I
-
0. Then \ fOJ be a vector with g(v, v) an (up reparameterisation) unique conformal null geodesic through x -y(O) with (O) v.
Corollary
3. 1. 1. Let
there exists -y
b c "adgbc) 2
have
91) =
v
E T,,M
=
to
=
=
Proof. By the fundamental
theorem for ODEs,
given
any function
A(t),
the differential equation
a has
a
+
gad '9bgdc
2
'9dgbc
b c
-
A a
=
0
unique solution for any v E TxM. It is easy to see that for any two A, the solution curves are identical up to a reparameterisa-
functions tion.
I
The
preceding corollary shows that at each point there is a unique con0 for all 9 E C geodesic in any direction Rv where g(v, v) This implies that there are exactly as many conformal null geodesics as there are light rays. In addition, it is easy to see that in the case of the Lorentzian conformal structure induced by Minkowski spacetime the light rays defined in Sect. 1.4 and the conformal null geodesics coincide. Hence we feel justified to link our infinitesimal Lorentzian conformal structure to light rays in spacetime by identifying them with conformal null geodesics. formal null
=
Postulate 3.1.2
formal
null
(Light rays).
geodesics of
The
its Lorentzian
light rays of spacetime conformal structure.
are
the
con-
154
3.
Space
and time from
a
global point
A Lorentzian conformal structure is all
causality. The following definition
is
of view
we
need in order to
straight
a
forward
investigate generalisation of
Definition 1.4.6. Definition 3.1.3. Let T be ture
(-, +,..., +),
a
Lorentzian conformal structure of signa-
, and A, U
g E
c
M.
A vector w is called spacelike, if g (w, w) > 0, timelike if (w, w) < g 0. The vector w is called 0, and lightlike (or null) if g(w, w) causal if it is timelike or lightlike. A vector field V is timelike (respectively, lightlike or null, causal, spacelike) if for each x E M the vector Vx is timelike (respectively,
(i)
=
lightlike
(ii) is
null, causal, spacelike). conformal structure global timelike vector field V. or
The Lorentzian a
(t is time orientable
Assume that (E is time orientable. A time orientation is lence class
if there equiva-
an
W if g(V, W, ) < 0 of timelike vector fields V where V at some point x E M. Let [V] be a time orientation of ( . A causal vector u is called future directed (respectively, past directed) if g(u, V) < 0 (respectively, -
g(u, V) > 0). (iii) A curve -y
is called spacelike (resp., timelike, lightlike, causal, directed, past directed) if all its velocity vectors are spacelike (resp., timelike, lightlike, causal, future directed, past directed). A timelike curve is often called a world line when one whishes to emphasise that it can represent the history of a (small) material object. (iv) The chronological future of a set A relative to U is
future
I+ (A, U)
x
The causal future
J+ (A, U)
There
are
Ix
=
M
cz
of
a
E M
I
a
future directed,
timelike
curve
-y C U
from A
causal
curve
-y C U
from A
to
xJ
to
xJ
set A relative to U is
3
a
future directed,
analogous definitions for
the
chronological past
l-
(A, U)
and the causal past J- (A, U) of A relative to U. If U M we omit the term "relative to M" and write T+(A), etc. If A fxJ is a =
==
single point
we
Definition 3.1.3 is
independent
Lemma 3.1.2. Let
I+ (x, U) is open.
write I+ (x,
x
E
U)
etc.
of the chosen
M and U be
an
representative
open
g E t
neighbourhood of x. Then
3.1
Let y E I+ (x,
Proof. a
(xo,
.
pact neighbourhood of line f from
straight
such that any
> 0
an a
supf0'(t)'Nt)) Hence these lines
.
G
VI
V n -/. Then the (coordinate) of V there exists
E
z
By compactness
straight line '
f'(t)
:
.
y, and let
to y is timelike.
z
from
z
with
Z(f, f') < a satisfies : f(t) G VI < 0.
lsupfg( (t)'i(t)) 2
M from
metric g e
to
z
not obvious
are
a
I+ (J+ (A, U), U)
c
A and
E
y
--
x.
E a
timelike
[0, 1]
--
a
z
curve
M of 4
from y to x which is timelike near x. ,' and let U be a timelike vector field along -y which
causal
(piecewise)
a
J+ (1+ (A U), U)
=
M from y to z and The concatenation - :
[0, 1/2]
M:
of U.
-
I+ (A, U) and J+ (1+ (A, U), U) C I+ (A, U). Let x E I+ (J+ (A, U), U). Then there exist
M,
However,
later, J+ (x, U) is closed if U is chosen small enough.
see
curve
(1). There is an 6 > 0 such that g( (t), (t)) < -,E for R+ be a smooth function which satisfies (I c, 1). Let 0: [0, 11 all t C- [0, 1 for 0 and (t) > 0 6]. For any (small enough) 0(1) 0(0) V The curves t s > 0 let f (s, t) f (s, t) all connect expy (t) (s 0 (t) (t)). We denote derivatives with respect to t by -1(t). y with x and f (0, t) satisfies all t E
U(1)
=
-4
-
=
=
-
--+
=
=
a
dot and with respect to
(g(f (s, t), f (s, t)))'
=
=
s
by
a
prime. We calculate
Vfas(g(f (s, t), f (s, t))) 2g(Vf if, f) 2 g(V, 2g(Vf f', f) 2g(V (OV), =
=
(g(j(s, t), j(s, t)))'(O,t)
for all t
c] and the F-] for s > 0 small enough. Since we have g( (t), (t)) < -E < 0 for all t E (I -,E, 1] the curves also satisfy 6 / 2 < 0 for sufficiently small I s 1. This proves that g (j (s, t), f (s, t)) < we have obtained a timelike curve t f (so, t) from y to x. This curve may have a kink at the parameter value 1/2 where the original curve -Y Hence
we
curves
t
have
-4f (s, t)
are
timelike
on
[0,
1
< 0
E
[0,
1
-
-
-
1-4
passes
through
z.
Using
difficult to smooth out t this
curve
-4f (so, t)
near
t
a =
coordinate chart it is not
1/2
while preserving that
is timelike.
The inclusion
gously.
Lemma 2.1.7 and
J+(I+(A,U),U)
C
1+(A,U)
can
be shown analo-
I
156
Space and
3.
time from
Lemma. 3.1.3. 1+ (A) Lemma
Proof. By
I+(A)
is open, there is
1+(J+(A)) Let
an
I+(A). J+(A), -y
bourhood of in
endpoint
I+(A)
y
be
a
1-(x)
J+ (A)
C I+
U.CAI+(x)
=
is clear. Let E
of view
is open. The inclusion
int(J+(A)). Since int(J+(A)) int(J+(A)). Hence x E I+(y) C c-
x
n
(A).
=
E
x
int (J+ (A)),
=
3.1.2,
int(J+(A))
C
global point
a
causal
curve
from A to x, and.U be
a
neigh-
Since any small enough deformation of -y has future we can deform -y thereby obtaining a timelike curve from
x.
U,
A to u.
I
Definition 3.1.4. Let
M, and
U be
C:(U)
an
fy
=
E
open
(M, Q )
be
a
Lorentzian
neighbourhood of x.
M:
3
a
conformal structure,
x
E
We call
future directed conformal null geodesic -/ C U
from
x
to
yj
the
integrated future light cone of x relative to U. There are analogous definitions for the integrated past light cone and the integrated light cone. If U M, we omit the term "relative to M and write Cx+, Cx-, and Q, "
=
(M, Q )
3. 1. 1. Let
Proposition
mal structure. Then each
x
(i) C (U) \ jxj sn-2
manifold with Lorentzian conforan open neighbourhood U diffeo-
is
a
smooth
hypersurface which
is
diffeomorphic
to
R,
x
(ii) C: (U) n I+ (x, U) i) CX+ (U) C I+ (X, U)
=
-'2
a
has
to Rn such that
morphic
(P.
be
E M
0,
=
J+ (X, U).
P. 158
3.1.1 will be
Proposition
a
corollary
to Lemma 3.1.4 below which is
result from Lorentzian geometry. Choose a representative g E V_' and denote the associated Levi-Civita connection by V. The exponential map a
allows x
us
to
E M with
space at
identify
the causal structure of
the causal structure of
an
open,
a
convex
convex
neighbourhood
set of the
of
tangent
x.
Lemma 3.1.4. Let
(M, (t)
be
Manifold with Lorentzian conformal strucneighbourhood of x E M with respect to
ture, g (=- ( , and U be a convex the Levi-Civita connection of g.
(i)
y E
where
I+ (x, U) v
a
(ii) J+ (X, U) The
(respectively J+ (x, U)) if and only if y timelike (resp., causal) vector. I+ (X, U),
=
expx (v)
future pointing =
complete proof of Proposition 3.1.1 requires the material from page immediately after the proof of Proposition 2.6.4 up to the end of Sect.
129 2.6.
3.1
g(u, u)
decomposed
vOu +
and
vo
scalar
as v
vi- here vi- E
T,,M. For
on
h(v, w)
=
>
E
every
u'
v
g(v, w) 0 let
TM
E
+
B,(O.,)
be
be
is
fv
=
01
=
Euclidean
a
E
"
M
obviously a neighbourhood of 0., in T.,M. similar argument as in the proof of Theorem 2.6.2 shows that U, expo.. (B, (0,,)) is a convex neighbourhood of x for small enough E.
h(v, v)
Ej.
U, is a diffeomorphism assertion (ii)
round
=
follows from Lemma 3.1.4 and the fact that every causal vector W E B,(Ox)q is either timelike or null. Assertion (iii) is a trivial consequence I of Lemma 3.1.4 (ii).
Proof of Lemma 3.1-4. (i): We prove the statement for 1+ (x, U) (the proof for J+ (x, U) is analogous). The exponential map is a diffeomorphism of a neighbourhood 0 of Ox E TxM onto U. For any geodesic -y we obtain V (g( , ) 2g(V , ) 0 whence the velocity vectors of =
=
their causal class. It follows that exp, maps geodesics timelike vectors into T+(x,U). We have to show that for each point
do not
y E 1+ (x,
U)
the vector
The double
cone
into 3 connected vectors
the U
change
3V E
Ox
=
v :
E T.,M is
g(v, v)
necessarily timelike. 01 divides 0 \ Ox
=
components: the future and past full
(Ox,+, Ox',-)
diffeomorphism
:
(exp,,) (y) fv E 1
with
and the set of expx: z
=
0
--+
U
exp., (v)
spacelike
we
see
vectors
cones
of timelike
(C-.',').
that the set
Applying
Cx(U)
divides U into the sets
CxO,
jz
E
CxO,
-T+ (x, U) there is a timelike curve co's, respectively. For every y which connects 1--4 t x and y. From g( (a), (a)) < 0 U, -y -y: [a, b] we know that -y must initially enter Cx,+. If the assertion does not hold X
then y E Cx+ (U) U Cx,'. Since 1+ (x, U) is open we can assume without loss of generality that y E Cx,'. Hence -y must intersect Cx at some point
7(to)
where -y leaves
Cxo,+.
Since
(to)
is timelike and future directed it is
and points into C',+. But this is to the construction of the point -y(to).
transverse to
For
(ii)
Cx
at
-y(to)
a
contradiction
it is sufficient to note that the set of causal vectors in
is the closure of the set of timelike vectors in
TxM.
TxM I
158
p. 3.56 p.
1
159]
Space and
3.
Since
a
relative one
time from
a
global point
Lorentzian conformal structure allows to
lengths (at
given point),
a
point. We just loose
So far
we
only
absolute calibration.
an
have considered
structure further
we
angles and spatial geometry at
measure
it is sufficient for
3.2 Inertial observers: the
our
of view
projective
light propagation.
structure In order to
specialise
must take into account other fundamental prop-
erties of nature. In
Chap. I inertial observers have played an important they are not subject to any physical forces one can physically implement inertial observers (or particles) by freely falling observers (or particles). We will use freely falling or inertial observers as the other input into our theory besides the Lorentzian conformal structure induced by light rays. The following postulate reflects that the movement of an inertial observer depends on his/her initial velocity and initial position. This is
theoretical r6le. Since
the main content of Galilei's law of inertia.
(Existence
Postulate 3.2.1
of inertial
observers). Through
any
for any timelike direction Ru there exists (up to point (M, g) and extension) exactly one inertial observer -Y: R -- M parameterisation x
and
E
which passes
through
Postulate 3.2.1
with
x
singles out
a
11
velocity
Ru.
collection of paths in spacetime. In Minkows-
ki space, inertial observers move along straight, timelike lines. This is again a global characterisation which we need to overcome by formuit
lating
i.e. in the
infinitesimally,
tangent bundle rather than in
space-
time itself. An infinitesimal
description of (unparameterised) inertial observers in Minkowski space is that their spatial acceleration vanishes. This property can be generalised as follows.
(Law
Postulate 3.2.2
(UxWx)
chart
of
coordinate system
we
inertia).
dt2 The chart maps W,,: Ux
x.
x
E
M there exists
a
x
d 27 a
at
For each
such that with respect to the corresponding have for all inertial observers passing through x
centered at
-->
R'
11 a. depend smoothly
on
the parameter
X.
its very formulation Postulate 3.2.2 is independent of the chocharts, we need to express it in an arbitrary coordinate system. In
While sen
by
Sect. 2.6
we
dependent ever,
have
seen
that the derivative of vector fields is coordinate
and therefore not well defined in
we can use
Theorem 2.6.1 and
a
Corollary
general
manifold. How-
2.6.1 in order to define
a
3.2 Inertial observers: the
projective
connection with respect to which Postulate 3.2.2
manifestly
coordinate free
can
159
structure
be formulated in
a
manner.
Let (U, ( ) be any chart and x E U. We define the Christoffel Fb', with respect to this chart at x by
symbols
192Xh
ap a
Fbc where
(x
x')
are
19xhq; bCq;; C'
the coordinates with respect to the chart
(Ux, ( x)
provided by Postulate 3.2.2. We will now prove that this construction gives a well defined connection 17. Let (1 ,O) be a second chart with x E
1.
Then
definitions
have
we
the
give
Pba,
same
a-: '
We need to show that these two
-
5X IT a93 a.-F
connection.
Indeed,
we
calculate
a2Xh
9. a a
Fb
C
axh q.,t-b,% c aj a a, k
axh 9.: l
a
TX-1 5
&7& axh qj b Cqj a
lq': k
2X h
9; k
9, l 0, m
5Xh
9., b
a., l 9._ m
a, _k
X,' 5y -57X-b k
which is
exactly
Thus
have
can
we
Given
a
q.;
,
9, ,- b
c
_5X__1,qj ca; b
^k
rim,
,
the transformation formula
provided by Corollary
2.6. 1.
as
follows. There is
a
connection V such that inertial
pregeodesics.
are our
3.2.2 holds. In
I(Vxi Ox)JxEM
of
compatible fact, Corollary
observers, 1A, Mj.,EM
is not the P,
charts such that the formula in Postulate 2.6.3
implies
that
exactly
those collections
which induce torsion-free connections that have the
as
V
are
also
possible choices.
This
implies
the
same
following
Corollary 3.2.1. Postulates 3.2.1, 3.2.2 determine a,projective structure 113 such that each particle is a pregeodesics with respect to any 17 E 93. Weyl characterised the connection as a field which forces a particle to be transported parallelly with itself in space and time. Thus we have arrived at a geometrical explanation for the law of inertia postulated by Galilei. 3
In order to understand the about
projective
PI
collection of inertial
only collection
pregeodesics corollary.
a,7 m -
X, k
92,;J
well defined torsion-free connection r and Postulate 3.2.2
be restated
observers
iqj a +
Yxh
+
a2; l
-5X---1 O' ca;-b
a2Xh
qj a
-57X_1 5_X _O b
axh +
corollary below
structures.
we
need to know
a
little bit
more
[I
1- 5,811
p.-127 129 P.
1
161]
160
Sp- ce ;---und
3.
3.3
O'nie firoin
a
global point
Compatibility: Weyl
We have obtained
of vif,- w
structure
Lorentzian conformal and
a projective structure but geometrical structures is still unspecified. We will now introduce a further postulate which links observers and light rays and therefore these two structures. It is an experimental fact that one can chase light rays with material observers arbitrarily closely, provided one uses enough energy. The following is a formalisation of this
the
relationship
a
of these two
idea. Postulate 3.3.1 Each
x
(Compatibility
M has
E
neighbourhood
a
with the causal U such that
for
structure). \ JxJ
all y E U
we
have y
=
As
-y(t) for
a
rays
an
inertial observer -y
through
--* y G
x
1+ (x, U) U I- (x, U).
first consequence of this compatibility axiom we can determine light connection instead of the Lorentzian conformal structure.
using the
Lemma 3.3.1.
geodesics
which
Let
The are
conformal null geodesics coincide with those lightlike.
pre-
somewhere
E M and let U be the intersection of the neighbourhoods of provided by Proposition 3.1.1 and Postulate 3.3.1. Let p be a conformal null geodesic from x to some fixed point y E U. Proposition 3.1.1 implies that y lies in the boundary of 1+(x,U). Hence there is a sequence of points yi E 1+ (x, U) which converges to y. For each i let 7i: [0, 1] F--4 U be the pregeodesic which corresponds to the inertial
Proof
x
which
x
are
observer which
moves
from
x
to yi
(cf.
Postulate
3.3.1).
Let
v
\ fOJ
be
accumulation point of the (bounded) sequence i(O) E T,,M and let v. By the continuous dependence of -y be the pregeodesic with (O) solutions of differential equations on initial conditions and parameters an
(cf. Theorem 2.4.1) there are for each point -y(s) (s E [0, 1]) and each neighbourhood V of -y(s) infinitely many pregeodesics -yi which intersect V and whose velocity vectors at s converge to :y(s). This implies that -y is causal and that -y C I+ (x,U). Since yi y and T! is compact the pregeodesic -y reaches y. The inclusion J+(I+(.T,U),U) T+(x,U) and y c -y n C: (U) imply -y c 1+ (x, U) \ 1+ (x, U) Cx+ (U). Since Cx+ is a null hypersurface and -y is lightlike (causal but not timelike) -Y must be a conformal null geodesic. That -y coincides with M follows now from the uniqueness of conformal null geodesics. For the converse we simply need to note that both pregeodesics which have an initial lightlike velocity vector and conformal null geodesics are uniquely determined by initial point x and initial velocity direction -*
=
=
R (O).
I
In the rest of this section existence of
a
natural
we
Weyl
will show that
structure.
our
postulates imply
the
3.3
Compatibility: Weyl
161
structure
a Lorentzian conformal structure and 93 be projective structure such that the Postulates 3.1.1, 3.2.1, 3.2.2, and 3.3.1 are satisfied. Then there exists a unique V E 93 such that for all I-P o (9 g. g E Q-" there is a one-form o with Vg
Theorem 3.3.1. Let (t be a
=
The
into several lemmas.
proof of this theorem will be split
Lemma 3.3.2. Let g be a Lorentzian metric and ' abc ': (ab,) be a 0 for all null vectors v. totally symmetric tensor such that zA (v, v, v) 7--
=
Then there exists
one-form V such that A(abc)
a
79(agbe)
':--
-
Proof. sym(d 0 g) clearly satisfies the condition of the lemma. We will verify that this is the only possible choice. 1. Then -1 and e E t--L with g(e, e) Let t be a vector with g(t, t)
now
=
=
t
e are
, Ab (t we
null vectors and from
e, t
e)
e, t
=
3tatb e'
A(abc) (tatbtc
+ 3te bec
e
aeb e')
=
0
obtain 0
' A(abc) (tatbtc
0
' A(abc) (3 tatb ec
3taebec), b + ea C).
(3.3.1)
+
e
(3.3.2)
e
Setting
(3,A(abc) tatb + 2A(t, t7 t)tdgdc)
19C
Equation (3.3.2) A (e, e,
e)
is
=
equivalent
-
3 GA (abc) tatb
Analogously, Equation (3.3.1) A
Finally,
(abc)
to
ea,btc
=
t9(e)
=
equivalent
g(e, e)?9(e)
Ve E
3
t-L.
to
1
A(abc) tatbtc 3
79(t)
=
9(ab?9c)
a
ebtc.
implies 1
1 =
=
_I
the definition of 79
A(abc) tatbec
is
c)
_
3
'0 (e)
=
3
9(t) 019(6)
=
9(abl9c) tatbec.
and
IA(abc) tatbtc
=
g(t, t)79(t).
By the polarisation identity for symmetric 3-tensors, we know that A(ab,) 1 coincides with 9(a0c) on a basis and our claim is proved.
4
The
proof of
Theorem 3.3.1 is rather technical. It
loss of continuity.
can
be omitted without
164
Proof.
3.
Space
Let
f Eo,
and time from
.
.
.
E,,- 1 1 be
,
be the dual basis. We
can
global point
a
an
orthonormal basis and
write L
La bc &
=:
La for all indices a, b, c. We consider .b COS (0) EA + sin (0) EB , where A -7 + Eo
2
L (N,
N)
=
=
-1(1 2
cos(20)),
-
L (Eo,
Eo)
c
(9
lightlike
(9
f wo,
Ea,
vectors
n-
1
.
where La bc
given by
B and 0 E
N
=
In order to as a
-1 2
Fourier
sin(20),
calculate
we
COS2 (0) L (EA EA)
+
Wb
[0, 21r]. c(N)N, we expand L(N, N) -1 + cos(20)), sin 0 COS 0 2 (1
exploit the condition L(N, N) polynomial in 0. Using COS2 0 and sin 0
of view
,
+ 2
cos(O)L(Eo, EA)
+ 2
sin(O) cos(O)L(EA, EB)
+
sin2(0) L (EB EB) I
sin(O)L(Eo, EB)
+ 2
I =
L(Eo, Eo)
+ 2
(L(EA, EA)
2L(Eo, EA) cos(O)
+
I(L(EA, EA)
+
-
2 =
for
some
c(O) (Eo
function
order 2 which
+
right
> 3.
cos(O)EA
we can
c
i3ABEA
+ 2
+
write
L(EA, EB) sin(20)
sin(O)EB)
side
c(O)
+
Fourier
polynomial of polynomial of order < 1 would be a Fourier polynomial of aAB + )3AB COS 0 + 'YAB sin 0. The a
a
Fourier
=
given by
1
aABEO
+
must be
implies right hand
hand side is then
2L(Eo, EB) sin(O)
The left hand side is
c(O).
that
Hence
L(EB, EB))
L(EB, EB)) cos(20)
since otherwise the
order
+
+
+
1^YABEB + (OABEO + aABEA)
COS
0
2
('YAB EO
+ 01AB EB) sin 0 +
cos(20) (2IOABEA (17ABEA 113ABEB) sin(20). 7ABEB
2
+
+
2
2
A comparison of coefficients gives
L(Eo, Eo)
+
I(L(EA, EA) + L(EB, EB))
2
aABEO
+
-
3ABEA
-7ABEB,
+
2
EA)
=
)3AB E0
+ CVAB EA ,
2L(E0, EB)
=
'YABEO
+
L(EB, EB)
=
L(EA, EB)
=
2L (E0,
L(EA, EA)
-
2
CVABEB
,
3ABEA -YABEB7
I'YABEA + 2I 3ABEB-
2
(3.3.4)
(3.3.5) (3.3.6)
(3.3.7) (3.3.8)
3.3
We obtain
Compatibility: Weyl
165
structure
linear system of equations for the components L' (b < c) bc c (g Wb (9 0 Ea. From Equation 3.3.5 we obtain immediately
a
-
of L
L'(bc)
=
LB OA =Of6rBVfO,Aj,
LOOA
A
L OA
OA B,
2
=
(3.3.9)
-aAB.
2
Further, neither OAB nor aAB can depend on B since the left hand sides Equations (3.3.9) are independent of EB. Equation 3.3.6 implies in addition that LB OB - aAB. Hence aAB cannot depend on A either. We will therefore write Ao := aAB and AA 13AB. Equation 3.3.8 implies LoAB 0 for A 4 B and LC for 0 pairwise different A, B, C. We AB of
=
=
=
also obtain
I
A
LAB
=
B
'YAB and
LAB
'AA.
=
These two equations are be used to eliminate
consistent if 'YAB AB. Equation 3.3.7 L (EB I EB) in Equation 3.3.4 resulting in
only
=
L(Eo, Eo)
+
L(EA, EA)
CeABEO
=
IOABEA + 1^YABEB + 1OABEA 2 2
2
2
implies LOOO
This equation
+
can
7ABEB
==
AOEO
+
AAEA.
Ao which is independent of A. Hence B yo := -LoAA, We also have Loo B A A -L AA for B =h A and L A If +LA set L all we coefficients AA. /.t 00 AA 00 are determined. It is now straightforward to check that La ja A ) + (b (be) /_,a gbc. Conversely, this tensor indeed has all the properties listed in the LoAA
LoBB for A =h B
=
+
LAA
=
set
we can
=
=
,
-
c
lemma.
Corollary sor
for all
L(., .): T,,M x .,M --4T.,M be a symmetric L(N, N) 11 N for all null vectors N and g(L(v, v), v)
3.3.1. Let
such that vectors
v.
Then
for
each g E Q
there exists
a
1-form
ten=
0
A such that
I
L(v, w) where
Proof.
/V
is
=
2
defined by A(v)
This follows
condition
(/\(v)w
+
/\(w)v)
=
=
g(v, w)1V
g(A0, v) for
immediately
g(L(v, v)v)
-
all
v
E
Vv, w
E
TM,
T ,M.
from Lemma 3.3.6 and the additional
0 for all vectors
v.
We are now ready to prove the main result of this section. Proof of Theorem 3.3.1. We will first show that there exists a unique V E T such that for each representative g E (t there is a one-form W
with
(VVW)
a =
VbabWa
+
gad(' (abgdc 2
+
acgbd
-
lydgbc)
166
Space and
3.
time from
a
global point
I
+
Wd9bc
2
-
of view
gd(00c) vbwc.
(3.3.10)
Recall the formula for A the
assumptions
La(bc)
ga(bk)
::::
(n + 1) Ab we
provided by Lemma 3.3-5. Since L'bc satisfies Corollary 3.3.1 there exists a one-form A such that 0 implies 0 /Nagbc. The property Laba ga'La(bc)
of
=
-1
Ab
-
(n
2
1) Ab
-
whence A
,
0 and therefore
=
0. Hence
have I -
1
-
abc
2
(19b9ac
19c9ba
+
Aa
aagbc)
-
bc
I 2
An
application
such that 0 From
=
-
gd(b(Pc) +
of Lemma 2.6.3
implies that there Equation (3.3.10).
0. This proves
Equation (3.3.10)
Vagbc
Odgbc
aagbc
a
unique V
G
q3
obtain
we
d c
d
1 abgdc
-
is
ga(bOc)
_
racgdb
I
19agbc
Pagcb
-
2
19agbc
2
+
+
19c9ab
19b9ac -
-
ac9ab)
ab9ac 2
Ocgab + gc(a(Pb)
2
(Pb9ac +
gb(a(Pc)
Wagbc-
Hence
(M, (t, V)
is
Weyl
a
structure and the theorem is
1
proved.
3.4 Reduction to the Lorentzian structure P. 161
[I
P.
1
169] There
are
experimental
facts which indicate that
ture has features which have
therefore necessary to
seems
no
specify
the
a
general Weyl
struc-
our
actual universe. It
geometrical
structure of space-
counterpart in
time further.
It is
plausible
of
2.7.4)
an
to
identify
The freedom t
server. zero on
an
affine parameterisation (cf. Definition a standard clock carried by the ob-
inertial observer with --*
at + b
corresponds
to the freedom to choose the
the time axis and to choose the unit in which time is measured.
An atomic clock
roughly
be found in textbooks
on
works at follows
(detail
of this mechanism
Quantum mechanics). Each
atom has
a
can
charac-
teristic minimal energy E which it can absorb. A very short while after the absorption of such a package of energy the atom will emit a photon
whose
by
E
can
frequency
=
v
with respect to the rest frame of the atom is given frequency is characteristic for each sort of atom it
hv. Since this
be used to build
a
clock.
3.4 Reduction to the Lorentzian structure
In Sect. 1.4.3
we
have
167
how to calculate E from the world lines
seen
of the atom and the
photon in the context of special relativity. Unfortunately, we cannot simply apply this calculation here since the number E did depend on the Minkowski metric n and not merely on its conformal class. But it is very suggestive to identify this atomic clock with the standard clock t given by the affine parameter of the world line of the atom.
We will now see that this identification gives rise to a global effect which has not been observed. Let x, y E M, y E 1+ (x) be two events and consider two atoms of the same kind which are moving from x to y
along different paths
(cf. Fig. 3.4.1). Y
'Y1
(al)
'Y2
-yi:
We will
[0) 011
assume
M in spacetime Mi 'Y2: [07 Ce2l they move initially along the same --+
that
^/2 (Ce2)
'Y1
Fig.
3.4-1. The world lines from
atoms which X
path
in
there is assume
'Y1
(0)
^f2 (0)
aA
x
initially and finally
to y of two at rest with
to each other
spacetime and that their clocks such that
1 (t) that just before reaching an e
> 0
For these observers
10
respect
are
we
V":YA (t) 9 ( A (t)
y
are initially calibrated at x, i.e., 2 (t) for all t E [0, 6). We will also they are again moving side by side.
obtain
7
A (t))
aA -,
9(
=
A (t)) A (t))
A
(t)9) (:YA (t)
A (t))
i
-dt
;-
JO
9( aA
+2
fo
g
A (t)
(V' /A (t) 9(
A (t)) A (t)
7
A (t))
A (t)) A (t))
aA
( A (t)) dt. 0
7
dt
168
Space- and
3.
Since the In
time from
frequencies
(g( , (a,),
were
, (a,)))
-
-
global point
a
initially equal
of
view
obtain
we
In
(9( 2 (a2)
in
(g ( l (al), 1 (a,)))
+In
,
2 (a2)))
(g( 2(0), 2(0)))
-
In
-In
(g ffi (0), 1 (0)))
(9( 2(Ce2)i 2(Ce2ffl
Let
Q be any 2-surface which is bounded by the curves -yl, plication of the theorem of Stokes (Theorem 2.5.5)5 gives
(g ( , (a 1),
In
, (a 1)))
In
(9 ( 2 (a2),
-y2. An ap-
2 (a2)))
0"
f ( i (t)) J dW w
==
=:
J
Hence the parameterisation of both the
curves
of both atoms
_
f012
W(
2 (t)) dt
0
F
-2
C Q
tification
dt
Q
and therefore
-
by
our
iden-
different at y, even though they were the same at x. As a consequence, the frequency of an atom clock would depend of the history of the atoms constituting it. This does not
-
frequencies
to be the
seem
Moreover, the spectrum of far away stars is aphistory of the atoms which constitute these 0. conclude that dW case.
parently independent stars.' Hence
we
Postulate 3.4.1 F
are
of the
=:
(No
second clock
effect).
The
length
curvature
1
=
-
of spacetime vanishes identically. 2 dW
Notice that the tation than the
justification of justification of
Postulate 3.4.1
requires
more
interpre-
other axioms. One may argue that it is the weakest link in the chain of arguments which leadsto general our
relativity. Corollary 3.4. 1. Assume that Postulates 3.1.1, 3.2.1, 3.2.2, 3.4.1 hold. Then spacetime is a Lorentzian manifold (M, g). Proof. This follows immediately We have
3.3. 1, and
from Theorem 2.7.2.
arrived at a geometrical structure which gives the framedescription of space and time. The arguments which lead to Corollary 3.4.1 may seem so compelling that the reader could ask herself or himself why we started with Newton's theory of spacetime work for
The
a
gist of our argument
is that the difference of the
frequencies at -Y1 (al) paths -yi, 'Y2 unless W can be chosen to vanish should be plausible even without appealing to the theorem of Stokes. It is needed for a strict proof though. This argument does not depend on the identification of atomic clocks with
'Y2(Ce2) 6
now
is
non-zero.
That this is the
case
the affine parameter of their world lines.
for suitable
=
3.4 Reduction to the Lorentzian structure
169
instead of
motivating our postulates directly. In fact, from a purely conceptional point of view it is, advantageous to analyze the measurement of space and time relations and to use this analysis in order to arrive at a Lorentz structure. This program has been carried out by Ehlers, Pirani, and Schild (1972) who also arrive at a Weyl structure and reduce it to
a
Lorentz structure via Postulate 3.4.1. With
our
preparation this
highly readable and certainly recommended to physicists who are interested in the operational approach. We have used a more historic approach for two reasons. Firstly, most readers are familiar with the classical description of spacetime (albeit less formalised, perhaps). Secondly, Newton's theory is also very compelling on first sight. So are Galilei's theory and the special theory of relativity. What is more, when these theories where still young and generally accepted it was very difficult to see how to improve them. In fact, most physicists and philosophers would have claimed that these theories are correct in an absolute way. There is no doubt that the Lorentzian description of spacetime will not be the last improvement either. It is even a prominent topic of current research to try to incorporate general relativity into a new general quantum theory of spacetime. It is generally believed that this new theory will be qualitatively very different from the geometrical theory we have presented here. The reader should recall that we started with macroscopic properties of rays of light which do not take into account the quantum nature of light. Also, we have always assumed that space and time are continuous rather than discrete. Hence there are several points where our theory of spacetime may prove inadequate. It must also be said, however, that a conceptionally satisfying theory of quantum gravity article is
does not yet exist. Our description of
and to date there is
no
of spacetime betten
spacetime is much better than previous theories theory which describes the global properties
other
[p. I
166
1]
P. 171
4. Pseudo-Riemannian manifolds
P.
We have learned that spacetime can be described by a Lorentzian manifold. In this section we will investigate the slightly more general case
of pseudo-Riemannian manifolds in detail. The development of the theory of spacetime will be continued in Chap. 5 where we motivate Einstein's equation, the central equation in general relativity which links matter to gravitation. Readers who wish to get to Einstein's equation quickly may skip most of Chap. 4. They only need to read Definition 4.2.2 and Sect. 4.3 up to and including corollary 4.3. 1. For mathematicians, this chapter contains the essentials of (pseudo)-Riemannian differential geometry. Almost everything we present here will be used in the following physically motivated sections. Prerequisitives of this chapter: Sect. 2.7.1 and Sect. 2.8 (up to but not including Lemma 2.8.2). Recall that
pseudo-Riemannian manifold (M, g) consists of an n-dimena symmetric, everywhere non-degenerate 20)tensor field g. We will often denote g by (., .). A pseudo-Riemannian (M, g) is called a Riemannian manifold (respectively, Lorentzian manifold) if g has signature is (+, +) (respectively, (-, +, +)). The simplest example of a Riemannian manifold is Euclidean Space, (Rn, d(xl )2 + .+ d(xn)2) and the simplest example of a Lorentzian manifold is Minkowski spacetime, (R', -d(x 0)2 + d(xl )2 + + d(xn-1)2). In this book, we are especially interested in Lorentzian manifolds a
sional manifold M and
(
.
.
.
,
-
.
.
,
.
.
,
...
mathematical models of spacetime. Riemannian submanifolds (cf. 4.4) of codimension I can be thought of as instants of time. They
as
Sect. will
play
an
important r6le when
we
discuss the initial value
in Sect. 5.4. Pseudo-Riemannian manifolds which nor
Riemannian
are
rarely applied
in
are
physics. However,
at any additional cost to widen the discussion to this
Unless
it does not
come
general case. geometrical objects are undermore
explicitly stated otherwise all from the metric and the Levi-Civita
stood to be derived
Remark 4. 0. 1. The
problem
neither Lorentzian
investigation of hypersurfaces
connection.
in Euclidean space has
tradition in mathematics and it has led to many important (mathematical) developments. (Pseudo)-Riemannian manifolds are the
a
very
long
We
only collect those facts which are essential to an understanding of Einstein's equation which will be presented in the next chapter.
M. Kriele: LNPm 59, pp. 171 - 254, 1999 © Springer-Verlag Berlin Heidelberg 1999
Pseudo-Riemannian manifolds
4.
172
of these
hypersurfaces and therefore of indepenwe will not push this angle, readinterested in differential ers primarily learning geometry should keep in mind the following example of a Riemannian manifold. Let M C R' be a hypersurface and consider for each x E M the R' tangent space TxM as a subspace of R'. To be concrete, let t: M be the natural inclusion and identify T,,M with t.TxM C T,(x)R' Pz R'. and define the We denote the standard scalar product of R' by Riemannian metric g of M by
natural
generalisation
dent interest to mathematicians. While
--->
g(V7 W)
:`
(t*Vl "W)R11
for all vectors v, w E TM. While this class of examples is rather
sualising
simple, it is sufficient for viimportant features of Riemannian manifolds. Whereas
most
Euclidean space is trivial in the sense that d(xl)2 + + d(Xn)2 is a constant tensor field with respect to appropriate coordinates, g is non-
constant in
-
-
and the curvature of its Levi-Civita connection does
general
not vanish.
given by the sphere by t: S2 -4 R 3, x f--> x the canonical inclusion. In this case we have T,,M ly E Rn 01. We can parameterise (a dense open subset of) the sphere (XI Y)R3
Example 4. 0. 1.
S2
=
fx
E
R3
The
simplest, non-trivial example
(Xl)2 + (X2)2 + (X3)2
:
11
=:
C R
3
.
is
Denote
=
=
using the chart (U, W) where
Cos 0 1
W_ (01 0)
sin
=
Cos
0
0 cos 0
.
sin 0
Let
f El, E21
be the standard orthonormal basis of R 2. From
-
-1
'9
( o-').Ej
490
-
=
-
cos
sin 0
sin
sin 0
ao Cos
-
=
0
0 cos 0 0 Cos 0
sin
Cos
0
we
obtain goo
=
COS2 0. We could now use 0, goo 1, goW g(ao, ao) the Levi-Civita concalculate to (Equation (2.7.7)) =
=
=
the Koszul formula nection and
given
we
through the formula study general submanifolds calculating these quantities.
could determine the Riemann tensor
in Theorem 2.8.1. In Sect. 4.4
and present better
techniques
for
we
will
A submanifold of Minkowski space does not necessarily inherit a Lorentzin fact, the example C,,+ (U) \ jxj shows that a hypersurface ian metric -
4. Pseudo-Riemannian manifolds
173
in Minkowski space may not inherit any pseudo-Riemannian metric. In contrast to the Riemannian case, it is probably not wise to try gain-
ing intuition for Lorentzian manifolds from studying submanifolds of Minkowski space. We should instead use the intuition we gained in the previous chapters. Lorentzian manifolds serve as models for space and time
as a
unit
and in this
-
physical
way their
geometry
can
be under-
stood best. a
have
a
pseudo-Riemannian manifold is everywhere nonisomorphism of vectors and one-forms.
Since the metric of
degenerate
we
canonical
(M, g)
Lemma 4.0.1. Let
be
pseudo-Riemannian manifold. The metv , where v (w) T ,M --* T,,*M, v
a
isomorphism all W E T,,M. for w) g., (v, ric,induces
-->
an
Proof. This follows immediately
from the fact that g is
isomorphism by (.)0: T*M naturally extended to tensor fields.
We denote the inverse
phism
can
be
Definition 4.0.1.
-
V)
.,v,+,)
-
completely
and the
Let
E
Then
T,,r(T--M).
we
T.,M. This isomor
define the completely
V) by
covariant tensor
V),(Vl,
.
-*
non-degenerate.
:-::::!
0(vi,
-,Vs,
contravariant tensor
OW.... Wr+S)
= -
I
-
-,
(V,+i) ...... (v,+,)')
00 by
(W,)O, w",
-
-
-,
r+sl)
components of V) are often simply denoted by V)j,...jjj i,jl j". components of 00 by 0" The
The
(.)0, (.)'5
isomorphisms
of indices". This
j,
and the
and
lowering
...
...
...
are
often referred to
terminology
is motivated
abstract index notation. We write
(90)ab
=
as
by
"raising
their expression in the
gab, (V )a
=
gabVb
=:
Va i
symbols " " and 5" should be easy to remember since there is an analogous notation in music. One of the most important inequalities in linear algebra is the CauchySchwarz inequality for positive definite scalar products (*7')R",' It states that for every pair of vectors v, w the inequality and
(.0)a
=
gab Wb
a
=:
.
The
(V W) Rn I
us,
7
(W W) R'11, I
inequality clearly generalises to Riemannian manifolds. To analogous inequality for Lorentzian manifolds is much more im-
holds. This an
(V V) Rn
portant.
4. Pseudo-Riemannian manifolds
174
(Inverse triangle inequality).
Lemma 4.0.2 ian
and let v,
manifold
u)
Let
be causal vectors. Then the
(M, g)
be
a
Lorentz-
inequality
VF-(VV)I VR-W*
(v, W holds.
Proof. The inequality holds trivially for null that both
sume
number
a
w
and
such that
w
v =
av
vectors. Hence
timelike. There is
are
+
This
e.
a
vector
implies (e, e)
>
we can as-
I
e
0, 0
(txl.... tx') which pass through a 0 for all x (xl.... x n) x0 are geodesics. Hence we get rb(c (tX)Xb x' =
=
small
enough.
This
implies
a
b
e
0
ax d
(29aeFbc(tX)X XC)
ad ((ab9ac(tX)
+
(adabgac(ft)
+
19dacgba(ft)
(19dgac(ft)
+
19cgda(ft)
,9x t
+ 2
Observe that
19dgac(O)
lim t_0
Hence
=
acgba(ft)
-
19agbc(ft)) X bxC)
-
adl9agbc(ft)) XbXc
-
aagdc(ft)) Xc-
0 and that therefore
119dgac(ft) t
=
d('9d9ac(0))(X)
dividing the equation above by
=
t and
19e'9d9ac(0)Xe-
taking
the limit t
0
we
get 0
=
(190bgac(0) + 2
=
+
Xb xc(4,9bC9d9ar_(0) this
(xl.... IXn) that
Gabcd
adaagbc(O)) X bXc
-
1909cgda(0) -
d x
we
=
(2190dgac(0)
is
arbitrary (for
:=
-
adaagbc(O)
equation with 0
Since
19d19c9ba(0)
(490dgac(0)
Contracting
implies
+
-
219(d19bgJa1c)(0)
a0agdc(0)) XcXe. +
2abacgda(O)
-
2abaa9dc(0))-
obtain
adaa9bc(0))X bxcxd small -
enough values)
a(dalal9bc)(0)
==
this equation
0. We obtain
188
4. Pseudo-Riemannian, rnanifolds
Gabcd
0
Gbcad
+
+
Gcabd
I
(2adabgac(O)
3
+
-
209c'Odgab(O)
adl9agbc(0) -
2abacgad(O)
+
-
abaagcd(O)
acaagdb(O))
1 +
-
3 +
(2adacgba (0) 2a,,adgbc(O)
-
-
adabgca(O)
+
2acaagbd(O)
+
2a,,abgcd (0)
-
C9cabgad (0)
lyaabgdc(O))
I
+
-
3 +
(2190agcb (0) 2ab19dgca(0)
(3adabgac(O)
3
adaagbc(O) This last equation 0
=
+
+
-
-
ad'Ocgab (0)
aaacgbd (0)
abacgda(O))
3ad19agbc(0)
adabgca(O)
implies
-
+
+
3acadgab(O))
19dacgab(0)-
(4.3.2)
now
Gdbea
1
3
(2aaabgdc(O) +
=
We
2acaagdb(O)
aa'9dgbc(O) -
+
=
2abacgda(O)
-
Nadgca(0)
acadgab(O))
+ 19bacgda(0) 2(OcAgde(O) 3
interchange
19caagbd(0) ad'90ca(0)
-
+
acaagdb(O))
-
b and d in this
equation and get Oa adgbc (0) + adacgba (0) + comparison with Equation (4.3.2) gives 19caagbd(0)
0. A
which is
It follows that
equivalent to our assertion 190bgcd Equation (4.3.1) implies
a Rbcd (0)
==
gae (0)
=
aclodg,,b-
(acabged(O) acaegbd(O)) -
and therefore
Racbd(O)
where
follows
+
we
Radbc(0)
=
190cgad(0)
-
=
a0cgad(0)
+
=
-3,9baagcd(O)i
a0agcd(0) 190dgac(0)
+ -
190dgac(0) -,90agdc(0) 19b0agcd(0)
have used
now
from
a
Equation (4.3.2). The assertion of the proposition Taylor expansion Of gab(x) around the point xo. I
Corollary 4.3. 1. Let A be a tensor field which is pointwise defined an algebraic expression of g and its first two derivatives. Then A is algebraic expression of g and the Riemann tensor R.
as
an
4.3 Curvature of pseudo-Riemannian manifolds
189
Proof. Proposition 4.3.2 implies that any tensor field A which is a pointwise invariant function of g and its first two derivatives only depends on g and R.
I
F[5pTT1W47j] I I
For 2-dimensional
manifolds, the Riemann tensor reduces to a function, a much simpler geometric quantity. To see this, let I e- I, e2 I be an orthonormal basis of T'M. Since (R(., .)., -) is anti-symmetric in the first two and in the second two entries the expression (R(ei, e2)e-2; ei) does not depend on
the chosen orthonormal basis and defines
is easy to
(For explicit
a
function K: M
that this function describes the curvature tensor
see
formulas cf. Sect.
R. It
-4
uniquely.
4.3.1)
higher dimensional pseudo-Riemannian manifolds such a simple relationship does not exist. However, it is possible to define a function K which maps the space of non-degenerate 2-dimensional subspaces of For
T'M into the real numbers. Let G nondeg (TM) 2
fspanfu,,,v,,l
=
:
Ux
be the set of all two-dimensional
spacelike
or
Vvx,9jsPan u_v,,,j subspaces
is
non-degeneratel
in TM which
are
either
timelike. The
Definition 4.3.2.
function -
K: G nondeg (TM)i--->R, 2
spanjux,vxjt-->
(R(ux, vx)ux, vx)
UX, UX) (VX' VX)
is
(Ux, Vx)
2
of M.
is called the sectional curvature
Observe that this expression
_
well defined since the denominator does
nondeg not vanish for any 17 C G 2 (TM) and both the numerator and the
denominator transform
Al'ux 1
+ A 2Vx, 1
fj.X
=
by
a
factor
det(A)2
pseudo-Riemannian manifolds
Those
ture reduces to
a
under
a
change
of basis fix
A2'ux +A 2VX. 2
function
on
for which the sectional
M should be
curva-
especially interesting.
Proposition 4.3.3. Let (M, g) be a pseudo-Riemannian manifold and M. If the sectional curvature satisfies K(Hx) x c K(flx) for all I-T, fIx E TxM then there exists a number c C- R with Rx (u, v)w c ((v, w) u (u, w) v) for all u, v, w E TxM. =:
-
Proof. Assume that K(Hx) does
not
depend
TxM. Then the definition of the sectional an
orthonormal basis
fel,
-
-
(R(ei, ej)ej, ej) for
some
constant
clude that
c.
-,
=
enf c
we
on
the choice of
curvature
implies
plane in given
that
have
( ej, ej) (ej, ej)
_
(ei,
From the tensorial property of
ej)2) R(-,
we
con-
p.
2091
4. Pseudo-Riemannian manifolds
190
(R(u, v)v, u) for
allu,v ET,,M. For
0
=
(R(u
=
(R(u, v)v, u)
+ w,
v)v, u
+
w)
E
w
every
-
((Uj U) (V, V)
C
=
T.,M,
(u
c:
(R(w, v)v, u)
+
u
(U, V)2
obtain
we
+ w,
_
w) (v, v)
+
(R(u, v)v, w)
+
U
-
V)2
+ W,
(R(w, v)v, w)
+
(u, w) (v, v) (w, w) v, v) c( (u, u) (v, v) 2 (U, V)2 (u, v) (w, v) (V7W)2) + 2
+
-
=
2 -
+
(R(u, v)v, w) (u, u) (v, v) (U, V)2
R(u, v)v
2
_
c( (u, u) (v, v)
+
(u, w) (v, v)
2
-
(u, v) (w, v)
+ 2
c((v, v)
-
which
R(u, v
=
R(u, v)v
w)(v
+
w)
c((v, v)
-
+
R(u, w)v
-
C((w, W)
R(u, v)w
+
-
U
-
u
(u, v) v)
-
c((w, v)
-
u
u,
(W, V)2
(w, w) (v, v)
(u, v) v)
-
c((v + w,
-
_
(V, W)2
+
=
=
(W, W) (V, V)
+
(u, v) v), w)
-
implies R(u, v)v c((v, v) polarise again and get
now
0
u
(U, V)2
_
-
+
v
u
u,v
-
R(u, v)w
+
(v, u) w)
-
w)
for all U,
-
E T.,M.
v
w) (v
+
c((v, w)
u
+
We
can
w))
(u, w) v)
-
R(u, w)w
+
(W, U) W)
c((v, w)
This
u
-
(u, w) v)
+
R(u, w)v
-
c((w, v)
u,
(v, u) w).
-
implies R(u, v)w c( v, w) u (u, w) v) R(w, u)v c( u, v) w (w, v) u) for all u, v, W G T,,M,i.e the expression R(u, v)w c((v, w) u (u, w) v) is invariant with respect to cyclic permutation. Since the cyclic R(u, v)w + R(w, u)v + R(v, w),, vanishes (cf. Proposition 4.3.1) and -
=
-
-
-
-
(u, W) the
cyclic
C((.V, W)
u
v
-
(v, W)
U
+
V, U)
R(u, v)w (u, W) v))
sum -
of
-
W
-
(u, V)
c((v, w)
u
-
W
(W, V)
+
U
(W, U)
-
(u, w) v) gives
-
0
=
3
v
=
0,
(R(u, v)w
Proposition
4.3.4
tional curvature K:
-
I
-
(Lemma of Schur). If dim(M) ! 3 and the secGnondeg R reduces to a function on M then (TM) 2 -*
it is constant.
Proof. Proposition
4.3.3
with R,, (u,
c(x) ((v, w)
v) w
implies that there
exists
a
function
x
--+
c(x)
u, w) v)
for all u, v, w E T,, M. Let W be vector fields whose covariant derivatives vanish at x and which
U, V, satisfy U.,
=
=
u, V
v,
Wx
=
(VtR) (u, v)w
u
w.
=
-
Then
we
obtain for any vector t E TM
dc(t) ( v, w)
u
-
(u, w) v).
4.3 Curvature of
dc(t) ((v, w)
=
be
Let
u
We
can
we can
u
u, w) v)
-
dc(u) ((t, w)
+
191
identity (Lemma 2.8.1) implies
Hence the second Bianchi 0
pseudo-Riemannian manifolds
v
+
(v, w) t)
-
dc(v) ((u, w) t
(t, w) u)
-
.
vector which is not
a
choose t also
assume
lightlike and w =h 0 be orthogonal to u. orthogonal to w. Since dim(M) > 3 u, v, w are linearly independent. The equation
and
w
=
that
v
to be
above reduces then to
dc(v) u
0
not
dc(u) v) (w, w)
.
0. Hence dc vanishes on implies dc(u) lightlike and therefore must vanish identically.
which in turn are
+
=
Observe that the
preceding proposition
all vectors which
I
is false for two-dimensional pseu-
do-Riemannian manifolds.
Proposition
4.3.4 motivates the
Definition 4.3.3. A
if
curvature
In
following
definition.
pseudo-Riemannian manifold (M, g)
nondeg the sectional curvature K: G 2 (TM)
Proposition
4.5.2 below
we
will
locally classify
all
-4
has constant
R is constant.
pseudo-Riemannian
manifolds of constant curvature.
4.3.1 2-dimensional
pseudo-Riemannian manifolds
formulas for the Levi-Civita connection and the curvature of 2-dimensional pseudo-Riemannian manifolds. These formulas will be used in Sect. 7.4. We collect
tensor
pseudo-Riemannian manifolds are the lowest dimensional pseudo-Riemannian manifolds which are not trivial. Let 71 772 E f 1, 11 and let f El, E21 be an orthonormal frame with 0, g(El, Ej) n2. By Corollary 2.4.2 g(El, Ej) qj, g(El, E2) 2-dimensional
-
1
=
=
=
there are coordinates (t, q) and functions A v e- (t, q) 19q. e- (t, q) 0t and E2
A(t, q), v(t, q)
such that El
=
Proposition 4.3.5. Let (M, g) be a 2-dimensional pseudo-Riemannian dv(E2)EI dA(Ej)E2 and the Levi-Civita manifold. Then [El,E2] connection is given by =
'7E,El VE2 El The curvature is
=
=
-
- 71772dv(E2)E2, dA(El)E27
'7E1E2
VE2E2
=
completely determined by
-
=
dv(E2)EI,
771772dA(El)El.
the scalar curvature
Seal,
192
4. Pseudo-Riemannian manifolds
R(Ej,E2)Ej=-22ScalE2, R(Ej,E2)E2=2IScalEj, 2 2 1
Ric 2
K(TM)
Scalg,
2
Scal,
where
Scal
=
-2
(77, (El
9
E,
*
A +
(dA(Ej ))2)
+ 772
(E2
*
E2v
+
(dv(E2 ))2)
)
-
Pro0f. We calculate first the commutator of the vector field El, E2, Using ordinary derivative of R 2with respect to the coordinates (t, q).
the
[El, E2]
D(e-A,9q)(e-'at) D(e-'at)(e-Ai9q) -e-'e-, dA(at)aq + e-e-ADaq(,9t) + e-'e-Adv(aq)c)t e-'e-ADat(aq)
=
-
-
-
=0
-dA(EI)E2
+
dv(E2)El
-
e-'e- A
[0t, Oq]
Our formulas for the Levi-Civita connection follow
'7E, El, El ) '7E2 El, El ) '7E, E2, E2 ) '7E2E"2) '7EjE2 E, )
=
El
9,qj
=
0,
E2
0
771
=
01
El
*
772
=
07
E2
'6
772
2 =
2 =
2 =
2 =
,
=
'7E1 E2 (dv(E2)El
'7E,Ej, E2)
from
0)
17E2El, E, -
dA(Ej)E2, Ej
=
?jjdv(E2)i
Ej,7E,E2) -Tlldv(E2)) '7E, E2 '7E2 El, E2 ) =
VE2 El, E2
-
(dv(E2)El
'7E2 E2, Ej) The curvature
directly
-
dA(Ej)E2, E2)
(E2, VE2Ej)
can now
be calculated
=
=
772dA(El)E2
-772dA(El)
straightforwardly.
R(El,E2)El =7EjVE2Ej -7E27E,El
='7Ej(dA(Ej)E2) dv(E2 )7E,El -
El
*
El
*
+ +
-7[E,,E2]El 771772VE2(dv(E2)E2) dA(El)7E2 El
A E2 + dA (El) dv (E2) E, + 771772 E2
9
E2
e
1/
E2
4.4 Subinanifolds
-
dv(E2)dA(E1)E1
(El
El
*
+ 771772
Since
=
=
Ric(El, E2)
Ric(E2, E2)
E2
(dv(E2
+
e v
obtain Ric
tr(R(., El)El) 772
=
))2))E2-
02 (R(E2, El)El)
(R(E2, EI)E1, E2)
Ric(E2, EI)
=
02 (R(E2, E2) Ei)
+
=
tr(R(., E2)E2)
ni
1
2
-77,
=
(R(El, E2)El, E2))
=
'
01 (R(El, E2) Ei)
=
0,
(R(El, E2)E2, EI)
(R(El, E2)El E2) 7
Scal g, where
Scal
-271,772 R(El, E2)El, E2)
=
-2(,ql (El
-
+ 772
(E2
*
the sectional curvature is
K(T M)
-772
=
tr(R(., E2)El)
=
=
Finally,
*
(dA(El))2 E2
(dA(El ))2
A +
(E2
+
have
we
Ric(Ei, Ei)
we
9
?11772(dv(E2))2 E2
+
193
*
El
E2
9
* v
A + +
(dA(El ))2)
(dv(E2 ))2)
given by
(R(El, E2)El, E2) =
Scal.
-
(El, Ei) (E2, E2)
-
(El, E2)
2
2
4.4 Submanifolds
pseudo-Riemannian manifold (M,g) and an (immersed) subM it is of interest which geometrical structure Z f: Z inherits from (M, g). Examples for physically especially interesting submanifolds are spacelike hypersurfaces (describing an instant of time, cf. Sect. 5.4) or the integrated light cone. In Chap. 9 closed trapped surfaces, a class of submanifolds of codimension 2, will play a central Given
a
manifold
-+
r6le. In this section
will
Remark
4.4-1.
f. Very
often the immersed submanifold is
we
explicitly a
refer to the immersion
subset of M and
f
is the
canonical injection. In this case it is more convenient to omit any references to f. We will often speak of tensor fields along Z instead of tensor fields
along f.
194
4.
Pseudo-Riemannian manifolds
We denote the space of vector fields along Z by To' (f) and will use the notation introduced ft-Alowing Lemma 2.9.2. In this notation, Lemma 2.9.1 is fundamental for the following. Lemma 4.4.1. Let
(M,g) be a pseudo-Riemannian manifold and f : Z ---> M be an immersed submanifold of M. Then the following holds for all vector fields U, V E To'(Z), X, Y Cz 'To'(f
0)
Vf" Uf. V
(ii)
d
Vf' Vf- U
-
X, Y) (U)
f. A V],
=
Vf.UX, Y)
=
X, Vf.UY).
+
Proof These properties follow immediately
(cf.
Lemma
2.9.1)
and the fact that V is
Definition 4.4. 1. Let
Z
f:
M be
--
an
a
from the definition of
Vf* UX
Levi-Civita connection.
immersed
I
pseudo-Riemannian
manifold and g be the metric of M. The map f (or the immersed submanifold
Z) is called non-degenerate non-degenerate (0) -tensorfield. A non-degenerate submanifold 2 is also called a pseudo-Riemannian submanifold, and a non-degenerate hypersurface is also called a pseudo- Riemannian hypersurface. Let Z be a non-degenerate immersed submanifold of (M, g) and denote by (T--Z)-L the set of all v E Tf(.,)M with g(v,w) 0 for all The decomposition Tf(.,)M w E f* 7:,,Z. induces f*T.,Z 0 (T if f *g
is
a
=
=
projections v
such that
T v
=
v
__>
+
vTE f*TZ v' for
Lemma 4.4.2. Let Z be
(M, g)
and U, V E
every a
To' (Z).
and v
is the Levi- Civita connection
Proof Observe that VUV
(i)-(iv)
2.7.1)
is
_L
Tf (,,) M.
non-degenerate immersed submanifold of VUV defined by
('7 f*Uf* V)
of (Z, f *g).
imply that
f
is
the Koszul
an
immersion.
Proper-
equation (cf. Equation
a
I
non-degenerate
immersed
submanifold of
The map 1:
symmetric
tensor.
(T. Z)
G
is satisfied for V.
Lemma 4.4.3. Let Z be
(M, g).
v_L
is well defined since
in Lemma 2.9.1
2.7.7 in Theorem
-4
Then
f*vuv:=
ties
E
v
IV (Z)
X
I
Tol (Z)
-
(TO, M)
(U, V)
1-4
I(U, V):=
in U and V and
(,7f*uf*v)
Junction- linear.
I is called the
shape
4.4 Submanifolds
Proof.
The map _Y is
clearly function-linear
in its first
195
entry. Recall that
the Lie bracket of any two vector fields tangent to Z is itself tangent to Z. Hence Y (U,
0, which
V)
-
_Y (V,
U)
(V f.
=
implies that
in turn
U f,, V
Y is
U)
V
f. V f,, symmetric. But then -
(f,, [U, V])
=
-L =
ff must also be
I
function-linear in its second entry.
have considered vector fields tangent to Z. We obtain similar relationships for vector fields normal to Z. So far
we
a non-degenerate immersed submanifold of field on Z and N be a vector field along Z (Tf(x)f,,Z)J- for all x E Z. Then Vf.VN is well
Lemma 4.4.4. Let Z be
(M, g).
Let U be
such that
defined
N(x)
and
a
E
vector
have
we
Vf,, VN where
0
denotes the
(Vf*VN)
=
(I(V, .), N)
lift of indices with respect
to the induced metric
f *g.
immediately from the coordinate expression that Vf VN does not depend on the extensions of V, N off Z. Hence it is well defined. Let X be a vector field along Z and XT, X-L its tangent and normal Proof
It follows
*
part. Then
we
have
(Vf*VN, X) (Vf*VN, X-L)
+
=
Vf*VN, XT) =0
((Vf* VN) _L,
=
-
=
11
X
+
Vf*V N, XT)
N, V f* VXT )
((Vf*VN)-J-,X-L) N, (V VXT) (Vf*VN) X--L) N, y(V,XT)) -
f*
-
Lemma 4.4.5. Let Z be
a
non-degenerate
immersed
Z be a smooth curve. For (M, g) and let -y: [a, b] -L (T_ (a) Z) there is a unique vector field N along f -Y --*
o
(i) N (a) (ii) N(t) (iii) and
=
E
n,
(f*Ty(t)Z)-L
(Vf* N)
1 =
for
0.
all t
c
[a, b],
submanifold of
every vector
with
n
196
4. Pseudo-Riemannian manifolds
We will write
transport of Let
Proof.
which span N
N(t)
P-L
=
Ej,..
-,
to these
-L
at every
Fj: [a, b]
in
point. We
(TE)
-L
(Vf* N)J-
=
can
N(t)
as
R such that
--+
decompositions we obtain
the equation
to it
as
the normal
be orthonormal vector fields
En-dim(_v)
(T,,(t) Z)
functions
refer
parallel
n.
along f o-y with values
are
and
n
decompose
(Vf N)Ja
o
-y
any vector field
Ni (t)Ej (t). Similarly, there
=
1
(Vf* Ej)
0 reduces to
along f
=
TjEj.
==
With respect
(- -N'+NjFj)Ej. Hence dt
first order system of
ordinary uniquely determined by its
differential equations and each solution is initial values N'(a) n'.
I
=
Proposition 4.4.1 (GauB equation). Let Z be an immersed non-degenerate submanifold of (M, g). Denote by ER the Riemann tensor of (Z, g) and let U, V, W, X E 701 (Z). Then we have f
*
g
(-'-R (U, V) W, X) =
Proof.
(R(U, V)W, X)
Since this is
a
+
tensor
(I(U, X), I(V, W)) equation
-
(I(U, W), I(V, X))
we can assume
that
[U, V]
=
0. We
calculate
(R(f* U, f* V) f* W, f* X)
'7f*U'7f*Vf*W' f*X) (,7f*V,7f*uf*W, f*X) Vf*uf* (VVW), f*X) ( ,7f* Uz (V, W), f* X) (vf*vf* (VUW), f*X) (Vf*V1(U' W), f*X)
=
-
+
=
-
-
f* (VUVVW), f*X)
=
f* (VVVUW), f*X)
-
=0
Vf*u (_Y(V1 W), f*X)
+
--;,0V
-
=
-
,
f*V (_Y(U1 W), f*X)
+
(f* (OR(U, V)W) f*X) ,
(YA W) I(VI X))
+
Since I is neither
I
a
tensor field
derivative of I is not defined
on a
(.Y(V, W), Vf*uf*x)
Z
-
(Y(U' W), Vf*Vf*X) (I(V, W), 1(U, X))
-
nor a
tensor field of
priori. However, it is
for any vectors u, v, w c T., Z and any vector fields w the expression u, V. v, W,, =
=
M, the
covariant
easy to check that
U, V, W with U.,
4.4 Submanifolds
(vf.wl) A V) depends only
on
Vf.W(I(U, V))
=
I(VWU, V)
-
the values of u, v, w. This of 1.
justifies
-
197
I(U, VWV)
to call V_Y
defined
as
above the covariant derivative
Proposition 4.4.2 (Codazzi Equation). Let Z be a non-degenerate immersed submanifold of (M, g) and let U, V, W E To' (Z). Then we have
(R(f,,U,f.V)f,,W)-L N be
Proof. Let
=
(Vf.U-ff) I(V, W) (Vf.V_iT) I(U, W). -
vector field
a
Z which is
along
orthogonal
to TZ.
(R(f, U, f,,V)f,,W, N)
Vf.UVf,,Vf.W, N) (Vf.VVf.Uf*.W, N) Vf*[U,V] f* W, N) Vf*Uf* (VVW) N) (Vf*UI(V, W), N) Vf*Vf* (VUW) N) Vf*V.Y(U, W), N) -
-
+
,
-
,
(1([U, V], W), N)
I(U, VVW), N) -
-
+
I(V, VUW), N)
I(VUV, W), N)
-
+
Vf*UI(V, W), N) Vf*VI(U, W), N) (I(VVU, W), N)
(Vf*UI) (V, W), N) The assertion follows since N in
was an
-
(Vf*VI) (U, W), N)
arbitrary
vector field with values
(TZ) 1.
For
some
1
purposes the
shape
tensor is too
complex and also too rich in simpler geometrical quantity one can the shape tensor over all direction in the
information. In order to obtain average at
a
given point
x
C-
Z
a
submanifold. Consider can
then
Sk-1
=
a
k-dimensional Riemannian submanifold Z
IV
G
Tx Z
*
:
(f g),, (v, v)
=
I J, This set is
a
submanifold of the Euclidean space
(TxZ, (f *g).,).
the volume form of Sk- Iconsidered
as
good
M. We
C
the set of directions in T,,Z with the unit
identify
definition for the average of I
Average
VOI(Sk- 1)
We denote
submanifold of
over
sphere
compact Riemannian
by *
psk-,
(Tx Z, (f g),,).
A
the directions in T.,Z is then
fS2
')I-tS2,
198
Pseudo-Riemannian manifolds
4.
where the vector space valued integral is defined as in Remark 2.5.2. This method would not work for pseudo-Riemannian manifolds which are not Riemannian since in this to
We will to
case
the set of directions does not
correspond
compact pseudo-Riemannian submanifold of T,,Z.
a
a
now
calculate the average in the Riemannian case. This leads can be straightforwardly generalised to arbitrary
formula which
pseudo-Riemannian submanifolds. For simplicity we only consider a 3dimensional submanifold Z C M. The general case is analogous but calculationally more elaborate. Recall from Example 4.0.1 that we can parameterise a dense open subset of S2 using the chart map o given by
Cos 0 I
(-7r, 7r)
(0, 27r)
x
S2,
_4
(0, 0)
sin
-4
Cos
0
0 Cos 0
sin 0
where T.,Z is identified with R 3 via the induced metric
noting 1, (gS2)OW
on
2
0, (gS2)00
:--
orthonormal basis
I Cos 01dO do. Let e4, .... e,, be are bilinear forms P with A
we
have
(gS2)00
:_-
0. Hence Lemma 4.2.3
COS
:--
an
S2 by gS2
orthonormal basis of
an
je-1, e2, e3j. De(gS2) (0o, ao) implies
(T,,Z)-L.
tLS2
There
n
Y Ii (V, W) ej
1" (V, W)
i=4
for all v, w E T,,M. Since the volume of S2 is obtain for the average of I.,
given by
fS2 ItS2
:--
47r 3
we
ej
47r
S2
i=4
Cos
7r/2
n
3
j j27r
4,7r
/2
i=4 n
+ 2 sin
0
Cos
0
Cos
0
cos(O)dodO ej
+ 2
Cos
2 0 sin 0 Cos offi12
0
0 Cos 0 sin 01i23
2
0 COS2 Oyi22
2
offi
+ sin
+ 19""
33) cos(O)dodO
ej
7r/2
3
Ef 4,7r i=4
( Cos 3 07r_yiI,
+
7r/2
n
(Yi1I
+
'ri22
+
-R'i33))
i=4 3
Ef j=1
,
sin
sin 0
0 COS2 0.yi1 I
0 Cos 0 sin 0 lil 3
Cos
Cos 0
2w
2
-7r/2
0 Cos 0 0 Cos 0
sin 0
Cos
i=4
n
sin
0
w/2
1: 47r + 2
fl,
*
g (ej,
ej).B' (ej, ej).
e-i
3 Cos
0,7r_yi22
+ 2cosO sin
2
0,r yi
33)
Cos
OdO ej
4.4 Submanifolds
This
199
justifies the following definition.
Definition 4.4.2. Let
f: Z -+ M be a non-degenerate immersed subpseudo-Riernannian manifold (M,g). The mean curvafield H is defined by
manifold of ture vector
the
dim(Z)
I
E
H.,
dim(Z) where
f el
7
edim(Z) I is
f *g(ei, ei)-Y(ei, ei),
i=1
orthonormal basis
an
of TM.
It is easy to see that H,, does not depend on the choice of orthonormal basis. The mean curvature vector field plays a prominent role in the
investigation of black holes (cf. Chap. 9) and is closely linked to the theory of minimal surfaces (cf. Lemma 4.4.8 below). The normalisation 1
factor more
is
dim(Z)
more common
in the mathematical literature than the
alternative 1. Since unusual normalisations
logical
then bad normalisations
we
are even worse
have retained this factor.
Definition 4.4.3. Let Z be
non-degenerate immersed submanifold of Z ---+ TZ--L along Z with (n, n) 1 is called a normal vector field or simply a normal The shape operator S,,: T, Z -4T Z of Z associated with n is defined by S,,u, v) .Y(u, v), n). Rn the orthogonal projection. Then the secDenote by Irn: (T Z) ond fundamental form k. corresponding to n is defined by ir,, (1 (u, v))
(M, g).
A
(local)
vector
a
field
n:
=
.,
=
,,
k (u,
v) n.
Definition 4.4.4. Let Z be
shape of (M, g). degenerate hypersurface The term second
non-degenerate immersed hypersurface fundamental form of a nonsimply denoted by S and k. a
operator and second
The
are
fundamental form
from the
theory of hypersurpredates Riemannian gea non-degenerate hypersurface Z is Idimensional. Therefore there exists an (up to sign) unique normal vector field n: Z --+ TM along Z. Hence even without choosing a normal, k and S are uniquely determined up to sign. In Euclidean geometry, the induced metric g is called first fundamental form since it f allows to determine fundamental geometrical quantities such as angles and lengths. The second fundamental form k is another (o)-tensor field. 2 Since the GauB equation reduces to
faces in Euclidean space (R3, ometry. The normal space of
comes
which
=
(ZR(u, v)w, t)
=
k(u, t)k(v, w)
the curvature of the surface Z is k also determines the
shape
-
k(u, w)k(v, t),
completely determined by k. Moreover, a hypersurface is
tensor and therefore how
200
4. Pseudo-Riemannian manifolds
curved in space. Hence k is indeed a geometrically fundamental quantity justifies the name second fundamental form.
which
Lemma 4.4.6. Let Z be Then the
shape operator
damental
form by k(u, v)
S is
a
is
non-degenerate hypersurface with normal n. given by Su -Vf*un and the second fun=
=
(Vf*un, f*v) (n, n).
-
The
self-adjoint and the second fundamental form k
Proof. Let V be
(ff (u, v), n)
vector field with V,
a
Vf*uf*V, n)
=
v.
=
Then
(f*v, Vf*un).
is
we
shape operator
symmetric. have
(Su, f*v)
The normalisation,
(n, n)
implies( Vf*un, n) fundamental form
k(u, v)
=
we
=
self-adjointness
ff (v,
U).
analogy
to the
average of the
.
of S and the symmetry of k follow from
curvature vector field
mean
one
can
_Y(u, v)
introduce the
shape operator.
Definition 4.4.5. Let
surface.
('7f*un, f*v) (n, n)
(ff (u, v), n) (n, n)
The
In
0 and therefore the first assertion. For the second
calculate
Then the
f:
mean
Z
--
M be
curvature is
non-degenerate immersed hypergiven by H(x) n'Itr(k,,). a
=
The most important mathematical application of this concept arises in theory of minimal submanifolds.
the
Lemma 4.4.7. Let
Then the
(M, g)
be
an
a
yz
(which implicitly defines Proof. -n
oriented
pseudo-Riemannian manifold pseudo-Riemannian hypersurface in M with normal n. volume form of Z is given by
and Z c M
It follows
an
directly
=
n-lym
orientation
of Z).
from Definition 4.2.1 that either nJ pm
J /,tm is the volume form of Z.
The rest vature
of this section is an illustration of the concept of primarily directed to mathematicians.
mean cur-
or
4.4 Submanifolds
(M, g)
Let
be
oriented, Riemannian manifold and
an
consider
a
201
compact
submanifold B C M of codimension 2. We denote the space of all smooth hypersurfaces Z C M whose boundary is B by 9X(B). Then the question
naturally arises which Z E TZ(B) has minimal volume vol(Z) f, Mz. In general, there may not be a minimising hypersurface or it may not be unique. Nevertheless, it is relatively easy to derive a necessary condition any minimising hypersurface must satisfy. Assume that Z is a minimising hypersurface and let U be a vector =
field such that for all
U(x)
h(x)n(x)
=
B then the flow
for
function h: Z
some
Ft of U generates
-4
R. If
h(x)
=
0
smooth
I-parameter family Ft (Z) of hypersurfaces in 9X(B). Hence a necessary condition that Z has minimal volume is given by (IT ) 0 /-t Ft, (Z) t 0 fFt, (Z)
Zt
x E
a
=
=
-
=
(M, g)
Lemma 4.4.8. Let
and Z be be
a
an
vector
oriented
field
in
a
be
an
oriented
pseudo-Riemannian manifold
non-degenerate hypersurface with normal neighbourhood of Z which satisfies
(i) n(x) fi(x) for 1 (ii) (ii, fi) =
all
E
x
n.
Let ii
Z and
=
and h be
a
function and d
)t=o
atwhere H is the
Proof.
mean
Since all
U
=
Mi. For the
(n
flow Ft of
U
we
have
qzhHpz,
-
Ft (_1)
J z
curvature
of Z and
77z
=
(n, n)
E
objects are smooth we can interchange differentiation integration over Z and obtain, using Lemma 2.5.2,
with respect to t and d dt
d
)t=o J
M Ft (_)
=
Ft, (Z
Ft
=
=
=
f E,.... Enj
be
an
=
-
J pm +
U
dt
+
we
t=O
J
n
J pm + div
n
J
J pm
=
=
ii and
get
n
=
X
=
orthonormal frame with E,,
be the dual frame. Then
div(ii)
"
,
=
Let
d
(p -) ( ( ) Ft"pfz -CU(Az) fz (n AM) JXUttm fz [U, n] (hii) ym) fz Qhfi, n] fz (-dh(n) div(hii))n fz hdiv(ii)pz.
n-1
E Oa(V Ea ii)
E Oa(V Ea ii) + on (,V EnEn
a=1
a=1
01,..., on
4. Pseudo-Riemannian manifolds
202
=0
n-1
(5Ea)
0
+
En, VEnE.,
a=1
(n
-
1) (n, n)
Since for each function h with
H.
obtain
I-parameter family of hypersurface can only 0. For if there would be a point x E Z \ JBI extremise volume if H with H(x) =A 0 then there would exist a neighbourhood Ux of x in Z and
hypersurfaces Et
E
9R(B)
hIB
0
=
with Zo
=
we
Z
a
a
smooth
=
a
function h with
(i) h(y) 0 for all y (ii) h(y)H(y) > 0 for (iii) h(x)H(x) > 0. =
This would
imply
E
B U
(Z \ U ,),
all y E B U
f,,qEhH1,tE 4
Z,
0 in contradiction to the
extremality
of
vol(Zo). The equation H 0 can be understood as a differential equation for a function f which describes the hypersurface Z. It is a classical =
problem in differential geometry to monograph on this subject (for M
solve this equation. A ==
R')
is
comprehensive
(Nitsche 1975).
Hyperquadrics
4.4.1
we study the simplest non-trivial class of hypersurfaces of Rn. These examples will be used in Chaps. 7 and 6.
In this section
Let V
,q,: Rn
Rn
x
--
n
E Xi Yi
(X) Y)
R,
pseudo-scalar product
Xk yk.
k=v+l
j=1
be the standard
E
+
of index
v.
Proposition 4.4.3. The pseudo-Riemannian manifold (R' \ 01, 71v) is I x E Rn \ 101 : q,(X, X) C1, foliated by hypersurfaces Quadv- (c) where c E R. The hypersurface Quadn,-'(c) is non-degenerate if and only if C =h 0. n
=
These
Proof.
hypersurfaces Quadvn-i(c) It is clear that every
Quadvn-i(c). c.
Since
df
x
are
G
called
Rn
hyperquadrics.
\ fOJ
lies in
ThesetQuadn-l(c) Ej=, xjdxj + Enk=,+, xkdxk
isthezerosetofx
V
exactly -4
f,(x)
=
0 the map
2.1.1
n
subset
77,(x,x)-
does not vanish unless
f has constant rank in Ix E Rn : x 7 1 implies that Quadv (C) is a hyp ersurface Proposition The tangent space at x is given by Jv G Rn : 71, (X, V) 01.
x
one =
01.
-
=
of Rn
Hence 01
4.4 Submanifolds
Assume that
c
0
=
Then
-
have 77, (x,
we
x)
T, Quae,
(c). Since for any tangent vector induced metric is degenerate. Assume
that
now
with 77, (w, v) 0, this would
=
0.
Tx Quad'v
c: -4
(c).
a
0 and therefore
have q, (x,
vector
v)
=
G
x
0 the
.,Quad'-'(c)
w
V
Since
0 for all y E
have also 71, (x, w) R'. This is impossible since we
0. The map
Quad'v-1(c)
-->
(xi I...Ixv, xV+1 I...IXn)
__,
t:
an
=
I
Lemma 4.4.9. Let
v we
Suppose there was
v E
imply qv (W Y) non-degenerate.
qv is
is
=h
c
0 for all
=
203
Quad,,:vl(-c), (Xv+1.... x-, xi n-v
I
I
...
Ix V)
anti-isometry.
Proof. The hypersurfaces QuadL.n-1(c) and 77v (X,
for all
x
X)
Rn. The map
E
vectors u,v
t
n-v
-?7n-v (44
:--
is an
TxQuad'-'(c)
E
Quad'-'(-c) 4x))
since for each
anti-isometry
TxQuadn:,'(-c)
=
coincide since
have
we
pair of
Tlv(v,w)
-?7n-v(t*V, L*W)Lemma 4.4.9
implies
hypersurfaces
Quad',,-'(c)
Definition 4.4.6.
that it is with
Possible
to restrict attention to those
0.
c
The
Pseudo-sphere of dimension n, index V, and rahyperquadric Sn-i(r) Quadn-l(r'). If v 0 we write Sn-i(r) instead of Son-'(r) and Sn-1 instead of Sn-1(1). dius
r
is the
=
=
V
V
Lemma 4.4.10.
Sn-i-v,
where
The pseudo-sphere Sn,-(r) Sk denotes the k-dimensional
is
diffeomorphic sphere.
to R'
x
unit
Mr: Rn --> Rn, x _, -lx maps Sn,-'(r) onto It is therefore sufficient to prove the lemma for r 1. ConS,"(1). sider the map f : R' X Sn I v -4 Rn' (y, Z) _4 (y, + Iy z). The
Proof. The diffeomorphism
=
-
equation 77,, (f (y, z), f (y, z)) set of Sn-
1
(1).
I
implies
that the
f has also
a
image of f smooth inverse map, f
hence it is in fact
a
diffeomorphism
The map
(y,,\/ll_+Jy _li),
V1
-
Lemma 4.4.11. Let
c
of
11c.
constant curvature
=k
=
0. The
-
hyperquadric
onto
Quadn-l(c) V
is 1
a
sub-
(y, i)
Sn-1(1).
is
a
=
I
manifold
Pseudo-Riemannian manifolds
4.
204
Proof.
The
is
shape operator
given by
( V_ICI x
Sv
whence the
shape
=
-D
v
M
V_ICI
tensor reads
("/-ICI ) u
1(u, V) and the GauB equation
=
-77V
,
c
x
Ici
V-jCj
v
(Proposition 4.4.1)
reduces to
C
77, (R(u,
v)v, u)
4.4.2 Umbilic and
=
ICII
( V (U, U),qv (V, V)
totally geodesic
_
nV (U,
V) 2)
submanifolds,
The investigation of submanifolds is a classical field in differential geometry. Naturally, one concentrates on submanifolds whose shape tensor is of especially simple form since only for these classes one has
of at least a partial classification. In this section we collect elementary definitions and results which should be covered in a course of differential geometry. Readers who are primarily interested in physical aspects can skip
a
chance
those
this section.
M be an immersed pseudo-Riemannian f: Z pseudo-Riemannian manifold (M, g). A point x E Z is called umbilic if there exists a vector n E T,,M with f *g(u, v)n for all U, v E T,,Z. The submanifold Z is called _Y(u, v) (totally) umbilic if all points in M are umbilic. 0. The submanifold Z is called totally geodesic if I
Definition 4.4.7. Let
submanifold of
--
a
=
=
M be an immersed non-degenerate subf: Z manifold of the pseudo-Riemannian manifold (M, g). Then the following statements are equivalent.
Lemma 4.4.12. Let
-+
(i) Z is totally geodesic; (ii) For every curve -y C and
only if f
o
Z
we
=
(iv)
For every
The
a
geodesic of (Z, f *g) if
a
For every v with (O) f,,v then there is
(iii)
have: -y is
geodesic with respect to (M, g). E T,,Z we have: If -y is the maximal
-y is
curve
-y:
[a, b]
a
6
--->
parallel transport of v along
> 0
Z and every -y
geodesic
in M
7Q-J,6]) C f (z) v E Ty[al Z we have:
such that
satisfies
f.P,,(v)
=
Pf ., (fv).
4.4 Submanifolds
Proof. (i)
=
=
Since expx is
Proof.
a
local
diffeomorphism
near
Ox
E
TxZ there
are
neighbourhoods )IV, V_
of x, i which are swept out by geodesics through of M by x, Jc. The image of these geodesics under f, f are geodesics of two sets of these Lemma 4.4.12. Since the tangent vectors geodesics at
is
f (x) a
each form
neighbourhood
4.4.3
U of
a
neighbourhood of 0
f (x)
=
f(x)
with U n
E
f,,TxZ
f (W)
=
=
fj;, , there f( V_). 1
U n
Warped products
Many standard spacetimes have a generalised product structure, the warped product structure. To write down curvature expressions for this class in general will save work in Chaps. 6 and 8. Definition 4.4.8. Let
(Z
x
F,,7rz*gz
+
(r
manifolds function. Then
be pseudo-Riemannian
(Z, gz) (F7 9F)
of dimensions nz, nF and r: Z -4 R+ the pseudo-Riemannian manifold
\ 101
0
be
a
smooth
7rZ)2 7rF*gZ)
,
where 7rE: Z x F --> Z and 7rF: Z x F -- F) are the canonical projections, is called the warped product of Z and F with warping function r
206
4. Pseudo-Riemannian manifolds
We
identify vector fields X an Z (respectively, V on F) with the k -7'(T,,,_k 0 (respectively with f7 satisfying We Cali X G' satisfying 7rz,,V V). (respectively, V) the lift of X (respectively, V). In the following, we will make use of this identification and denote both vector fields by the same letter. Further, for on Z x F there are unique vector fields X on Z and any vector field can
vector field
=
V
by
F with
on
=
X + V.
For every x E M we denote the submanifold Z Z., and the submanifold 17rz(x) I x F by F ,.
X, Y be vector fields on Z x F which U, V vector fields on Z x F which
Lemma 4.4.13. Let
fields vector fields
vector
f7rF(X)j
x
are
Z and
on
of Z
are
x
F
lifts of s of
li
F. Then
on
0) 7rz*vyx::= v7rz*y7rz*xl (ii) IrF*'7yX O (iii) VXU VUX d(Inr)(X)U, (U, V) grad(ln r), (iv) 7rz* VUV (V) 7rF*VUV VITF*U7rF*V. =:
=
=
=
-
=
Proof. These equations
(2.7.7)). (i), (ii): Since
be verified
can
using
the Koszul formula
(Equa-
tion
VXY, V)
that
F. From
=
therefore to 2 function
on
Z is
--+
2.4.4
get (7r-,) [X, V]
we
(VXY, V)
=
-V
=
=
=
*
X, Y)
-
van-
-
tangent
to the fibres
for all vector fields Y which
2
0 and
to show
to the fibre
Z the first summand vanishes. The second summand
=
formula
tangent
0. The Koszul formula
[X, Y] is tangent to Zx (iii): From [X, U] 0 we get VUX is
only have
we
are
(7F) [X)'VI 0 (2.7.7) reduces V, [X, Y]). Since (X, Y) is a
*
(7rF)*X
0 and
ishes since
VXU
isometry,
an
0 for all vector fields V which
Proposition
(7rz)*V
since
Z ,
-/rz:
=
=
are
The covariant derivative
VXU.
Fx since
Y, VXU)
=
lifts of vector fields
VXY, U)
-
on
=
0
Z. The Koszul
implies
(VXU, V)
==
X
(U, V)
=
X(r2 9F (Ui V))
2rdr (X)gF (Ui
=
V)
2 =
-dr(X) (U, V). r
(iv):
equation follows from
The
V'Vux)
(VUV, X)
=
-
(
1
dr(X)U
V, r
-1 (U, V) (grad(r), X)
.
r
(v):
For each
metric 91F is
a
x
E
Fx is a submanifold of M whose induced 2 r (X)gFmultiple of the metric on F, gjF,,'
M the fibre
constant
=
4.4 Submanifolds
207
Hence their Levi-Civita connections coincide and the assertion follows. I
Corollary 4.4.1. Let -y a geodesic if and only if
(i) V (ii) V
z F
Proof. We
F
can
at all t. Then
write
(t)
X(7(t)
is
we
have
assertion follows if
of
==
be
a curve
in Z
x
F. A
X (-y (t)) + V (-y (t)), where X, V
curve
-y is
V,;/
==
are
vector
and
V(-y(t)) is tangent to F,,(t) Zy(t) + VXX VXV + VVX + VVV and the
tangent
to
project this
we
Lemma 4.4.14. Let vector
(,yz, IF)
( F) F) gr ad (In r), -2d(lnr)( -v) F.
Z
fields such that
=
vector to
I
TZ,, and TF,
X, Y, Z be vector fields on Z x F which are lifts of U, V, W be vector fields on Z x F which are lifts
Z and
fields on fields on
vector
F. Then
(i) irz,R(X, Y)Z R-r(7-r.X, -7rz.Y)7rz,,Z, (ii) 7rF,, R(X, Y) Z 0, (iii) R(X, Y) U 0, (iv) R(X, U)Y -IVVr(X, Y)U, (U, V) VXgrad(r), (v) R(X, U)V (vi) R(U, V)X 0, (vii) 7rz,,R(U, V)W 0, (Viii) 7rF,, R(U, V) W RF (U) V) W +-' rT (grad(r), grad(r)) ((U, W) =
=
=
r
-
=
=
=
V
Proof. Assertions (i) and (ii) follow directly from Lemma
(ii).
-
(V, W) U).
4.4.13
(i)
and
-
(iii):
We may choose
R(X, Y) U
=
=
=
X,
Y such that
[X, YJ
=
0. Then
VXVyU VyVXU VX (d In r(Y) U) Vy (d In r(X) U) (VV ln(r)(X, Y) dlnr(VXY) + dln(r)(Y)d ln(r)(X)) -
-
-
(VV In(r) (Y, X) d In r(VyX) + d In (r) (X) d In (r) (Y)) U -
-
=
(iv):
dlnr([X,YI)U
Since
[X, U]
R(X, U)Y
=
=
=
0
we
=
0.
have
VXVUY VUVXY VX (dlnr(Y)U) dlnr(VXY)U -
-
U
208
4.
VV In
=
dlnr(VXY)U dlnr(VXY)U In d (VV r(X, Y) + In r(X)d In r(Y)) +
dlnr(Y)VXU
+ =
r(X,Y)U
-
U
I
vvr(X, Y)U. r
(vi):
We
R(U, V)X
directly calculate
can
=
=
VUVVX
-
VVVUX
VU(d
In r(X)V)
-
-
V[U,V]X
VV(d In r(X)U)
-
d In r(X) [U,
V]
2
=
VV(In r(X)) (U, V)
+ d In
2
-
r(X)VUV
-
VV(In r(X)) (V, U)
2
d In r(X)VVU
-
d In r(X) [U,
V]
0
where
we
(v):
have used that
Since
[U, V] (R(X, U)V, W)
=
VUV VVU. (R(V, W)X, U) -
=
0, the
vector
R(X, U)V must be
tangent
Z,,. The
to
(R(X, U)V, Y)
assertion follows from
-IVVr(X, Y) (U, V).
(R(X, U)Y, V)
r
(vii): (viii)
This follows from
R(U, V)W, X)
-
R(U, V)X, W)
Observe first that the Levi-Civita connection induced
=
on
0.
the fi-
bre F,
equals the Levi-Civita connection Of 9F since both metrics differ only by a constant factor r2 (x). The result follows from the GauB equation (Proposition 4.4.1) since by Lemma 4.4.13 (iv) the shape tensor is I given by 1(U, V) (U, V) grad(In r). =
-
Lemma 4.4.15. Let
fields vector fields vector
Z and
on
F. Then
(i) Ric(X, Y) (ii) Ric(X, U) (iii)
X, Y be vector fields on Z x U, V be vector fields on Z x
on
Ric (U,
V)
=
Ricz(irz,,X, 7rz,,Y)
=
are
F which
are
lifts of lifts of
'F'7'7r(X, Y), r
0,
=
x (U, V), (iv) Scal Scalz =
-
F which
Ar
RiCF (Ui V) + 1ScalF r2
+
r
2njr
zAr
(nr-1) r2
(grad(r), grad(r))
nl,,(nF-1) -2
(grad(r), grad(r)).
4.5 Isometries and
Killing
vector fields
209
trz(R(., X)Y)+trF(R(X, .)Y), (i), (iv) while assertion (ii) is a (i) Formula (iii) is implied by and 4.4.14 Lemma of (v). (iii) consequence Lemma 4.4.14 (v) and (viii) and assertion (iv) is just the metric trace of (i) and (iii). Since
Proof
follows
Ric(X, Y)
=
tr(R(., X)Y)
=
from Lemma 4.4.14
directly
4.5 Isometries and
Killing
vector fields p. 189
diffeomorphism which preserves the metric. pseudo Riemannian manifolds with many isometries are especially simple. The relevance to the theory of space and time comes from the fact that observations indicate that our universe is well approximated by Lorentzian manifolds with many isometries (cf. Chap. 6).
An isometry is
a
Definition 4.5. 1. Let
and
(M, g)
(1 1, j)
I [I
be ps eu do- Riemannian mani-
folds. An isometry is a diffeomorphism 0: M -- 1 1 which preserves the such that metric, (O*j) g. A local isometry is a local diffeomorphism 0 =
=
g., at all
points
Lemma 4.5. 1. Let
and U C then
0
=
11
be
0 if
x
(M, g)
M.
and
(1 1, )
be
pseudo-Riemannian manifolds
connected open set. If 0, 0: U --+ 1 1 are local isometries, and only if there is a point x G U with TO T,,V). a
=
The two isometries
Proof.
E
clearly coincide
on
the closed set V
ly
=
G
Ty,01. Since V is non-empty and U is connected, we only TyO need to show that V is open. Let y e V and W be a normal neighbour-
U
:
=
hood of y. Then for every z E W there is a vector expy(v[z]). But this implies O(z) O(expy(v[zl))
=
=
exp,b(Y)(TyO(y[z]))
=
V)(expy(v[z]))
=
O(z).
v[z]
E
TyM
with
expO(Y)(TyO(y[zJ))
Hence
01W
=
01-,V
=
=
and I
therefore W C V.
Definition 4.5.2. A
defines
z
Killing
vector field is
a
vector
field
whose
flow
local isometries.
manifold (M, g) is stationary in a region U c M if there Killing vector field in U. It is static in U if this Killing field is orthogonal to spacelike hypersurfaces.
Lorentzian is
a
timelike
vector
Clearly, only very special pseudo-Riemannian manifolds can have nonKilling vector fields. A simple example is given by a metric which does not depend on one of the coordinates. Then the corresponding Gauffian vector field is a Killing vector field.
zero
Lemma 4.5.2.
only if V '
is
A
vector
field
is
if.C g 0 Ve - ' de.
Killing if and only
antisymmetric. In this
case we
have
=
=
if and
p.
11
210]1
210
4.
Proof.
The first
Pseudo-Riemannian manifolds
is clear since the Lie derivative
equivalence
derivative
along the integral equivalence we calculate
Gc g)(U' V)
we a *
V
add to
-
of
.
(qu' V)
V
V 5 (V) U).
(u, V)
zero.
+
+
(vu ' V)
U' V V)
-
Levi-Civita connection the terms marked
a
It follows that
anti-symmetric.
U' X V)
-
V U' V)
-
is the
X
In order to prove the second
V (U, V) + (U1 VVO
have used that for
Here
with
is
-C A V)
=
=
curves
is
a
Killing
vector field if and
Now the assertion follows from
(de)ab
=
only
if
2V,,, bj.
[Fp.-209--q] p. 255
be Killing vector fields. Then [6, 61 is Proposition 4.5. 1. Let also a Killing vector field, i.e., the Killing vector field on a pseudoRiemannian manifold form a Lie algebra.
X
only to show that the commutator of two Killing vecKilling vector field. From Proposition 2.4.3 we know that , X, ] V) for any tensor 0. In particular we obtain I [ ' Mg
We have
Proof.
tor fields is
0
X 'C?7g
a
[X X77'CO
=
-
=
0
be
Lemma 4.5.3. Let
1,y
is
Proof
a
,
Jacobi
7(s) (s,t)
vector
field,
s
and
a
Killing
is
1--4
-4
geodesic,
Ft(^ (s))
-dt-Ft-y(s)
note that '7
)
a
+
is
vector
is
field
by Ft. Since Ft the a
curve s
F-->
variation of is
=
a
geodesic. Then
a
Ve ( ' )
is
an
isometry for each
Ft(-y(s))
is also
geodesics and
a
t
geodesic.
its deviation
Jacobi field. For the second property
anti-symmetric by Lemma
'V )
and -y be
is constant.
s F-+
Denote the flow of
Hence
and
field
0'
0
-
=
4.5.2. Hence
V ,-y) 1
0.
of this section we will investigate highly symmetric pseudomanifolds. These results are of independent mathematical interest and will be used in Chaps. 7, 6.
In the rest
Riemannian
pseudo-Riemannian manifold (M, g) 0. cally symmetric if VR Definition 4.5.3. A
is called lo-
=
implies that the components of R with respect allelly propagated frame are constant functions.
This definition
to
a
par-
4.5 Isometries and
Lemma 4.5.4. Let
be
(M, g)
Killing
vector fields
211
pseudo-Riemannian manifold. It is loonly if for every curve -y and all vector fields cally symmetric if wich W are parallelly propagated along -y the vector field R(U, V)W U, V, is also paralley propagated along -y. a
and
The
Proof. fields
equation VR
0
=
implies
for
parallelly transported
vector
U, V, W =0
V (R(U, V)W)
(V R) (U, V)W
=
+
R(U,V V)W
+
+
R(V U ,V)W R(U, V)
V W
0
along -y. Hence R(U, V)W is also parallel along -y. Let Conversely, let , u, v, w G T,,M and -y be a curve with (O) and that of W be the assume w along -y parallel propagation u, v, U, V, the vector field R(U, V)W along -y is also parallel. Then the assertion follows from =0
(17 R) (u, v)w '1'7 (0)(R(U, V)W) -R(V =
-
=
R(u, V
(O)V)w
-
(O)U, v)w
R(u, v) V,
(Off
0.
I
(M, g)
Theorem 4.5. 1. Let
and with
x
E
M,
,
E
AR(u, v)w
1 1. If =
and
(1 1, j)
there exists
f?(Au, Av)Aw
neighbourhoods U, 1 of x,.:
and
a
a
be
locally symmetric manifolds
linear isometry A: TxM -- T=1 all u, v, w c TM, then there are
for unique isometry 0:
U
with T
A.
Proof.
We
only need
to prove existence since
uniqueness follows from
neighbourhood U of x exp.,-' is a local isometry. First sufficiently small. By Proposition U for 2.6.5 there is a unique w., c T,,M with exp(w.,) every y E for every uy E TyM there is a unique iiw. G Tw.,,,. (TxM) y. Further, with Tw. exp,,(iiw.) Twx T,,M C Tw.TM there is a uy. Since U, Lemma 4.5.1. We will show that for
some
the map 0: U --* 1 1, y F--* exp, oA note that 0 is well defined if U is
normal
o
=
vector Ux such that
i1w.,,
Proposition 2.9.5 that
dt
uy,uy)
It=0 =
Ux). It follows therefore from J(1),J(1)), where J is the unique
(wx
+
4. Pseudo-Riemamnian.
212
Jacobi vector field
V J(O) 0. (uy)
=
=
along
the
-y: t
curve
ii,,. From the definition of 0 T expj
TATexp.,
-
1
(uy)
=
f (0, t) with J(O)
-4
we
0 and
=
get
T exp;z, TA
=
T exp,;,- (AUAw.,j
and
by the same argument as before it follows that there is a Jacobi j along the geodesic : t i--> exp(tAw,,) which satisfies J(O) 0, and j (0. (uy), 0,, (uy)) V J (0) Au, (J(I), j(1)). j ly
field
=
=
Let
=
f El, f kj,
E,, I be
parallelly propagated frame along -y with El orthonormal, parallelly propagated with Ei(O) frame along AEj (i E fnj). With respect to these frames the Jacobi equations for J and J are given by and let
.
.
.
.
,
.
.
,
&I
a
=
be the unique =
n
d2 ji +
dt2
n
d2 ji
Y R'IkI jk
and
-Tt-2
+
k=1
Y_ -I ilkl jk. k=1
is the unique geodesic with (O) we have used that A (O) which The functions R'1kj and R1kj are each constant implies that ti by Lemma 4.5.4. Since we assume AR(u,v)w R(Au,Av)Aw for.all R'1kI u,v,w E T,,M, the definition of our parallel frames implies R'1kI jk (0) for all i, k. Further, the functions ji, jk Satisfy jk (0) 0 and
Here
=
=
=
=
(by
the definition of
our
frames) Ajk(O) dt
=
_dt _jk(O).
mental theorem for differential equations 2.4.1 all k and we get
=
Hence the funda-
implies jk (t)
=
jk (t)
n
(j(j)
j(O.U Y, 0. UY)
,
f(11: ji(j)jk(j)j(t,(j)j kk (1)) i,k=l
n
=
E ji(j)jk(j)j(ki(0), kk (0)) i,k=l n
ji (1) jk (1)j (AEj (0), AEk (0)) i,k=l n
=
1: ji(j)jk (1) g (Ei (0), Ek (0))
=
9 V(1) 1 J (1)
i,k=l
g(UY' UY). and the assertion follows from the
g(u, V)
1(g(U + 2
V,
U
polarisation identity +
V)
-
g(u, U)
-
g(V, v)).
for
4.6
Length
and energy functionals
213
Proposition 4.5.2. Let (M, g) and (1 1, j) be pseudo-Riemannian manifolds with constant curvature c and a. They are locally isometric if and a. only if they have the same dimension and signature and satisfy c =
Proof. Observe first that the conditions are necessary. We show now that a pseudo-Riemannian manifold with constant curvature is necessarily locally symmetric. Let t, u, v, w E T'M and U, V, W vector fields which satisfy U.., w and whose covariant u, V v, W derivatives vanish at x. From Proposition 4.3.3 we get =
=
,,
(VtR) (u, v)w
=
Vt(R(U, V)W) R(VtU, v)w R(u, VtV)w R(u, v)VtW -
-
=
If
(M, g) x
any
If
(k, )
and
Tx M, 5
G
(M, g)
E
cvt ((V, W)
have the
same
T k there exists
(1 1,
and
-
have the
a
U
(U, W) V)
-
Corollary non-zero
constant curvature then this isom-
same
k(Au,
A
global
classification of
dicates that
Hyperquadrics have
orthonormal bases
TyQuadn-l(c)
:
c
Let
:
f,,.
0
4.6
0
to
Rn It is
--+
an
Quad,
an
(Wolf 1977). a
very
manifolds with constant The
following Lemma large isometry group.
in-
Quad,n-1(c). For any pair of TxQuadvn-1(c) and ffj,...'fn} C
and x, y E C
isometry 0:
Quadn-l(c)
--->
V
Quad'-'(c)
with
V
Rn be the linear map which maps x, ej,...' en to isometry of (R',,q,) onto itself and therefore also
In this section
functional mannian many
itself. Since
is the desired
(c)
Length and
for
1
fi (i Ell,..., nj).
onto maps-Quadvn-1(c) n-1
of
0.
fel,...,enj
there is
V
Proof. Y, fl,
TxM and the
(M, g).
pseudo-Riemannian
difficult
more
Lemma 4.5.5. Let
0,(ei)
E
pseudo-Riemannian manifold with a hyperquadric Quad, n-1(c)
a
isometric to
locally
curvature is much
=
be
w
constant curvature. Then there is
which is
(c: 0)
(M,g)
4.5.1. Let
0.
dimension and signature then for linear isometry A: Tx M -- T;-
Av)Aw for all u, v, etry satisfies AR(u, v)w assertion follows from Theorem 4.5.1. =
=
0*(ei)
=
0(ej)
=
fi, the
restriction
isometry.
energy functionals will
study the problem of extremising the length sufficient conditions in the Rieand the Lorentzian case. Here we will lay the foundation surprisingly strong global theorems in differential geometry and
we
find
necessary and
1
214
4. Pseudo- Riemannian manifolds
(e.g. the Theorem of Myers,cf. (ONeill 1983, theorem 10.24)) and for the singularity theorems in general relativity (cf. Chap. 9). This section is
mathematically more involved than the other chapter and can be skipped on first reading. section uses material from Sect. 2. 9.
sec-
tions in this This
In Riemannian
geometry the length of
would need to model the
one
curve
a curve measures
in space. It is
how much wire
fundamental geo-
a
metrical experience in Euclidean geometry that for any given (not too distant) pair of points there is a curve of shortest length which connects them. In Lorentzian geometry the
length of a causal curve can be interthe time observer needs in order to traverse this an preted proper world line. Since in special relativity moving clocks are slower (twin paraas
doxon)
expects that for any
one
point there is
curve longest For other signatures the problem
to non-trivial results
(cf.
(not
two
causal
a
Lemma
of
curves.
The
distant), causally
related
extremising length does
not lead
4.6.9)
For the discussion in this section it is
widen the class of admissible
too
which connects them.
curves
to the
technically advantageous continuous, piecewise
to
smooth
advantage lies in the fact that in many situations it is much a continuous, piecewise smooth curve with certain
easier to construct
properties than
smooth
a
Definition 4.6.1. Let -y:
[a, b]
--
M be
Then the
curve.
(M,g)
be
a pseudo-Riemannian manifold and continuous, piecewise smooth curve in M.
a
length of 7
This definition makes defined derivative
is
b
defined by L(-y) since
sense
everywhere but
a
piecewise smooth
on
a
set of measure
of the chosen parameterisation. In the
pendent
V/j__ g ( (t), (t)) I dt.
case
has
curve
a
well
It is inde-
zero.
of Euclidean space
it coincides with the
length one would define through the approximation of -y by polygons. The following lemma guarantees that there are no repercussions in considering piecewise smooth curves instead of smooth curves.
Lemma 4.6.1. Let -y: [a, c] --+ M be a piecewise smooth there is a sequence of smoothly immersed curves -yi: [a, c] converge
pointwise
Proof Assume that M and \:
[b, c]
(Xl,...,Xn) that b
-
2-'11
2.1.7 there
---
to -y and
are
a
Then
M which
limi-,, L(-Yi)
=
L(-Y).
-y is the concatenation of two smooth
M where
such that A is >
satisfy j(t)
curve. -->
and let tj
ft(b)
=
A(b).
given by =
b
-
smooth functions
t
curves M: [a, b] ---> coordinate system (t,0,...'0). Let io (E N such
We choose
2-' where i
Wj,,0j: [a, c]
>
a
io and i
-+
[0, 11
E
N.
By Lemma
such that
4.6
Oj (t)
>
Oi (t)
We define the
I
for all t E
1
for all t
-1
for all t >
1
for all t E
2
pi with
curve
and energy functionals
Length
respect to
215
[a, ti- I], ti+j], [b, cl,
E
[a,ti-1], ti], [b, c], coordinate system
our
by
t
/-t'(ti-i)
+
I
(W,(S) (Ak(s)
ti
cik
where the constants Notice that
1,t(t)
smooth there is
jy"(t) t E
=
Iti+14
-
c
c
for
-
by
C O,(S))
+
jkO, (s)) ds,
the condition
for t < ti-I and
number
a
/-t(b)l < c1lb tj [ti, ti+,] we obtain -
determined
are
-yi(t)
+
I
-
A(b)
-yi(t) A(t) for t > b. Since /-t is > 0 such that I Ak (t) I < c and I yk (t) -A (b) I all t E [a, b]. Hence using W(t)oi(t) > 1/4 for =
=
b
ti
c Wi(s)Oi(s)ds
M
that -y is the concate-
M and A: [b, c] M with [a, b]. A(b) and A(b) point into the same future mu:
--
sequences of timelike
curves
--->
pi and
satisfy Mi(b) Ai(b) p(b). For provides a sequence -yi,j of curves ==
=
Ai which pair
each such
converge of curves
such that -yij converges
4.
216
,nanifolclms,
to the concatenation of Mi and
future directed
so
is 7ij for
Ai. Since both, Ai and
j enough.
We
can
pi
assume
are
timelike and
without loss of
generality that
all - ij are timelike. It follows that the sequence consists of timelike curves and converges to -y.
4.6.1 Variation of
length
f-Yi,ijiEN
and energy
In Euclidean space, the shortest curve between two points is the straight line connecting them. In Minkowski space, the longest causal curve between two
points
x, y E
I+ (x), is also the straight line connecting them.
Fig.
4.6.1. A broken
smoothed out
In
a
by
a curve
lightlike geodesic can be of arbitrarily small length
Riemannian manifold "without holes" it is
general
intuitively
clear that any two points can be joined by at least one shortest curve. In a Lorentzian manifold, the infimum over the length of all curves which connect
x and y is always zero since we can join any two points lightlike geodesic which then can be smoothed out to give a smooth curve of arbitrarily small length (cf. Fig. 4.6.1 and Corollary 4.6.1). It is also clear that there does not exist a curve of maximal length connecting x and y since we can always choose a spiralling spacelike curve of arbitrarily large length (cf. Fig. 4.6. 1). However, we will see below that in many situations there exist curves connecting causally related curves
by
x
a
broken
and y which maximise L in the class of all causal curves. 5 pseudo-Riemannian manifolds which are neither Riemannian
rentzian do not admit any non-trivial solutions to the
problem,
even
if
one
restricts to
spacelike
or
nor
Lo-
length extremising
timelike
curves.
These
ar-
guments will be made precise in Lemma 4.6.9 below. 5
Our a
examples
solution if
also
one
imply
that the
restricts to
length extremising problem does not spacelike curves instead of causal curves.
have
Length and
4.6
Fig.
energy functionals
4.6.2. A
curve
distance between two
Z1
217
minimising the spacelike sub-
manifolds Z, and Z2
actually investigate the slightly more general problem where endpoints x, y are replaced by submanifolds without boundary Z1, Z2 (cf. Fig. 4.6.2). In order to solve the length extremising problem in the Riemannian and in the Lorentzian case we will study 1-parameter families of curves f : [a, b] x (-E, E) --+ M, (s, t) -4 f (s, t) such that E Z, and f (s, b) (E Z2 for all s. If 7 extremises the length f (s, a) 6 functional L for all smooth curves which connect Z, with Z2, then we have A, L(f (s, .)) for all such I-parameter families with - (t) f (0, t). ds We will
the
=
=
ls=o
Through
the
of
investigation
d
2
dsyls=o L(f (s,
.))
we
will arrive at sufficient
conditions. Definition 4.6.2. Let be
a curve
Z1, Z2 be submanifolds of M and
which connects Z, with
A continuous variation
-y:
[a, b]
--
M
Z2-
f: (-E, E)
x
[a, b]
--+
M, (s, t)
called piecewise smooth if there are numbers tj.... ) tk E is smooth, where to := a, tk+1 := b and i x [t,,t,+,] f,
-->
f (s, t) of -y
(a, b) f 0,.
is
such that
Q. (continuous, piecewise smooth) variation f of -y connects Zi with Z2 if f (s, a) E Z, and f (b, s) G Z2 for all s c (-E, e). We denote the vector field T(,,t)f (as) along f by fs, the vector field T(s,t) f (,Yt) along f by ft (where defined), and call the (piecewise smooth) vector field (t) : (f,)) I s=o along 7 the variation vector field. C=
..
,
A
=
Lemma 4.6.2. Let -y: [a, b] ---+ M be a smooth curve which connects two submanifolds Z1, Z2. For any vector field along -y with (a) E Ty(a)ZI, (b) E Ty (b) Z2 there exists a variation f of -/ which connects Z, with Z2 and which has variation vector field .
Proof.
A2(0) a
Let M, C Z, and /LL2 C Z2 be smooth curves with pi(O) -y(a), (a), and A2(0) (b). We can now extend to -y(b), Al(O) =
=
=
vector field -";' such that yj
denotes the flow of E 6
In the Lorentzian
we
case:
set
all
(i
E
f 1, 2
f (s, t)
=
are
integral
F, (-y (t)).
smooth, causal
curves
curves
of
If F
218
4.
If -y:
Pseudo-Riemannian manifolds
[a, b]
M is
--
piecewise smooth then
is discontinuous at those
points where
-y fails to be smooth. We will therefore need the technical definition.
following
Definition 4.6.3. Let -y be a continuous, piecewise smooth curve and V be a piecewise smooth vector field along -y. For each to E [a, b] we set
,,AV(to)
lim
V(t)
V(t).
t_tO't 0 for all t
in
Then
function exists C-
[a, b]
by
Lemma
2.1.7).
and does not vanish in
a
neighbourhood of to. Taking a variation with variation vector field < 0. This implies that there o i we obtain therefore (Tdds L(f (s, -))) ,=o =
shorter
are
We
orthogonally. Let
be
6(a)
is
than 7 in contradiction to our assumption. that -y is a pregeodesic but does not intersect Z, Then there is a vector v E Ty(a)Zi with ( (a),v) < 0.
curves
assume
a
now
variation vector field with
tangential such that
field
(jdds L(f (s, -)))
to
Zi there is
f (s, -)
connects
< 0 in
a
(a)
variation
Zi with Z2
contradiction to
,=o
=
f
our
v
and
6(b)
=
0. Since
of -y with variation vector for all s. Again we obtain
assumption that
-Y is
It follows that -y intersects Zi orthogonally. observe that the same argument holds equally well for
length
minimising.
Finally
The discussion above does not square root of
This
apply to null curves problem can be avoided
Z2. I
since L involves the
if
one
considers the
b
1 dt. 7 energy of the curve -y, E(-y) := f' E ( (t) (t)) a Unlike the length functional this integral expression does
the parameterisation of the curve. While it is not true that timelike curves which extremise L also extremise the energy will
see
below that this property almost holds.
Lemma 4.6.4
(First
variation of
continuous, piecewise smooth
f (s, t) field .
depend on spacelike or integral, we
be
curve
M be energy). Let -y: [a, b] x : f [-(E, 6] M, (8, t) [a, b] --4
and
---*
a F-->
continuous, piecewise smooth variation with variation vector by t,.... tk E (a, b) the points where -y fails to be smooth. Then the derivative of E with respect to s is given by
The
name
"energy"
I
comes
from the fact that in the Riemannian
just the kinetic energy of a mass point of mass the integrand has nothing to do with energy.
tegrand case
a
Denote
is
case
the in-
1. In the Lorentzian
220
4. Pseudo-Riemannian manifolds
k
d
(ds
)
E(f (s, .))
Proof. Consider
fb
=
1'=O
"
E (J (ti),
(ti))
+
i=1
piece of
a
)
dt +
-y where it is smooth. The assertion follows
from I d
(f" f')
2 ds
ft
It follows that
merely
a
,
If7 a ' ft
a curve
ft,
Vf a, f,
which extremises energy is
a
geodesic (and
not
pregeodesic).
We will now, derive sufficient conditions for
between submanifolds without
length
If7 at ft, f '
(ft, f ')
curves
to extremise the
But first
boundary.
need
we
a
tech-
nical lemma. Lemma 4.6.5. Let 7 be a spacelike or timelike pregeodesic and denote the orthogonal projection to the orthogonal complement of -Y by (.)-L.
Then
for
every vector
V
field
along
the
-
formula
(V V)'
=
V (V)
holds.
Proof. The vector field V can be decomposed into its part orthogonal to , W, and its part tangent to , W , where W is a smooth function. From V =W +W and ( ,W) =Oweget -L
(V V)-L where
we
=
dW( )( )
have used that
The assertion follows
1
+ W
(VA
a curve
now
is
a
+
W' V ) implies In the
(V W)
following we
Lemma 4.6.6
like
V W
will
=
only
if
V
=
0
-
0
=
0
V (V').
freely interchange
(second
if and
pregeodesic
since
variation of
timelike
I and
arc
V
when 7 is
length).
a
Let 7 be
geodesic. a
space-
G 1-1,11 and let f: [-E,E] x geodesic with 77 [a, b] -- M, (s, t) --+ f (s, t) be a continuous, piecewise smooth variation of -y. If we denote by ti, tk E (a, b) the points where f (s, -) fails to be smooth then the second derivative of L o f with respect to s is given by or
=
.
(
..
d2 ds 2
L (f (s,
77 15=0
7
jb
+
(R( ,
dt
4.6
(
+
Length
and energy functionals
221
f
fs
V, V,541
+
1
R(
dt
k
as denotes the variation vector
where
jection
()
to
Proof. Using 4.6.3
b
Is=O, field
a
-J-
and
the
orthogonal
pro-
' .
the formula for the first derivative in the
proof
of Lemma
obtain
we
d2
-
V77 (ft ft) d82 I
d
( V7-77ft,
f
ft
ds
,V
ft)
ash f
f
77
/?__3 (ft ft)
V I
ft,
V
I
f
77 +
Vf a, f
a, ft,
V
-If (Y t)
aft) ) ( Vf
ft, V
a,ft
t
ft,
+
'9,
If7 9, f
t
(4.6.3) f
f
From V
a ,ft f v
=V at f, and f
f
f we
V a, ft,
get
f
f
f
a., V 9, ft =V as V at f , =V at V 9, f,
Vf as ft
f
f
ft, V a,
V
f
f
V
atf,,V
ft,
V
f
a"ft
Since for every vector field V
f
atfs) )
9t V 95 f,
along
we
-y
+
R(f, ft) f,
and
+
have V
(ft, R (fs, ft) f,) =
V-L
(V'f") + 77 (ft, ft)
ft
and
I
1
(V, V) V VJ-) Equation (4.6.3) simplify to
therefore
=
,
the second and third summand in
+
f
Vfn- (f
t
I
ft)
atfs)
f
atf,)
222
4.
Using
Pseudo-Riemannian manifolds
the
product formula for the
f
ft, Vf
term
a,,f,)
V at V
we
finally
obtain b
d2
(
L(f(s, dS2
-)))
=
1 '=O
I
and the first with
(, )
equality in the
=
,
t
f
+
( (V aif') (
*' :7:_-If=t):: ( ft, V a,
+
'
f
77
f
fS)
(ft, R (f, ft) f,)
)
f
V at ft,
-
f
V
f'
dt
assertion follows since
f (0,
-Y is
a
geodesic
q.
Since for every smooth piece of the variation f the equation
( V4-L "74 ) 1
holds, the
There is
second
an
equality
-L
V -L'
=
follows from
)' ( V 74-L' -L ) -
an
integration by parts using
formula for the second variation of the energy
analogous
integral. Lemma 4.6.7 and
f : [-,E, E]
x
(second variation M, (s, t) [a, b] --
smooth variation
f (s, -) fails to be to s is given by
of
7.
Denote
energy) Let -y be a geodesic f (s, t) be a continuous, piecewise by tj tk c (a, b) the points where -
...
7
smooth. Then the second derivative
d2
d82
of
-*
E(f (s, Is=0
fb ( V4, V4) f.')
+
f
+
V
a,
I
,=,,
f
(R( ,
)
+
'a'.
k
f
+ 11=0
of E with respect
)
dt
4.6
Proof. The first equation
(ft (S' 2 j_S2
and energy functionals
ft (S'
f
f
V
Oft' V Oft
f
f
f
V
atf" V atf,
f
f
+
f
atf
V
"
f
equality follows
from
V
+
V '9 ' V at fS,
ft
f
f
+
V at
V'9'fS'
ft)
ft
f
an
) V' atft) (R(f, ft)f,, ft)
atf. f
V 7 a
proof
V 0, V C'), ft I
(R(f, ft) f, ft) f
the
f
+
f
V at fS,V at f,9
The second
223
follows from
d2
1
Length
f, V
a 'f
+
"
ft
+
integration by parts exactly
as
in
1
of Lemma 4.6.6.
M be Z1, Z2 be submanifolds of M and -y: [a, b] which both intersects E2 submanifolds orthogonally. from Zj Assume that f is a (continuous, piecewise smooth) variation of 7 which connects Z, with Z2. Then
Lemma 4.6.8. Let
--->
to
a curve
( (V a'f') 1'=O' ) f
holds, where lz,,
=
(.Yz2( (b), (b)), (b))
denotes the
Proof The
assertion follows our
(
tive
form
shape
tensor
of Zi.
immediately from the definition I Z1, -y(b) 1 Z2-
implies
.))
)
that for
a
geodesic
(respective 1 y 1,9=0
tively, a
LEY
I
d2E(f (s, .)) )
-d-s-7
the space of all variation vector fields
-y).
It is
an
shape
variation the second deriva-
) is a quadratic jS=O the central geodesic
along form
-y. The associated bilinear form is called the index
of
of the
assumption -y(a)
d2 a_S2 L(f(s,
on
.Yz, ( (a), (a)), (a))
a
tensor and
Lemma 4.6.8
-
infinite dimensional
analogue
1,L
of -Y
(respec-
to the Hessian of
function.
Definition 4.6.4. Let
Z1, Z2 be submanifolds of M or points in M and which connects Z, and Z2 and intersects both geodesic [a, b] I submanifolds orthogonally. Denote by TZ'l,Z2 -y be the space of piecewise smooth vector fields along -y which are tangent to El at a and to Z2 at --
M be
a
b.
The energy index form is the bilinear
form
224
Pseudo-")
4.
E, -Y
-nnamfolds
I
Ti
R,
'T ,,.27
X
( (V 6 V 6 ) i
either
(R( l
('Z2( l (b), 2 (b)), (b))
+
If -y is length
+
spacelike or timelike defined by
and
dt
(1z, ( l (a), 6 (a)), (a))
-
satisfies
1, 11, the
77 E
index form is
IL11'' Z2 -1 11' Z2 'Y
X
-
/
1 -Y -r,'l,-2
V41j-, V 62-L )
77
+ 77 -
R,
-->
Tj
(R( l, )6,
+
dt
(Iz2( l (b), 6 (b)), (b)) (1zj (61 (a), 6 (a)), (a))
Corollary 4.6.3. Let 7 be a geodesic from Zi to Z2 which intersects these submanifolds orthogonally. The index form IL,-y is positive semiZ1, Z2 definite if -y minimises length and negative negative semi-definite if -Y maximises length. ,
Proof.
For L the assertion follows from the
d
L(f (s, -) The
L(-I)
=
proof
+
s( is-
for the energy
Taylor expansion
d
8
L (f (s,
-)
integral
)'=0+2 is
exactly
dS2 the
L(f(s,
.))
+O(S3). 8=0
I
same.
following lemma summarises in which cases there is a non-trivial extremising problem for E and L. In particular, it implies that the extremising problem has only in the Riemannian and Lorentzian case nonThe
trivial solutions. Lemma 4.6.9. Let
(M, g)
be
a pseudo-Riemannian manifold, 'Ply Z2 M be a geodesic boundary, and -y: [a, b] which connects Zl with Z2 and intersects both submanifolds orthogonally. If -y is a null geodesic assume in addition that,,7y(a) Z Ty(a)ZI and (b) V T-y(b)Z2-
be
submanifolds of M
without
--
E'-Y
is positive (respectively, negative) semi-definite then signature (+ +) (respectively,
N if 1 , Z' has
(ii)
Let
E,-y, -L
1 1,Z2
be the
bilinearform
E'-Y
1 1, Z2
positive (respectively, negative) definite +) (respectively, g has signature (+
is -
restricted to then either
or -
g has
signature (-
+
+) (respectively,
-L. If
g
TE,'Y,-L L
Z, Z2
4.6
7 is causal
-
53b 52
-
-y(b) (iii) If
are
(respectively, -y is spacelike spacelike (respectively, 5'b
the index
signature (+ g
form IL,-y ZI,Z2 ,
+)
energy functionais
or
null)
32
timelike)
are
-Y(a),
at
positive semi-definite then
is
225
g has
or
L,-y
index
(iv) If the
Length and
form has signature (-
is
negative semi-definite then
either
+)
+
7 is timelike
Z1, Z2
are
spacelike
-y(a), -y(b)
at
or
g has
+)
signature spacelike Z1, Z2 are timelike
-
-y is
-
-
Proof. Let J
E
every k E N
we
R
and
\ 101
v
be
vector with
a
parallel translation
vector field vanishes at the
The
TI(a)M
E
consider the variation vector field
where V is the
a) b-a
-y(a), 7(b)
at
of
v
Wk (t)
along
J. For (v, v) 1 V (t) sin ((t k =
=
-
-y. This variation
endpoints
of -y and is therefore in El "'Z' 'Y
-
equation
jb V Wk, V Wk (R(Wk, ) a) b-a) jb b-a) ) sin2((t -a) (R(V, ) )))
E'-Y
1 1, Z2 (Wk, Wk)
Wk,
+
a
2
Ir
J
k7r
Cos2
a,
k7r
I
V,
+
V
implies
that for
sufficiently large k the integrand
sign(IE"y,Z2 (Wk, Wk)) Assertion
(i)
b
=
follows
has the
-
a
sign
as
6,
("Y(a), v)
=
0.
same
sign(6). immediately
from 6
=
(v, v).
For the rest of the prove we will assume in addition that 0 for all t E [a, b]. Consequently, we have (Wk (t), (t)) =
(ii):
Assume that
IE,'_I,Z2 'J-
is
positive semi-definite and suppos6e either
that -
-
2-dimensional
there is
a
negative
definite
there is
a
or
I-dimensional
definite and that -y is In both v
I
subspace of T,,M restricted
cases
there is
subspace restricted spacelike.
a
vector
v
E
Ty(a)M
to which g is
with J
=
that either g is Riemannian
or
negative
(v, v) < 0 and TE,-,, Icannot be positive :=
-I and ,,. Hence sign (I E"'Y'-L ,Z2 (Wk, Wk)) ZI,Z2 our assumption, and therefore
semi-definite. This contradicts
-
to which g is
that
it follows
226
-
4. Pseudo-Riemannian manifolds
g is Lorentzian and -y is causal.
Since
(a)
TI(a) -"I ( (b)
the submanifold must be semi-definite _T E,-y,l
sics. For
1 1,Z2
large enough
at
completely analogous.
is
Z1,Z2
The index form
and Zi (Zb) is orthogonal to -y(a) (-y(b)). The proof for negative
Ty(b)Z2)
spacelike is
defined for spacelike
only
k the relation
L,-y IXE (Wk; Wk)
Wk
,
L,-y
Sign( 1 1, -, (Wk, Wk))
=
2
or
timelike
geode-
implies
771EEl"Y
2
and therefore
J_
Z2
(Wk7 Wk)
sign(,qJ).
Assume that g is not definite. Then v can be chosen such that L,-y This implies that 1 1, _,2 is not positive semi definite either. -q. There is to (iv): nothing prove if (M, g) is a 2-dimensional Lorentzian
(iii): 6
=
manifold. -
-
Suppose
g has not
signature (-
dim(M)
Then
v
+
-
-
-
+) (respectively (respectively, timelike).
or
that
: 3 and -y is spacelike
can
sign(,q6)
that either
be chosen such that
I for
=
Tj and
large enough k implies
L,-y
sign( 1 1, Z2 (Wk) Wk))
that I L,-y is not Z1,Z2
negative semi
definite.
I
4.6.4. Let
Corollary
(M, g)
be
a
Riemannian
or a
Lorentzian
manifold
and 7: [a, b] -- M be a spacelike or timelike geodesic which connects Z, with Z2 and intersects both submanifolds orthogonally.
(j) If with
L, -y
is definite then for all variations f: [-e, E] 2 non-vanishing variation vector field there is a 6 r
x
E
[a, b] (0, c)
-->
M
such
that
L(-y) L(7)
-
-
< >
L(f(s, L(f(s,
Vs E
Vs E
[-6,6] [-6,6]
in the Riemannian case, and in the Lorentzian case.
L,,y is not Z1,Z2
ii) Ifi
[a, b] L(7)
>
L(-y)
-
-
Proof.
M
L(f(s, L(f(s,
To prove
vector field
.
semi-definite, there of -y such that
(i)
let
Since
Vs
f be
and
1 15 2( _' _)
< 0.
variation vector fields
L(-y)
+
jS2 1 L'I,Z2
2
In Lemma 4.6.15 below
f: [-E, E]
x
in the Riemannian case, and in the Lorentzian case.
variation of -y with
L(-y)
L
7
proof of (ii) let
L
a
[-E, El [-c, c]
L(f(s,
assertion follows from
For the
E
Vs E
exist variations
+
non-vanishing variation IS2,L,-y + O(S2) The 2 Z15Z2 ( ' ) .
0 and Lemma 4.6.9.
variation fields with
"Y IL Zl,--2(
+' +)
> 0
By Lemma 4.6.2 there are variations f of -y with . Now the assertion follows from L(f(s,
+0(82). we
will extend this result to null
geodesics.
4.6
Conjugate and
4.6.2
focal
Length
and energy functionals
227
points
points is closely linked to semi- definiteness of that Z1, Z2 are points. Before investigating this relationship we will need to collect a few important facts about Jacobi vector fields in pseudo- Riemannian manifolds. The existence of conjugate
the index form in the
case
Lemma 4.6.10. Let -y:
fields which
(V J' j
vanish at
[a, b]
--->
M be
point to
some
a
geodesic
[a, b].
E
Then
and
have
we
J be Jacobi
J,
(J, V j)
-
It follows from
Proof.
(R(J, (R(j,
V J,
that
J,
V j)
V J' i
-
diate consequence of
)J,
+
) J,
+
V J, V J) V j, V J)
J,
V j)'
is constant. Hence the assertion is
J(to)
=
i(to)
=
0
an
imme-
I
.
that it is possible to split Jacobi fields into a tangential orthogonal part unless -y is a null geodesic. Moreover, the tangential part is always trivial. We show
and into
now
an
[a, b] all for 11 (t) (t)
Lemma 4.6. 11. Let -y: -y with
along
(i)
The vector
along
field
-->
M be
t E
[a, b],
-y is
a
a
be
geodesic,
and J be
Jacobi
a
Jacobi
vector field field along -Y.
a
field if and only if there
are
numbers a,,3 with (t) (at + 0) (t). (ii) The following statements are equivalent. =
(a) (J(t), (t)) 0 for all t E [a, b], (b) there are two different numbers c, d E [a, b] with (J(c), (c)) and (J(d), (d)) 0, (c) there is a number c G [a, b] with (J(c), (c)) 0 and 0. (c) J, (c) =
=
0
=
=
'
)
(V
Proof. (i):
If
11
=
we can
write
(t)
=
O(t) (t).
equivalent
(ii):
to
(t)
=
R(O(t) ,
0.
The assertion follows W (t)
satisfies
Since
0
0 which in turn is
the Jacobi equation reduces to
once we
=
have shown that
(J(t), (0)
0. But this follows from
4. Pseudo-Riernannlaji mauiiold,,
228
-0
(J,
V V
V
(V V J, )
=
7
=
)
J,
(-R(J, ) , )
J,
+
=
V )
0.
I
Corollary 4.6.5. spacelike and J be jT
and J
to
1
Let y: [a, b] --> M be a geodesic which is timelike or Jacobi field along -y. Then the orthogonal projections
a
J-
to
Proof. Without loss jT is given by
of
Then
,V
are
q
generality
(J, ) .
(V J) T.
J)
also Jacobi
fields along
assume
Since 7 is
In the
same
a
way
that
-y.
(, )
geodesic we
=
1, If.
77 c-
obtain V jT
we
1:Y
get T
V ,7 jT From
jT
we
obtain
=
(V VA
R( jT, )
=
0 and therefore T
,V V jT + R( (jT,
V V jT
=:
(V V J) +
=
where j
we
have used
jT and
_
R(.,
V V jT
assertion follows
now
=
from
=
0
we
0
get
R(J-L, )
(R(J, ) )T
=
0,
(cf. Proposition 4.3.1).
From J-L
V V J`
The second
=
=
R(J, ) .
V V J.
I
Let -y: [a, b] --> M be a geodesic and J be a Jacobi field which vanishes at a and c E (a, b). There is a geodesic variation f : (s, t) F-4 f (s, t) of -y with variation vector field J
(cf. Proposition 2.9.1 and Corollary 2.9.1). geodesics pass through -y(a) and "intersect -y(c) to first order" though they may not actually meet this point. Up to second order the geodesics segments f(s,')[a,,] have all the same length as 71[a,c]- It is therefore plausible to expect that 7 will not extremise length beyond These
7(c). A
typical example where the geodesics meet in both points y(a) and given by those great circles of the unit sphere sphere S2 which intersect both the south pole at -y(a) and the north pole at -y(c). Since in this example all neighbouring curves intersect -y(c) at an angle different from zero, they can be deformed so that they meet 7(b) and have length shorter than -y (cf. Fig. 4.6.3). This is also true in the general case, even if the neighbouring geodesics do not actually intersect 7(c)
-y(c)
(cf.
is
Theorem
4.6.1).
4.6
Length
and
energy'functionals
229
For
arbitrary submanifolds Z1, Z2 an analogous statement cannot simply no canonical way to compare different submanifolds along a curve. Hence we cannot speak of a "first point where a given normal geodesic fails to minimise length between Z, and Z2". However, we can ask at which point a geodesic -y orthogonal to be formulated. There is
a
submanifold Z fails to minimise distance from Z. In order to
this question
points
to
we
will need to
pair which
a
consists of
a
Definition 4.6.5. Let Z C M be a
with
geodesic
called
(i)
E
V J(a), v)
(ii)
and y: [a, b] --> M be Ty(a)Z. The point -y(c) is
submanifold
E
Ty (a) -", J(C)
J (a) E
a
Z, (a) (Ty(a)Z)-L \ of Z along -y if there is a Jacobi field
7(a)
focal point
a
+
Observe that in the
J
along
-y with
0,
: --
(I(J(a), v), (a))
case
answer
the concept of a pair of conjugate submanifold and a point.
generalise
that Z is just
=
a
0
for all
v
E
single point,
T- (a) -"condition
(ii)
is
empty and the definition reduces to the definition of a pair of conjugate points. The following lemma explains why we also demand condition (ii). Lemma 4.6.12. Let Z be
a
submanifold of
M and -y:
[a, b]
--+
M be
-y(a) Z, (a) (T-y(a)Z)-L \ T-y(a)ZM is a variation of -y through geodesics If f : (- E, E) x [a, b]
geodesic with
E
-
to Z then the variation vector
thogonal
Jacobi
(i) (ii)
(a)
(iii)
V, 4(a), v)
is
a
E
a
thogonal
is
or-
satisfies
field,
be
+
(.Y( (a), v), (a))
=
0
for all
v
E
Ty(a)Z-
vector field along -y which satisfies (i)-(iii). Then f : (- c, E) x [a, b] -- M of -y through geodesics orwhich has variation vector field . a
variation
to Z
Proof. "=*": Assume that f That
field
T-y (a) Z
let
Conversely, there is
a
E
a
is
follows from
f (s, a)
c
Z for all
s.
V, 4
V
through normal geodesics. Proposition 2.9.1 and property (ii)
variation
a
Jacobi field follows from
The equation
f
implies that
=
f
atfs
for every vector field V
=
V
C%ft
tangent
to Z
=0
V)
=
f
f
ft,
V
=V
S
(ft, 1 (fl, V)) holds. This proves
(iii).
(ft, V) S'-'
a
f
-
ft, 9SV)
230
4. Pseudo-Riemannian manifolds
Let y: a
vector field
fine
(-c, e)
along
--->
Z be
a
V(s)
I
p with
with
curve
T,(,)Z
A(O) V(O)
=:
and
=
(a) and V be (O). We de-
f(s,t) exp(tV(s)) thereby obtaining a variation of -y through geodesics normal to Z. This variation has variation vector field if and onlyif f,(O,a) (a) and7 (a)fs '7 (a) - We clearly have f,(O, a) =
=
A(O)
(a)
=
=:
have to choose V such that
possible,
let
Then
have
we
A(O)v
=
for any choice of V. Since
V(s)
=
=
V(O)
17A(O)V
PjYJ[O'S] =
=
'7 (a)fs Vfft '7A(O)V V (a) . To that this is always =
see
(7 (a) )
(a)+sPj-
pqo,s]
(a) and, using
'7A(O)P1-Ltj[O'S]
(a)
+
we
=
(cf.
Lemma
4.4.5).
Lemmas 4.4.4 and 4.4.5,
P/_'-tl[0,0]
( V (a) )
O
('7A(O)Pit1[O'Sj (-Y(f,, '), ft)'
where the last
inequality follows
(1 Yll W), ft)
=
=
for all vector fields W which
(a)
(7 (a) )
+
from f
-
(7
+
V
-
f
W
OSW' ft
(W, Vf atfs) tangent
are
V'%ft
W,7 (a) )
to Z.
If -y is not a null geodesic then the neighbouring geodesics provided by Lemma 4.6.12 are of the same causal type as -y. However, if -y is null then this property is not guaranteed. The following lemma clarifies the situation for null
geodesics.
a submanifold of M and -y: [a, b] -4 M be a I Z, (a) E (T-Y(a) Z) \ Ty(a) ZLet be a vector field along -y which satisfies properties (i)-(iii) in M of -y Lemma 4.6.12. Then there is a variation f : (- c, c) x [a, b] through null geodesics orthogonal to Z and with variation vector field 0 for all t E [a, b]. if and only if ( (t), (t))
Lemma 4.6.13. Let Z be
null
geodesic
with
-y(a)
E
-+
=
Proof. Assume that f
is such
a
variation
through
f
equation (ft, ft)
0
therefore
(a)
imply
0 for all t E
implies
)
=
0
V
alft,
0. Lemma 4.6.12
[a, b]
-
ft) (ii)
null
geodesics.
The
Vf atf, ft )
and
and Lemma 4.6.11
(ii)
4.6
Conversely,
that
assume
is
(iii)
of Lemma 4.6.12 and is
f
in the
as
proof of Lemma
Length
and energy functionals
vector field
a
orthogonal 4.6.12 but
along
-y which satisfies
231
(i)-
to -y. We construct the variation
will choose V
now we
so
that it is
null at every point of /,t. To do so, let s -4 W(s) E Ty(,,)M be a curve with 0 for all s. As in the proof of Lemma W(O) (a) and (W(s), W(s)) =
=
4.6.12 let M: (a). We set
and
-4
s
V(s)
M(s)
Z be
with
tt(O) -y(a) and A(O) P'AJ[O'S] W(s). Clearly, V(s) is normal to Z at all
=
VA (V (s), V (s))
E
=
2
a curve
(V AV (s), V (s)
=
2
( (V V(s))
=
s
V (s)
0 for all s. Hence f (s, t) implies (V(s), V(s)) exp(W(s)) is a null through geodesics normal to Z. We have to choose W such that the variation vector field of f coincides with . Since we have f,(0, a) #(0) (a) we only have to arrange W such that 0
=
=
variation of -y
=
=
f
at f, (0, a)
17
=
174.
,
From
(I (f, .), ft)o
-
(V*) )
=
(cf. proof
of Lemma
4.6.12)
we
obtain f
atfs)
f
=
(0,a) =
( '9"ft)
(0,a)
(0,a)
(VAV(S))-L(O,a)
(I (A (0), .), V (a))
-
T
(VAV('5))-L(O,a) + (7 (a) )
(VA(O) (P J[O'S] W(S) I
(V
A( PI/-'1 [0, S] W(O))
+
+
(V (a) d
P-L AJ[O'S]
18=0
( Fs W(S)) JS=O)
T
+
(a)
)
is- W(8)
)
(7 d
(V4,
We have
(a)
Since
I
:
)
=
0 since
is
Ty(a) I' we get therefore
space of the null
T-y(a)
(a)
(7
+
( (a), X)
C-y(a)
cone
=
01.
(a)
)T
JS=O
C
Hence
orthogonal
to -y
( (V4) (a))
Ty(a)M
(V4)
,
at the
is
a
point
tangent
(Lemma 4.6.11). =
0. The
(a)
tangent
just fX
is
vector to
C-y(a)
E
at
I
(a)
and
we can
choose W such that
d ds
W) I 8=0
f
implies
tfs (0, a)
V q
=
V4
and
we are
done.
V4)
.
But this
4. Pseudo-Riemannian manifolds
232
Fig. 4.6.3. Conjugate points sphere
We need another technical lemma Lemma 4.6.14. Let Z be
[a, b]
-y:
J1, J2
--+
are
M be
Jacobi
==
0
for
J1 (t) t E
Proof.
Rom the
,
the
to Lemma 4.6.10
pseudo-Riemannian submanifold of M and orthogonally at -/(a). If
which intersects Z
fields which satisfy Ji (a)
(I(Ji (a), v), (a))
for all
a
geodesic
a
analogous
on
all
v
E
C-
Ty(a),E,
V (t) J2
Ty(a) Z
then
and
(17 Ji (a), v)
+
J1, J2 satisfy
V (t) JI
i
J2 (t)
[a, b]. proof v
of Lemma 4.6. 10
J1) J2
)
-
JI
we
i
V
know
J2
))
0
-
Hence the assertion follows from
(V (a)jl, J2(a)
-
Ji(a),7 (a)h
(I(JI (a), J2 (a)), (a))
+
I(J2 (a), J1 (a)), (a))
=
0.
I We
can now
present
a
theorem which links focal
points
to
length
extrem-
ising geodesics.
(M, g) be a Riemannian or a Lorentzian manifold, M be a geodesic which submanifold, and -y: [a, b] orthogonally at -y(a). If (M, g) is Lorentzian we also assume
Theorem 4.6.1. Let Z be
a
Riemannian
intersects Z
that -y is timelike
--*
4.6
(i)
The
submanifold
Length
Z does not have
and energy functionals
focal points along
-y
if
233
and
only
if the index
form
L,-y
positive semi-definite in the Riemannegative semi-definite in the Lorentzian case. If L, -y 1 ,J-y(b)} (VI V) 0, then there exists a function vo with V vo , i.e., V corresponds to a reparameterisation of . (ii) The point -y(b) is the only focal point of Z along -y if and only if the index form is positive semi-definite in the Riemannian and negative semi-definite in the Lorentzian case. there is a non-vanishing vector field V: [a, b] ---+ M along 7 which L,-y 0 and has values in satisfies 1 , ,y (b) II (V V) is
nian and
=
-
-
=
(iii) if
There is
the index
focal point -y(c) of Z along form is not semi-definite. a
-y with
c
b
2.
E
have
we
the
We
independence of the Jacobi fields i)n-1 VI ( V ) > 0 unless i)2 -y (b) I linear
at every 0. Since
=
therefore V
E,-y,l
' ,J-y(b)j (V V)
0
=
7
v'J1
=
which is
implies that V2
parallel
237
point
a
V(b)
0
Vn-1
0 and
=
to
(ii) and (iii) (ii), (iii).
of assertions
proof
of Theorem 4.6.1
proof
and energy functionals
'Y,-L
equation The
Length
is
to the
completely analogous
I
geodesics in the case that one Z1, Z2 degenerates to a point. This is achieved by there is a variation of -/ through timelike curves from the
extend Corollary 4.6.4 to null
can now
of the submanifolds
showing that spacelike submanifold that
we
cannot
use
Z to
the
-1(b)
if
Z,J-y(b)J
argument
same
as
semi-definite. Observe
i 11 ot 'S in
Corollary
4.6.4. While
we
f with E(f (s, .)) < 0 for all 8, be clear that these varied curves are everywhere timelikel.
would obtain the existence of it would not
a
variation
(M, g) be a Lorentz manifold, Z be a spacelike subM be a null geodesics which intersects Z orb]
Lemma 4.6.15. Let
manifold, and
thogonally. Z to
from
-y: [a, E, -y, I
ff ,J-y(b)j
first focal
a
(0, b
J c
not
-y(b) arbitrarily
Proof. The strategy there is
is
c)
-
semi-definite then there
a
timelike
curve
close to -y-
of proof is as follows. Theorem point 7(c) (c E (a, b)) of Z along
will construct two vector fields
we
is
of -y with 71(a,c+61 such that for every variation f
4.6.2
that
implies
-y. For
small
some
(t) and A(t) along (t) and f,(O,t) =
f
(0, t) f,
V
=
A (t)
we
have
(ft (t, s), ft (t, s))
to
f (s, a)
join Z
curve can
E Z
from Z to
curve
for
s
> 0
and
a
variation
which,
sufficiently
addition, f (s, c + 6) 7(c + 6) for all s. It is then possible and -y(c + 6) by a timelike curve arbitrarily close to -Y. This in turn be slightly deformed in order to arrive at a timelike
small. We will show that there is such satisfies
< 0
in
=
-y(b).
Observe that A cannot be chosen
completely independently
of
.
f
fact,
-y(a)
at
1( , ). I 2
E
Z
From the
(
we
proof
d' ds
2
( ft ft ,
have must have A'
))
Lemma 4.6.7 f
we see
f
V at fs, V
at
atf,') (
f
f
a 'fs
Vf atfs + R(f,, ft)ft,
f,)
+
,
+
V at f",
f.
)'
LI)
that
(R (f,, ft) f, ft)
Vf
_Y (h I
V at V
1 '=() +
(V a 'f')
=
ft)
.
+
f
f
V'9t
V
a'fs
,
ft)
In
238
4. Pseudo-Riemannian manifolds
holds. If
we can
construct -vector fields
, A
with
?
V, V
+
) >0_ V4,
R( , ) ,
__
V
)
A,
> 0
then any corresponding variation f satisfies (ft, ft) < 0 for small 8 > 0. Let -y(c) (c c (a, b)) be the first focal point of Z along -y and let J be a Jacobi field according to Definition 4.6.5. It follows from Lemma 4.6-11
(ii)
that this Jacobi field is
If there would be
field
J(t)
-
t
'
(t)
d
a
point d
would have
everywhere orthogonal
(a, c)
C
with
J(d)
at d and
a zero
of Definition 4.6.5. Hence there would be
=
a- (d)
to -Y.
then the Jacobi
all the conditions
satisfy
focal point -y(d) of Z along in contradiction to the definition of c. We have therefore
-y before
-y(c) J(t)
shown that
( (t))-L \ R (t)
E
The derivative of J at
c
a
for all t c-
satisfies
V.,J(c)
(a, c). E
T7(,)M \ R (c)
since J
is non-trivial and not
such that
is
c
parallel to . This implies that there is a J > 0 the only point in (a, c + J] where J is tangent to - . Hence spacelike vector field U along -y with value in ( )' and a
there exists
a
function p:
[a, b]
-
-
-
-
R such that
I for all t E [a, c + (U(t), U(t)) J(t) p(t)U(t) for allt E [a,c+J], for alltE (a,c). >0 W(t) for all t E (c, c + 0 W(t) < =
=
We will
be
-4
a
now
by slightly stretching U. 6 (W + O)U. From
construct
function and consider V
V4 + R( , ffzy
=
=
Let
b: [a,
c
+
J]
-->
R
=
V V, (OU) + OR(U, OU
+
20V U + O(V V U + R(U,
(o
+
0)
We get
(V V4 There is
a
for all t E
+
R( , ) ,
)
=
numb er A 1 > 0 such that
[a,
c
+
J].
Let A2
( -
+
0
(A 1) 2
> 0 and
0(t)
(V V, U
called
are
and t E
geodesic
a
equivalent,
v
-
+
w
[a, b].
Two
if there is a number of equivalence classes
w,
a (t). We denote the space ( (t)) I by [ (t)] ' and set [ ] -L UtE [a,b] NO] _L
E
v
(t))
[a, b]
-L
=
*
*
A map R
along along 7.
[ (t)]'
[A](t):
x
...
-L
[ (t)]
x
-y which is linear in each
(mol-L)
x
of
x
its entries is called
From the definition it is clear that any tensor A of [A] at t via
(mol
x
...
tensor class
a
( (t))-L
induces
a
tensor class
[A]([vi], where
w ( )
cp' =
.
-
-,
is defined
[v,,', k,,'], by
[W']([v])
0). Conversely,
[ o'])
-,
=
V'i,(v)
=
A(vi....
for all
Wil
...
(P r)
I
( (t))-L (In particular,
E
v
vs,
i
any tensor class is induced
by
tensor in this
a
way.
The metric and the covariant derivative in direction gous
objects
Lemma 4.6.16. Let -y be
andAbe
induce analo-
for tensor classes.
any tensor
geodesic
a
field along
and
[A]
be
a
[A]= [A].
-y with
tensor class
along
Then
V A is
[A]
:=
well
defined. If -y is a null geodesic
[g] Qv], [w])
:=
g (v,
The operator
Proof.
defined w) [R]: [ (t)]-L [ (t)]-L, [v]
and positive
-4
For any one-form W
(Vxt),P) Mt)
[g]: [ftL
then the metric
is well
+
0
-4
definite.
[R(v, ) ]
1--4
satisfying W( )
R, [v, w]
x
is well
defined.
have
we
f (tMt))
V (t) (w (V (t)
f W (W)
+
(V
-
(t) (V W
+
f W W)
V (t)(w(V(W) -
(V (t)V(t)
W
V (t)MVW) where have
we
have used that
[V, (V(t)
+
+
-
V,
f (t) (t))]
=
df (t) (t)
w
+
(Vxt)V(t))
f
V (t) (t)) =
V WMWI
0. For any vector field V
=
[V, V(t)
+
df (ffy
+
fV
]
along ==
-y
we
[V V(t)]-
Hence the first assertion holds for
1-forms,
for functions
derivation this proves the first claim.
along
-y. Since
V
is
a
The second claim follows since there is
such that k E
g(ei,ej) n 11.
=
6ij
for
i,j
E
vector
basis el, 21 and
a
11,...'n
fields,
-
.
.
.
,
a
consequence of
R( ,
(trivially)
e,,- 1 of
9(en-liek)
-
The third assertion is
and
0.
(t))
-L
0 for
4.6
and energy functionals
Length
Definition 4.6.7. A Jacobi tensor class is
[]
J-
along
which
for
7
[A]+ [R] [A]
if
and
there is
only
[01
a
tensor class
a
[A]:
holds.
[A]: [ ]J-
Lemma 4.6.17. A tensor class tensor class
=
245
[ ]' along
-->
A
field
tensor
and has the property that t i-+ AV (t) is ' vector field V with values in ( )
[A]
a
along
-y is
Jacobi
a
-y which induces
Jacobi field for every
parallel
.
Proof. Suppose that A is a tensor field along -y such that AV Jacobi field for any parallel vector field V with values in immediately from 17
V [A]
[R]
+
are
is then
[A]
exactly (2n
a
+
Jacobi tensor class.
.
v
T,'-'v'Ej
=
and the
observe that there
Conversely
-
.
.
,
and define the tensor field A: where
a
+
=
2) linearly independent Jacobi values in ( ) -L. Let f El, E, I
which have
is
It follows
[A]) [VI [ (V V A) V] [R(AV, [17 V (AV) R(AV, ) )] =
that
( )'
E
vector fields
Ji along
be
frame of
parallel
a
by Av
Aj
are
=
E'_1 i,j=i AiviJj i
(constant)
real numbers. It is
( )J-
the vector field AV
clear that for every parallel vector field V is a Jacobi field with values in ( )-L. Hence
[A]
G
[A]
is
a
Jacobi tensor class.
Since the differential equation 0 implies that the space of + [R][A] Jacobi tensor fields is 4(n 1)2 -dimensional if -/ is timelike or spacelike and 4(n 2)2 -dimensional if -y is null every Jacobi tensor field can be =
-
-
generated by
some
It is clear from the tensor class with
fields
proof of
constructed above.
I
Lemma 4.6.17 that the columns of
to
respect
as
parallel
a
basis of
( (t))-L
are
a
Jacobi
just
Jacobi
in this basis.
expressed
Corollary
tensor field A
4.6.6. Let
-/:
[a, b]
M
--
be
a
geodesic
and to, t,
E
(to =A ti).
[a, b] (i)
For any
pair of
[Ao]: there is
a
tensor classes
[,,:y(to)]J-
-4
[ (to)]-L ,[Aol: [ (to)]
unique Jacobi
[A] (to) Ao; (ii) Assume that
tensor class
[A]
with
[ (to)]--L [A](to)
=
[Ao]
and
=
[Ao]: be
a
-y does not have
[, /(to)]J-
given pair of
tensor class
[A]
--+
conjugate points and let
[ (to)]J-,
tensor classes.
with
[A](to)
=
[Aol
[A,]:.
[ (ti)]J-
Then there is
and
[A](ti)
=
---+
a
[ (ti)]-Junique Jacobi
[Afl.
4. Pseudo-Riemannian manifolds
246
Proof The sitions
assertions follow
immediately
from Lemma 4.6.11 and
Propo-
2.9.2, 2.9.4.
1
The
following lemma is clear from the definitions and the fact that non-vanishing Jacobi field which is parallel to has at most one zero:
a
Lemma 4.6.18. Let -y be a geodesic. Two points -y(c), -y(d) are conjugate if and only if the Jacobi tensor class [A] which satisfies [A](c) [0], ==
[A]](c)
=
id is
singular
at d.
Definition 4.6.8. Let -y be a geodesic and a tensor class along -y. Then the adjoint of
denoted
by [B]
[a, b] [A] (to)
-y:
=
M
and
be
[A]
assume
is
a
Jacobi tensor class
that there is
( AV, 17 AW)
non-singular
-
=
v,
w
E
=
( )-L.
geodesic
[a, b] with [A] [A] -I is
at t then the tensor class
0
( )--L.
-y with values in
The
17 AV, AW) (17 AV, 17 AW) AV, V V AW) (17 17 AV, AW) 17 AV, 17 AW) +
-
A(to)
a
E
at all t.
=
and
is
along
number to
a
Proof. Let V, W be parallel vector fields along equations
'7
be
to
-
[0]. If [A]
self-adjoint
[ (t)]J[g].Y(t)
-+
*
Lemma 4.6.19. Let --+
[B](t): [ (t)]-L [B] with respect
-
(-R(AV, ) , AW)
(Av, 17
imply that
(AV, -R(AW, ) )
-
17
Aw
Av, Aw
)
=:
0
for all vectors
It follows that
17 AA`v,w
(17 A(A'v),A(A-1w)) (A(A-1v), 17 A(A-lw) v, V AA-lw) .
self-adjoint tensor class [A][A]-' has a direct geometrical interpretation in terms of congruences of geodesics. Let -y: 1a,b] -4 M be a spacelike (in the Riemannian case) or timelike (if M is Lorentzian) M, (81,...,sn-l't) 8n-1, t) geodesic9 and f: R'-' x R f(81
The
-4
,
)
'
The
interpretation
in for
lightlike geodesics
is
slightly
I
*
I
less direct.
1
4.6
be
a
smooth
that each
(n
247
1)-parameter geodesic variation of -y. We may assume t sn- 1, t) satisfies (ft, ft) E I 1, 11 and f (s', the vectors I ft, fi, a f,. 1 1 are linearly independent. ft(s',...' sn-1, t) is a well defined vector field near -y(a). i-+
-
.
.
.
,
.
Then U
energy functionals
-
geodesic
that at t
Length and
.
.
,
-
The covariant derivative of U geometric properties of our congruence of geodesics. The function 0 div(U) measures the divergence of neigh=
dO is the infinitesimal rotation bouring geodesics. Analogously, w and a, the traceless, symmetric part of VU, the infinitesimal, volume sn- 1) 0 preserving distortion of neighbouring geodesics. At (sl, =
=
.
we can recover
The Jacobi field
V ft fs, ft)
=
an
such that
fk(O,..., 0, t)
a
arbitrary function,
--->
( )'
It follows that
( )J-
(V A)A-lf,i
=
is
(V ':Y A)A-1v
(V A)Ej
for all
f fl, E
v
Definition 4.6.9. Let Then the
=O for all
tensor
by
t +
h(sl,
.
.
.
,
sn- I ) I
always parameterise our geodesics parallel along
alongy that
maps
f,,,,, I
.
=
f Ej(t)jj=j,...'n-1 -/ and denote
by
Ej into fi, (0.... 01 t). 1
span
Vftfi
( )--L
-J-.
This motivates the
[A]
be
expansion 0 of [A]
vorticity
t
for all k and t. Let
V (AEj)
0 the
0.
=
Jacobi tensor class and that
a
Since the vector fields
1
which is
the tensor field
[A]
we can
( (t))
E
orthonormal basis of
( )'
,
=
where h is
A:
V fS ft, ft )
=
0 implies (fi.(0,...,O, a), (a)) [a, b]. Since we can replace the parameter
Hence
be
.
fi satisfies
Vft Us, ft) t E
.
this information in terms of Jacobi classes.
a
is
we
=
Vfsift
VfSi U.
=
conclude that
following
Jacobi tensor class
along
VvU
definition. the
geodesic
-y.
defined by
QA] [A] 1), -
=
tr
of [A] by
w
2
and its shear tensor
a
by
([,4][A]-' ([A][A ([A] [A] ([A] [A] +
2
The is
following singular.
+
lemma
implies that O(t)
0(t) id
if
0 n-2
if
;T-I
id
tr([AI [A]-')
E
0.
diverges
where A
4. Pseudo-Riemannian manifolds
248
Lemma 4.6.20. For any
0
Jacobi vector class
a
-_
[A]
we
have
-(det([A]))*, det([A])
where det is any parallel determinant function. (In particular, choose the determinant function induced by the metric [g]). Let
Proof.
r
n
=
-
1 if -y is
spacelike
timelike and
or
r
n
=
-
one
can
2 if -y is
null. Since the space of parallel determinant functions it is clear that the formula in the assertion is
along -Y is 1-dimenindependent of the
sional,
choice of det.
Assume that
[A](to) is non-singular. There is a parallel linear tensor [A] (to) [B] (to) id. Let [C] [A] [B] and f [Ei] I j= be a parallel orthonormal basis of [ ] 1 We choose the determinant function defined by det([D]) where Det is the standard deDet(QDji)i,k), k terminant in Rr and [D] [Ek] At t to we have [C] [Ek] [D]ik [Ei]. [Ek] and Det([El],..., [EI) 1. This implies class
[B]
such that
=
=
.
=
=
=
=
=
(det[C])'It=t,,
(detQC][Ej],...,
=
[C][E,,-,])'It=t,,
r
[i ] [Ei], [Ei+,],..., [Er]) It=tO
det([El],..., [Ei- 1], i=1
n-1
E [C]i det([El],..., [Ei-I], [Ei], [Ei+,],..., [Er])It=t,,
=
i=1
tr([C])lt=t,,. Since
[A]
(det [A])'
=
=
[C][B]-1 (det [C]
we
obtain therefore at t
det [B]
-
=
At to
we
==
(det [Cl)'
1
tr Qi ] [B] ([i ]) det [B] tr Q-A] [B]) det [C] -det [A] -
=
1)
tr
have
-
1
=
=
[C]
Lemma 4.6.21
=
id and therefore
tr
=
to
det [B]
-
1
[B]) det [C]
([A] [A]
(det [A])
*
=
(Raychaudhury equation).
-
1
tr
-'
det ([C] [B]
[C]) det [C]
-
1
det [A].
QA] [A] -') det [A].
Let
(M, g)
be
a
I
Lorentz-
Riemannian
manifold and -y be a causal geodesic if (M, g) is Lorentzian and a spacelike geodesic otherwise. If [A] is a Jacobi tensor class then its expansion satisfies ian
or
-Ric( ,,:y) -Ric
-
tr(W2) tr (w 2)
-
-
tr(,72) tr (U2)
_
_
n1102 n1202
if
if
E
f -1, 11, 0.
4.6
Proof. Let Let
r
=
Jacobi class -
n
[R]
spacelike or timelike and r parallel orthonormal frame of (
a
have
we
QAJ [A] 1)'
Q'41 [A] 1) 2and
=-
-
1
-
tr
=
r
=
vector
n
-
-
0
Y g (R(Ei,
find
) , Ej)
tr
-
w
+
+
a
-
Ei'=, g (R(Ei, ) , Ej) vector field
parallel spacelike field E, such that I El, E, I
E,,. Then
we
[Al
is
a
[A] [A]
tr([A] [A] 1)2
1 it is clear that
we can
2 if -Y is null.
Since
(([A] [A]-')
i=1
is null
-
-
r
If
n
)-L.
[A] (- [A]
+
r
=
=
249
therefore
(tr ([A] [A]-'))' -tr([RI)
[A] [A]
-
-
_
energy functionals
1 if -/ is
-
be
jEjjj=j,...,,
Length and
a
=
id
)2)
-Ric( , ).
E,_1 and
a
orthonormal and
are
If -Y
timelike
En
+
have n
Ric( , )
=:
E g (R(
, Ej) , Ej)
n-2 =
E g (R(
, Ej) , Ej)
i=1
+ g -
g
(R (En- 1
+
E, E,,- 1) (E,,- i
+
(R (En
+
En, E,,) (En
En), E,)
-
I
-
I
+
E,,), En- i)
n-2 =
E g (R(
, Ej) , Ej)
+ g
(R (E, E,,-,) E, En-1)
i=1 -
(R (E,- 1, En) E,,- 1, E,,,)
g
n-2 =
E g (R(,:y, Ej)
, Ej)
,
i=1
where
we
have used the symmetries of R and the fact that n-1
tr(B)=
E g(BEj, Ej)
-
g(BE, E,,)
i=1
for every linear map B. Hence in either case,
r
=
n
-
I
or r
=
n
-
2,
we
get
Ric( , -:y)
-
tr
w
2+
2 a
0
2
id +
+
r2
(W+U)
+WU+UW
r
definition we have tr(w) 0. For any tensor (B] we have tr(a) tr([B]2) tr(([B]* )2) + tr([B]*[B]) [B]*)) tr(([B] + [B]*)([B] tr QB] [B] *). Since the definition of the trace implies
By
=
-
-
250
4. Pseudo-Riemannian manifolds
n-1
tr([B]2)
n-1
E[91 ([B]2[Ei], [Ei])
=
=
E[g] ([Eil, QBI *)2)[Ed)
i=1
i=1
tr(([B]* )2) and tr ([B]
conclude that
we
*
[B])
tr QB] QB]
=
tr(wa)
tr(aw)
=
Lemma 4.6.22. Let -y be Jacobi tensor classes is
parallel along
is
(R(v, ) , w)
=
[v], [w] implies [R]
[B]
[A]
-
*
[h])
=
In
=
[R]*.
Hence
+
([A]*), [B]
-
[A]* [B]
QA]) [B]
-
position
a
obtain
[h]
QA]
QR] [A]) [B]
-
-[A]*[R]*[B]
=
[g] Qv], [R] [w])
-
QA]
[B]
[A]
*
=
we
==
[A]* [_b] *
in
[B]
fact,
(R(w, ) , v)
([A] *) *'[B]
=
=
are now
tr (B
spacelike geodesic and [A], [B] be
or
self-adjoint.
-
We
=
-y.
[g] ([R] [v], [w])
(([A]
-
-y. Then the tensor class
Proof. First observe that [R]
for all vectors
[B]) [B] 1) 0.
=
timelike
a
along
*
to prove
+ +
Proposition
*
[A] [R] [B]
[A]*[R][B]
=
0.
4.6.3
Proof of Proposition 4.6.3. Let -/: R --+ M be a complete geodesic and 1 if -y is spacelike or timelike and r n 2 if -y is null. We n choose to E R such that R(-,, (to)) (to) 4 0. The symmetries of R imply then that the induced operator [R]: [ (to)]' [ (to)]J-, [v] 1-4 r
=
=
-
-
--+
[R(v, (to)) (to)] of
conjugate
which
satisfy
does not vanish. We need to show that -Y has a pair points. Let J be the space of all Jacobi tensor classes [A] w
0, [A] (to)
=
=
and tr
id,
We will first show that each
[A]
E
QA] (to))0 J satisfies det[A](t)
=
0 for
loss of
generality) that [A] E J_ Since t>to. Suppose (without a is self-adjoint, tr(a2) > 0 and the Raychaudhury equation implies 6 < _102. If there is a t, > to with O(ti) < 0 then an integration some
the shear
-
implies
1
0 (t)
vanishes for
r
> -
1
0(ti)
some
t
+ =
t -t'
n-1
t2
>
for all t > ti. Since the
t, the expansion
Consequently, det([A]) vanishes
O(t)
right
must
at t2- If there is not any
hand side
diverge ti
>
at t2. to with
4.6
0(ti)
then the
< 0
imply 0(t) that
a
=
is
inequalities 0(to)
=
-[R]
([A][A] -1)2
-
to in contradiction
to
this would
imply R(., to. The proof
on
completely analogous. For each i > to let [Bj] the unique Jacobi
[Bi] (i)
Assume for the moment that tensor class
[B]
[B] (t)
with
0 and
=
=
[B]. Moreover, [B] (to)
id
=
r
[Bil
0 for all
have
From the
the existence of
a
J+
Jacobi
Vt 1 >
0
vanishing vorticity
W so
the Jacobi tensor class
(still
E
id.
=
proved
implies that
we see
[A]
for
limi __,.[BE](t)anddetQB1(t1))=7
to. Since all Jacobi tensor classes
-.102
to. Because of
=
assumption
our
:5 0 and
Raychaudhury equation
[A] [A]-'
0 and therefore also
energy functionals
tr([A](to))
=
0 for all t > to. From the
=
([A][A]-')* t >
Length and
[B]
has
must
proven) fact that [B] (t) J+ \ J-. It follows that tr([b] (to)) > 0 and therefore that there is a i > to with tr(jB ] (to)) > 0. id this implies that the expansion O[B,] of [BJ at to is Since [Bi] (to) strictly positive. From the inequality 6[B,,J :5 n-1 (OB f')2 we obtain lie in either J_ is
J+.
or
non-singular for
t >
to
we
to be
infer that
[B]'
E
=
1
I
to
>
O[Bf (to) Since
O[Bf W
tt
(t det([Bi] (ti))
this
oo)
oo
--
n-1
satisfies
-
t
for all t < to.
+ n
-
I
the existence of
implies
0. Hence there is
a
a
ti which
non-vanishing, parallel vector
field V such that
V(t)
(, (t))-L
E
for all t
and
BjV(ti)
Since the non-trivial Jacobi vector field defined
i
both t, and
by
J
0.
BiV
vanishes at
geodesic -y has a pair of conjugate points. We still have to show that [B] does exist and that [B] (t) is
singular
[Bil
for all t
which
Let
[A](to)
[A] ==
f
>
to. In order to do
depends only
on a
so we
will first obtain
single, given Jacobi tensor uniquely
be the Jacobi tensor class which is 0 and
t > to since
Let
our
[AI(to)
[A] (to) be
=
a
respect to this frame
=
non-
formula for
class. determined
by
id. This tensor class is
0 and -yl [t,),,,) does not
parallel orthonormal define
we
a
a
non-singular for all have conjugate points.
frame of
tensor class
[ffj- along
-/. With
[C] by
t
([C] [vl)'
=
[A] (t)
Observe that this definition is two
parallel orthonormal
(QA] [A]) 1) 3k (s) ds [V] k. *
independent
frames
are
-
of the chosen frame since any
related
by
a
constant orthonormal
4. Pseudo-itierriani-iianniaiiifolds
252
matrix D and since such matrices
[C]
that
satisfy D*D
=
id. We will show below
[Bj].
=
But first
we
need to check that
(14 1 (t) Io
i
is
Jacobi tensor class.
a
(([A]*[A])-')j (s)ds[V]k
1,41 (t)
=
[C]
k
t
[A]3 (t) (QA] [A]) 1) 3kk (t) [VI k -
[A]3 (t) QA]
=
IV])'
(([Al')
([C] IV])
and therefore
[i ] (t) IV]
[A] [A]
[C] IV]
[A][A]-1
+
[A] [A]
[A] [A]
[A] [A] [A] [A]
-
*
1
-
QA] [A] 1) *) QA] -
[A] [A]
is self
[A] [A] [C] + [R] [A] [A] QA] + [R] [A]) [A] [C]
[R] [C]
=
[C]
is indeed
Now
[Bi] and
we
a
show
[C] (([A] [Bi]
follows
=
=
[Bj].
once we
Since
[A] [bi]) I t.
=
id
[a, b].
In
hand, -
[Qi) 1 (i)
([A] *) We
-
can now
0
[( ](i)
implies
This
employ
=
[Bi](i)
[( J(i)
the
equality
Lemma 4.6.22
[A] [bi] *
-
id
[A] (i) [bil (i) *
particular, we get id equivalent to [Bi](i)
[bj (i).
0
imply
*
at all t E
[Qi)
have shown
([A] *) [Bi]
0. This in turn is
0
Jacobi tensor class.
[C] *
-
adjoint by
[C]
1
and
IV]
[C] IV],
where in the last equation we have used that Lemma 4.6.19. Hence we get +
[C] IV]
([A][A]-'[C] ([A]*)-) IV]
[Al QA] *) 1 IV] [C] IV] + ([A] [A]
-
([A]
+
-
[Bil (i)
On the other
[A](i)[A]-'(i)[C#)
=
since
completes the proof
of
[C]
-
=
QA]*)-1(i) [Bil.
the formula t
([Bil)
(([A] [A]) 1) 3k (s) ds *
[A]j'3 (t)
-
t
in order to show that
[B]
limi_,,,, [Bi]
exists if for
some a
[g] (A+ I (to) IV], IV])
[g] (QA]*)-1(t)[v], IV])
Igj A- I (to) IV], IV]
-
t+
6ii
k 1 (([A] [A]) 1)'k (8) IV] IV] ds. *
-
t+
[g] (([A] [A]) 1(S) IV] k[Ek](s), [v] [Ej](s)) *
The last
expression *
[g] (([A] [A])
is
-
non-negative
IV], IV])
since *
[g] (QA] [A])
=
-
1
and
[g]
is
[g] ([A] QA] [A])
([bi] (to) IV], IV])
give
an
0
implies that the operator QA]-'[B])(t) composition of non-singular operators
is is
non-singular.
and therefore also
Thus B is the
non-singular. I
5. General
relativity
p. 210
Einstein's equation is of the form Dg T, where D is an operator acting on the Lorentzian metric g and T an expression which describes the distribution of matter in the universe. In Sect. 5.1 we motivate
1
=
tensor field which symmetric (0) 2 find an expression for Dg.
that T should be
a
and in Sect. 5.3
we
is
[I
divergence-free,
5.1 Matter
Chaps. 1 and 3 we did not explicitly consider gravity. However, one of the main insights of Einstein was that gravity and the geometry of spacetime are closely linked. His argument is very simple and runs roughly as In
follows.
subjected to a fixed external "force field" depends on its initial location, its initial velocity, its mass, and its charge (i.e. its "sensitivity" to the force field). For instance, a particle in an electric field which is initially at rest will move to one side if it is positively charged, to the opposite side if it is negatively charged and not at all if it is neutral. To be more concrete, consider a reference frame (-r, t) in a Galilei spacetime and suppose that there is a The movement of
non-relativistic has the electric
a
which is
particle
particle (m, -y) which is located charge e. Then the formula Y
an
electric field
and
-
:4
M
in
=
e
-
(t(t,!)
Similarly, let 6 be a gravitational field and charge"i of the particle. Then
holds.
g be the
"gravitational
M.
holds. It is
an
experimental fact that the quotient -L depends on the analogous quotient A- is a universal constant and M
particle can
be set
mere
than 1
whereas the
M
=
1
(EUv6s 1896).
Einstein concluded that this fact is not
coincidence but reveals that a
It is
a
gravitation is an acceleration (rather and therefore geometrical. He therefore replaced something force)
usually called the passive gravitational
M. Kriele: LNPm 59, pp. 255 - 286, 1999 © Springer-Verlag Berlin Heidelberg 1999
mass.
p.
270]
5. General
256
relativity
14
the equation
6 by
=
the
geodesic equation
V
0 and the "force
=
the connection V of
field" 0
by spacetime. This point of view physical interpretation of inertial observers: They are simply observers which are freely falling. to
leads
those
a
experimental fact that the matter distribution 2 spacetime determines gravity. Hence we have to look for an equation of the form It is
an
Dg where
(M, g)
is
=
T,
(5.1.2)
n-dimensional Lorentzian manifold 3, D is
an
kind
some
of operator acting on the metric g, and T contains the information on the matter distribution. The "correct" form of T cannot be derived. First
all, it is beyond doubt that matter cannot be described by a smooth object in spacetime but instead demands a quantum description. This implies that we can hardly expect a description from fundamental, physically suggestive principles. T will therefore be a classical approximation, i.e. something phenomenological. Consequently, our final form for Equation (5.1.2), Equation (5.3.11) will appear to be grounded less firmly than the spacetime structure. However, the reader should recall that in the derivation of the Lorentzian structure of spacetime we already assumed that light can be described in an entirely classical (i.e. non-quantum) of
way.
The
matter models
only
relativistic point
had considered
we
particles (cf.
44)
p.
so
which admit
far where a
specialstraight-forward
generalisation. Definition 5. 1. 1. A
particle
is
pair (m, -y), where
a
particle and -y is a curve in M with g( of t E M, representing the history of the particle. the
Exactly v
at
x
as
=
in the
7(0)
momentum serve as a
=
Lemma 5.1.1. Let
for
mass
for
all
x
T.
defining =
-y(to)
Cz M
.
Since
the
-1
=
and
Jw1,...,w,,J be n linearly 1. Then m (to) and m
=
by
=
-
dependent timelike vectors with (wi, wi) determined by the numbers E,,,, Ew.,,,. Proof.
(t), (t))
M
guidance
mined
> 0 is
special-relativistic analogy, an infinitesimal observer the energy E, -mg( (O), v) and the spatial 1 -"v. The (O) following simple observation will
measures
(O)-L
m
19(wi, Ew,,,
is
and
.
.
Tx*M, m
be calculated from
m can
inare
,
basis of
a
-
is -m
uniquely 2 =
deter-
g (M , M ).
1 2
Here
we use
the term "matter" in
a
rather wide
sense
encompassing
all forms
2 of energy. This is motivated by the special-relativistic equation E mc 1: velocity of light) which asserts that (rest) mass is simply a form of (c =
=
energy 3
(cf.
Sect.
1.4.3).
spacetime we live in appears to be a 4-dimensional Lorentzian manifold. 4. However, in this book we will not specialise to n The
=
5.1 Matter
In other
words,
we
only need
E:
fv
E
in order to
T, M ,,
the
recover
to know the energy function
g (v,
:
257
v)
1
R,
w
complete information about
a
1-4
E,,
single particle.
Since g is a smooth object, we would expect T to be smooth also. This indicates that point particles which are not depending smoothly on the coordinates of M cannot be used to constitute T. The to obtain
a
smooth matter distribution from
to consider averages instead of individual
Definition 5.1.2. A congruence of c: M --+ R is a function and U is
g(U, U) The
=
a
simplest way particles is
collection of
particles.
particles is a pair (,E, U), where future directed vector field with
a
-L
integral
curves
of U
are
identified with the world lines of the particles 6 with the energy density, measured by
and the energy density function comoving observers. To keep the
presentation simple we will restrict to a A0 0, i.e. that there exists locally a special to U (cf. Theorem 2.5.4). which Z is orthogonal hypersurface spacelike If B c Z is a compact region then an observer flowing with the particles measures for the energy of those particles which pass through B the quantity case
and
assume
dO
that
E
=
=
JB
'EME,
where jLZ is the induced volume form single observers must be identified with
Definition 4.2.2 4
(cf. a
timelike
curve
Since
a
rather than
a
congruence of curves this expression should be understood as an approximation for small B. It is clear that we recover the definition of a point
particle if the compact set supp(,E) n Z shrinks to a point and the energy density E increases adequately. A different congruence of observers, represented by a vector field V 1 and dO A 0 with g(V, V) 0, will measure a different energy =
content,
EV
=
JB
EV/-tzv
I
v
where -
-
-
4
Ev
is
a
spacelike hypersurface orthogonal
pzv the volume form induced
BV
=
fx
E
EV
:
3
a
V, particle through
to
V,
on
x
which intersects
ZJ,.and
Readers who have not read Sect. 2.5.4 may wish to do so now. Alternatively, they may (for the time being) refer to the footnote in Definition 4.2.2. In the
following
we
in order to avoid
will make
clumsy
use
but
of calculus for differential forms
straightforward calculations.
(Sect. 2.5)
258
-
5.
Ev is
a
General
relativity
function which
depends
on
the congruence of particles
(6, U)
and the observer .el j V, We will
with
now
motivate
transformation law
a
E
-4
ev
through comparison
special relativity
Fig.
5.1.1.
A
localised
congruence
Consider Minkowski space
(A', TI) V:
where
x
and the inertial observer field -4
V
-1. Assume also that the congruence is a vector with q(v, v) i.e. restricted to any spacelike hypersurface Z, Ev has compact support (cf. Fig. 5. 1. 1) The inertial (or freely falling) observers t F-4 x + tv v
is
=
localised,
with
common
rest space
EV
=
x
JX+V-L
+
vj-
EV
/-tx+V-L
measure
the energy
=1.+V-L
EV
(V IPA-),
where MA,,. is the volume form of q. We assume now that the congruence of particles is freely falling, i.e. the field U satisfies the geodesic equation
17UU
=
0. In the limit that
supp(Ev)
n Z
shrinks to
a
point
we
would
the energy associated with a single freely falling particle. Since 17UU 0 we can choose linear coordinates (x 0 xl .... Ix n-1
recover
=
such that
U=00,
This
(aO 11V1101)
V=
+
gives EV
(aO ilvim) J[1'? +
'EV +V
V-1:T,1 V71
-
Matter
5.1
f"+v X
1-ITe
JjVjjdxO A dX2
-
(dxl
F- V I
fx+v
A
...
A
...
A
259
dx"'-'
dx'-1
Evdxl
A
...
A
dXn-
I
I
Evdxo
dX2
A
...
A dx-
X+V-L
X1+
f.XO Evdxo, +
Evdx
V-1 Fv(x 0, x1) (minimal) values
where
hyperplane,
we
VI-1
I ive =:
fEvd X2
of x
A
...
A dx
n-1
and
denote the maximal
4
restricted to the support of ev in
have xO
JIVIlx'
x
In this
+ V
and therefore X1
Let -/: t
-
rest frame
sured is
by
Xn-')
(t, X1,
falling particle -
with rest is
m )
Evdx
(m, -y)
and let
mass m.
to
+
fX
I
Ev
be the
Then its energy =:
the infinitesimal observer
-m
v
(,Oo, ao)
( fl
corresponding freely
-
=
measured in its
m.
The energy
(ao
-
own
mea-
+
given by
to (Vm )
Ev
An
=
-
-F,
analogous relationship should also hold for our smooth congruence can be used to smoothly model a point particle.
U since this congruence Hence we should have
E,
+F&I. f 1 IFVF X
Since
x
-+
Ev
(x)
was
arbitrary this equation implies
transformation law indicates that EV Postulate 5.1.1 The map T is measured by an
(Tensorial
'Ev
T7-,FV1-11-1,11
depends quadratically
This
V.
character of energy momentum). field and the energy density
symmetric Q-tensor 2 infinitesimal observer v is given by T(v, v). a
on
E.
260
.5.
elierai
4
'.1"Vity
In the is
a
special case of our congruence of (non-interacting) particles there simple, well defined tensor field TU namely ,
TU Observe that cv
TU(V, V)
=
U,
c
=
U1.
0
is in accordance with the transformation
law derived above. The
lemma indicates that it is
following
energy densities c, in order to
Lemma 5.1.2. Let T be
determined Let
Proof. all
u
with
by
the values
T, S be
(u, u)
two
a
v
2
all vectors
u
tensors with
symmetric
-I and let
=
(')-tensor.
symmetric
T(u, u) for
enough
the tensor T
recover
be
to know all
possible
(cf.
5.1.1).
Lemma
Then T is
with
g(u, u)
T(u, u)
v/ V_--(v, v, ) satisfies g (v/ V"----(vv, ), v/ V _--(vv,)) T(v, v) S(v, v). Since the space of all timelike vectors =
for every vector w Hence we obtain
a
J > 0 such that
I
T(w, w)
=
2 1 =
2
(dd-t T(v (d S(v
+
v
tw,
-1.
S(u, u)
for
timelike vector. Then the vector
a
=
=
uniquely ==
-
1 which
yields
is open there is + tw is timelike for all t EE [-J, J].
+
v
tw) jt=0
2
dt2
and the claim follows from the
tw,
+
v
+
tw)
)
S (w,
W)
1t=0
polarisation identity.
Conservation of energy and momentum is another fundamental propmatter which we wish to encode in our theory. We will find an
erty of
infinitesimal formulation which momentum
(in special cases) recovers conservation of (cf. Equation (1.4.12)). In Sects. and 1.2.1 and 1.4.3 we have
simply stated conservation of momentum. These conservation laws can actually be derived within the theory of point particle mechanics. This is the content of the Noether Theorem which is covered in textbooks on mechanics. The main non-mechanical input for the Noether theorem is the Galilei group (in the non-relativistic case) and the Poincar6 group (in the relativistic case). Recall that the Poincar6 group is the set of all
isometries of Minkowski
spacetime. In order
mulation of conservation of momentum
Killing
vector fields which
1-parameter
a
Killing field.
Proof.
be
regarded
to find
an
infinitesimal for-
will therefore have to as
infinitesimal
employ analogues of
groups of isometries.
Lemma 5.1.3. Let T be be
can
we
since T is
a
Then div
symmetric
(T( ,
divergence-free
(') -tensor with div(T) 2
0. we
have
=
0 and
Matter
5.1
div(T( , -) )
=
(TabQ
V,,
=
(VaT ab) 6
+ T
ab
(Va6)
Now the symmetry of T and the anti-symmetry of imply that the second summand also vanishes.
lEtItER
Let
be
a
foliation of M into
=
Ve (cf.
T
ab
261
(VaCb)
Lemma
-
4.5.2) I
spacelike hypersurfaces with future
normals nt. A world tube with respect to lEtItER is an open subset )/V of M with piecewise smooth, timelike boundary such that the intersection
WnEt is connected for all then the
we
t. If W is
a
world tube with respect to
I-TtItER
denote th& subset
boundary
UtE [ti,t2l )/V n Zt by IlVt,,t, and the part of which is not contained in Zt, U Zt, by )/Vti,,,,.
0 -tensorfield with div(T) Corollary 5.1.1. Let T be a symmetric (0) 2 and be a Killing field. Let tj < t2 and VV be a world tube with respect to jZtjtER such that supp(T) nl/Vtim, 0. Then the following conservation =
law holds.
(nt2
(nt,, T
nt, _j pm
2nwt,,t2
t,
nwt,
nt2 -j AM
"2
Proof. We have
(nt, T( , -) ) (nt ipm)(VI,..., Vn_j) 0) ym (nt, V1 (nt, T -(T( , for any
Zt,/, Wtim.
=
Vn-1)
1.tm)(Vj,...' V'_j)
I)-tuple of vector fields tangent to Zt. Hence pulled back to -T( , .)0 Jym. This (and supp(T) n get Ot, T( , .)0) nt Jym 0) imply
(n
we
_j
1
-
=
I (nt,T( ,-)O)ntJpm=j
ntV T( , .)0) t2
nw,,
nt, J ym
t2
nt2 'j AM t,
where
we
nw,,,, t2
have used that the future and past boundaries Zt2 and Zt, have orientations. The assertion follows now from the Theo-
opposite induced rem
of Stokes 2.5.5 since
d( nt, T( ,
(ntipm))
=
-div(T( , -)O)pm
o.5
fZ, Ot, T
Hence the
quantity
parameter
t defined
Readers who have of GauB.
by
0) (nt J pm)
the foliation if
skipped Sect.
div(T)
independent
is =
2.5 may instead
0. We will
apply
the
of the time
now
identify
integral theorem
262
5.
this
General
relativity
quantity with
a
component of special-relativistic
momentum in the
context of Sect. 1.4.3.
We will
that spacetime is isometric to Minkowski spacetime represented by a spacelike hypersurface Zt, and after a time t2 represented by a spacelike hypersurface Zt, We will study a matter model which consists of k freely falling congruences of particles in the region before Zt, and after Zt, In between these hypersurfaces interactions or collisions may take place. Hence in this region we will (at this point) neither make an assumption on the matter model nor on the before
assume
time tj
a
metric.
To be concrete, consider the set A', a point 0 E A', and a nonvanishing constant 1-form -r. This 1-form defines a foliation of A with affine hypersurfaces Zt Ix E A : 7r(x-o) tj. Let 77 be a Minkowski metric such that Zt are spacelike hypersurfaces and let V be the timelike, future directed constant vector field which is orthogonal to all Et =
and satisfies M
q(V, V)
=
-
=
1. Assume that the
spacetime (M, g) satisfies
A' and 9jfxEM:7r(X) '-[t1,t21j 7711XEM: W(X) 'E[tl,t2l 1. Let (Ei, Ui)i=l,...,k be the congruences of particles defined at all points x with .7r(x) =
==
(tl,t2).
Assume that
isfy supp(Ei)
n
Let T be
a
Ix
VUiUj
E M
:
7r(x)
G
(0)
symmetric
0 and that the energy densities Ej sat-
=
Itli t2jj
is compact.
tensor field with
2
div(T)
=
0 and
k
Tx
and
=
that
assume
E ci (x) (Uj)
(M, g)
0
X
admits
(Uj)
for
Killing
a
7r
X
(x)
(t 11 2)
vector field
. Corollary
5. 1.1
implies that
f
(V,
,
f
V J pA,,-
T
(V,
,
T
V _j /-tA,,,
t2
Conversely, ity
it is clear that
is valid for all such
div(T( ))
=
0 must hold if the
integral equal-
particle flows.
Since the vector fields Uj
constant for
are
x
Zt (t
E
[tl t2l) i
we
I:k dEi(Ui)Ui Uj are get div(T) point linearly independent then this equation implies dEi(Ui) Oj.e. the =
=
0. If the vector fields
at each
=
energy
density
following are
we
of each
will
particle
assume
flow is constant
that this is also the
(pointwise) linearly dependent. (mi),, (Tni)2 defined by
Then the
along case
where
(Zi)a
=
f(_,').
is chosen such that
+ WO
if these vector fields
in-going and out-going
masses
(Tni)a
its flow lines. In the
,6i Ui
J tLA-
rest
5.1 Matter
-
-
supp(Ei) ?T((Zi)l)
are
n
t2 Ix
E
M
:
=
7
263
0 and
well defined constants.
Fig.
5.1.2.
tion of the
T ransforma-
mass
density in
special relativity
is constant for t > t2 now that the Killing vector field ,. For each vecti, and denote these constant vector fields by 6 0 field Uj and a E f 1, 21 there are vectors (e-i) a with (V, (ei) a)
Assume t < tor
:::::::::
1
((ei)ai (e-i)a)
=
1, and I
A)
The
integrals
(V, T( ,
(V
z- "-
+
Oj (ei),,).
in the formula above reduce to
V J PAII
Ei
(V, Ui) (Ul a)
V J /-tA,l
k
a)
6i
fz
V _j ttA,,,
t(II
k
(Ui7 a)
Ei
Ui
J PAn
k
(Tni) (Ui a) a
where Hence
we
we
have taken the
contraction into account
conservation of
special-relativistic demand the second matter postulate
recover
motivates to
length
i
(cf. Fig. 5.1.2).
momentum. This
264
5.
Postulate 5.1.2 T has
(Infinitesimal conservation law). 0. vanishing divergence, div(T)
The tensor field
=
Postulate 5.1.2 is
interpreted
as an
infinitesimal formulation of
conser-
vation of energy and momentum. That these quantities are conserved is intuitively clear from the absence of a perpetuum mobile. However, the
infinitesimal formulation
implies a true conservation law only if spaceKilling vector field. In general, this is not the conservation of energy can only hold infinitesimally.
time is endowed with case.
It follows that
a
(0)
Definition 5.1.3. A symmetric -tensor field with div(T) 0 is called 2 an energy momentum tensor. It is sometimes called stress energy momentum tensor
5.2 Some If T and g
or
stress energy tensor.
matter models
specific
are
=
simultaneously diagonalisable,
-1
0
0
1
0
...
6
0
0
P,
T=
g
0
...
(5.2-3)
0 0
then
...
0
1
0
0
0
...
Pn-1
the energy
density with respect to the flow of interpreted as principal pressures. (i 11, 11) To motivate this interpretation we will simplify to a perfect fluid, i.e. a matter distribution for which all principal pressures are equal. E
is
interpreted
matter and pi
5.2.1 The
E
perfect
as .
.
.
,
n
-
are
fluid
Definition 5.2.1. Let E,p: M --+ R be smooth 1. Then vector field with g (U, U) =
T
functions and
U be
a
-
=
(c +p)0
0
is called the energy momentum tensor with p 0 is called dust.
0 +pg
of
a
perfect
fluid. A
perfect fluid
=
EW & U5 considered in Observe that the energy momentum tensor T the motivation of Postulates 5.1.1 and 5.1.2 describes dust. =
Lemma 5.2. 1. Let T be the energy momentum tensor Then div(T) 0 is equivalent to
Of a perfect fluid.
=
dc(U)=-(E+p)div(U), where iru-L denotes the
projection
(,E+p)VUU==-7rU_Lgrad(p), to the
orthogonal complement of
U.
5.2 Some
It is
Proof.
straightforward
specific
matter models
265
to calculate
(divT),, =
9C bVcTab cb
=9
(ac(E+P)Uaub+(E+P)((Vcua)ub+ua(VcUb)) +OcPgab)
=
d(E+p)(U)U,,+(E+p)(VUU),,+(E+p)div(U)U,,+grad(p),,
=
(dE (U)
(E
+
+
(E
+
p) div U) U,,
A (VUU)a
+
+
The assertion follows since
projected
to U
((0
0
VUU
0
+
g) (grad(p),
(Ub
1 U and
(9
Ub
+
is the metric
g)
1
I
-
The vector field U is the
velocity
of the fluid
particles
and
the
e
en-
ergy a comoving observer would measure. If the divergence of this vector field is negative then the particles are getting closer together and conse-
quently the
energy
density should
increase. This is
in the first
expressed
equation. The second equation states that the spatial acceleration of the fluid particles is proportional to grad(p). This indicates that p should be
interpreted
as
the pressure exerted
Perfect fluids
are
the fluid.
phenomenological
models and the equations im-
develop shock waves. It is therefore often fluids exhibit properties which are not shared by real
0 tend to
plied by div(T) argued that perfect matter. However, perfect fluids =
on
are
prevalent
in
cosmological
models of
the universe.
5.2.2 The collisionless gas
An attempt to arrive at a more realistic matter model is to consider a relativistic gas. The idea is that we do not have a congruence of particles but that each individual
particle
can move
energy momentum tensor is then obtained
velocities. Let
the canonical defined can
by
a
where
=--
JP+
P+(x)
over
all
particle
(xo.... Xn-1) be a coordinate system of M and choose coordinates (xo.... Ix n-1 ,Po Pn-1) of T*M which are ....
==
Pa(a)dxa
be described
Tab (X)
in any direction. The
by averaging
by
an
for every 1-form
a
E
T,,,M. A relativistic gas
energy momentum tensor
PaPbf(x, P) (- det((gcd)c,d=0,-..,n-1))_'! dP1 2
A
...
A
dPn,
(-) C
T*M
denotes the set of future causal 1-forms and
f:
P+(x)
--4R+
is a density function. We assume that for lp,, I --- oc the function f (x, -) is sufficiently rapidly decreasing so that the integral is well defined. Observe
that the n-form
266
General
5.
relativity
det;((gcd)c,d=O,...,n -1))_'! dP1 2
does not
depend
the choice of coordinates
on
A
...
(xl,
Xn- 1)
tivistic gas is collisionless if the Liouville equation where I ab
XH
PaaXb
=9
-
2
dp,
A
df (XH)
-
The rela-
==
0
holds,
19x,ga6PaPb19pc .6
Using a system of normal coordinates it is easy to see that df (XH) 0 0. implies div(T) If U is a vector field and f was replaced by the delta distribution =---
=
j(,1-CUa "gas"
_
pa)
would obtain dust. Hence dust may be viewed as a are all aligned and move into a preferred direction
one
whose molecules
determined
by
the vector field U.
Analogously,
relativistic
a
photon
gas is
given by
an
energy
momen-
tum tensor of the form
Tab (X)
fPO
::::
PaPb A (X
P)
+
Po+(x)
where
C
Vpl 0 (X),
Tx*M denotes the submanifold of non-vanishing future v 1-form on PO+ P+ (X) is a non-vanishing, oriented n 0 (x),
null Morms at x,
-
0
and fo:
P+(x) 0 We
R+ is the photon density function with respect
-+
assume
that for
IPal
oo
the function
fo
vp+(x). rapidly decreasing so that the integral is well defined. The following lemma implies that the energy density a relativistic gas is always positive. --->
is
0
to
sufficiently
associated with
Lemma 5.2.2. Let T be the energy momentum tensor gas (respectively, photon gas) with f ':: 0 (respectively,
T(u, u)
> 0
for all timelike
spectively, fo)
vectors
u
unless the
vanishes.
Proof. This
is clear since for each vector
for
is
T(u, u)
of a relativistic fo ! 0). Then density function f (re-
u the integrand in the definition I positive unless f (respectively, fo) vanishes.
Lemma 5.2.3. Let T be the energy momentum tensor Then tr(T) 0.
of
a
photon
gas.
=
Proof.
The assertion follows from
tr(T)
==
gab
,
+
PaPbfO(XiP)1,'p(+ (x) 0,
0
since the 1-forms p,
are
JP0 91 (p, A +
A (X, A VP,+ (X)
=
0
0
null.
Readers who have
knowledge of mechanics will notice that XH is just the Hamilton vector field to the Hamiltonian function H (x, p) IgabPaPb. The =
2
equation df (XH)
(Ehlers 1973)
for
=
0 expresses then conservation of mechanical energy
details).
(cf.
5.2 Some
5.2.3 The
An
field
can
matter models
267
field
electromagnetic
electromagnetic
specific
be described
by
a
2-form F which satisfies
Maxwell's equations,
dF
div(F)
=
0,
(5.2.4)
=
J,
(5.2.5)
where J is
interpreted as an electromagnetic current one form. The first be geometrically explained within gauge theory (A small equation volume which contains the essentials of gauge theory is (Bleecker 198 1)7). The second equation does not have any content without a prior interpretation of J. For our purposes it is sufficient to note that J is linked can
to other forms of matter.
Remark 5.2. 1.
instead of If there is
fields,
Using
div(F) no
=
the
Hodge
star
operator
we can
write *d
interaction between
i.e. if matter is
neutral,
electromagnetism
then
1
(Tel)ab
we
are
leading.
=
J
and the other matter
dF
=
0,
(5.2.6)
div(F)
=
0.
(5.2.7)
=
4-7r
(gcdF
I
ac
-
Fbd
many mathematical texts
on
-
one
(F,
F)gab)
may
(or
lack of
replaced by a
given by
(5.2.8)
-
justify these formulas.
"gauge theory"
For "mathematical convenience"
the Lorentzian metric of spacetime is This leads to equations which are of
_
4
We will sketch in Sect. 5.3.1 below how There
F
have the set of equations
These equations are called the source-free Maxwell equations. The electromagnetic part of the energy momentum tensor is
7
*
J.
a
which
are
very mis-
physical knowledge) Riemannian metric.
very different nature from those
which describe
physics. Only in very special cases (a prerequisite is that all is it possible to convert results of the Riemannian theory to the Lorentzian theory using an analytic extension argument according to which one can "rotate" a Riemannian theory into a corresponding Lorentzian theory, where both theories are embedded in a complex theory. In the literature on quantum field theory this rotation is known as the Wick rotation. The Riemannian analogue of gauge theory is mathematically (but not necessarily physically) of interest because it is linked to the well developed theory of elliptic partial differential equations. Gauge theory, on the other hand, is linked to hyperbolic partial differential equations. To sell the Riemannian analogue as gauge theory has presumably the advantage that functions
are
analytic)
it makes it easier to
get funds for research in
pure mathematics. On the
other field
hand, it does confuse people. A pure Mathematician who worked in a closely related to this Riemannian analogue and who saw work using
the Lorentzian metric instead of a Riemannian metric once even asked whether this Lorentzian approach would also be useful to physics!
me
5.
268
General
relativity
Lemma 5.2.4. Let F be
by Equation Proof. Since 1 Fac VbFca
5.2.8. Then
dF =
47rdiv(T)b
=
0
we
a
we
closed
cd =
gcd FacFbd
Va
9
Fbd VaFac
4
+
assume
F(.,
that Tj is given
'-,div(FO))
41r
V,Fab
=
F(., JO)
0 which
=
implies
gives
-
19ef9cd FecFfdgab
4
gcdF
, a,
ga Fbd
-
-
-9
4
ef
cd
9
Ffdgab Va F.
9ef 9cd Fecgab Va Ff d gcdFacVa Fbd
==
F(., div(FO))
+
=
F(., div(FO))
+ F
=
=
have VaFb, + VbF,, +
-F ac Vo Fb,. This
=
2-form and
div(Tj)
have
a,
VaFbc
-
-F ec VbFec 2
+1 Fec VbFce 2
F(., div(FO)). I
Corollary
5.2.1. Assume that the
source-free Maxwell equations hold.
Then
div(Tel)
=
0.
Remark 5.2.2. Recall that in the derivation of the Lorentzian struc-
spacetime we assumed that light rays can be described by null geodesics. Since light is electromagnetic radiation we should now check that this identification is consistent with the description of electromagnetism in this section. However, this would require a proper discussion of electromagnetism which is beyond the scope of this book. Readers with knowledge of electromagnetism may consult (De Felice and Clarke 1990, section 7.8) for the identification of light with lightlike geodesics. Here we can only say that nullgeodesics can be taken as a description of light
ture of
rays in
an
limit.
(observer-dependent)
5.3 Einstein's
equation
Recall that the equation which links geometry and matter should be of the form
Dg In the
preceding
two sections
we
=
T.
have motivated that the
of this equation should be a symmetric, divergence-free Now we will find an expression for the left-hand side.
right-hand side
(0) 2
tensor field.
5.3 Einstein's
theory
In the Newtonian
of
gravitation, gravity
equation
269
is described
by the
equations
where
0
I
==
0
A0
=
ko,
is the Newtonian
=
potential
grad(o),
(5.3.9) (5.3.10)
for the
gravitational field. Equation
a second order partial differential equation for the Newtonian potential 0 and describes how it is related to the mass density
5.3.10 is
varrho of the universe. Recall that
17
=
we
0 which is
5.3.9
have
replaced Equation equivalent to a
a It follows that the Christoffel
by the geodesic equation
Fbc symbols rba,,,
have
r6le similar to the
a
force field 6. One obtains the Christoffel symbols from g differentiation, just as one obtains the gravitational force field 0 from the Newtonian'potential 0 through differentiation. This indicates that 0 corresponds to the metric g. Since the Newtonian potential is related to the matter distribution via a second order partial differential equation, we expect that g F-4 Dg is likewise a second order operator.
gravitational
via
(Gravitation
Postulate 5.3.1
is determined
given coordinate system, D: function Of ged; aaged, and '9a'ybgcdIn any
Theorem 5.3.1. Let
08 and Dg be
a
(0) 2
Postulate
If,
in
Dg
is
a
a
2
nd
-ord.
pde).
pointwise smooth
be a Lorentzian manifold such that dScal: field which satisfies Postulate 5.3. 1 and
(M, g)
tensor
div(Dg) (cf.
--+
g
by
=
0
5.1-2). is linear in
addition, Dg
aaabg
then there exist constants
A, y
E
R such that 1
Dg
Proof. By Corollary
=
/-t(Ric
-
2
4.3.1 and the
Scalg)
+
Ag.
linearity assumption Dg
must be of
the form
Dg Lemma 4.3.1
implies 0
=
clRic + C2Scalg +
C39-
now
==
div(Dg)
cl
=
8The condition states that dScal is vanish identically.
(2 C2) +
not the
dScal.
null-function,
i.e. dScal does not
270
5. General
relativity
Hence the result follows
dScall., =h
by
assumption that there is
our
an x
E
M with
0.
1
Remark 5.3. 1. The assumption that
is linear in its
Dg
highest derivatives
is rather awkward. Lovelock not in
(1972) has shown that in 4-dimensional (but (!)) Lorentzian manifolds this assumption is
higher dimensional Unfortunately, his proof is
not needed,
far too involved to be reproduced
here.
Remark 5.3.2. Observe that
did not
we
even
need to
assume
symmetry
of
Dg, i.e. Postulate 5.1.1 is superfluous. However, the symmetry assumption was important to prove the conservation property Corollary 5.1.1 which motivates the
In
conclusion,
equation
as
our
requirement div(T)
postulates imply
that
:=
0.
gravity is governed by Einstein's
defined below.
Definition 5.3. 1. Einstein's equation R is given by
(with cosmological constant)
A E
I
Ric
-
2
Seal g +
Ag
where T is the energy momentum tensor tion. In the above
form, Einstein's equation
the Gravitational constant and the
=
87rT,
(5.3.11)
describing
is valid in
velocity
geometrical
of light
1984, appendix F) for explicit translation rules
the matter distribu-
are
units where
(cf. (Wald units).
set to I
to other
Remark 5.3.3. Einstein's equation itself does not indicate any
special
value for A. In the past, astronomical observations seemed to very
of
small,
if not
zero.
gravitation arises
if A
imply that JAI is theory (c: velocity of light) if and only
It should also be noted that the Newtonian
as a
limit for
c -4 oo
0. This
implies that A must be very small if non-zero (cf. (Hawking and Ellis 1973, p. 362), (Sandage 1968)). On the other hand, I have been told that to present day cosmological =
data point to
a non-zero
value for A.
Some of the theorems which will be presented do require A
=
0,
1
Ric
-
2
[p.
255
and
in
-9
1
p. 270
Seal g
=
87rT,
much of the literature Einstein's equation is used
(5.3.12) synonymously
with equation 5.3.12. 9
Our
guide ends with Einstein's equation. For what follows we will also use skipped in order to get to Einstein's equation
the material which has been
5.3 Einstein's
5.3.1 The
formulation of Einstein's
Lagrangian
In this section
equation
271
equation
alternative way which leads to Einstein's equation
an
approach also aids in finding an appropriate energy momentum tensor. Unlike the rest of this book, this chapter rests on an underlying principle which is difficult to verify directly. This section can be omitted on first reading and is not required for any other part of this book. is sketched. This
It appears that all fundamental
dynamical equations in physics admit a Lagrangian formulation. According to this formulation, a physical system R where E is an appropriate is described by a Lagrange function L: E generalisation of a vector bundle over spacetime M which contains the possible physical states of the system. Such a setup is motivated by classical mechanics. One can calculate the movement of a point-particle -y: [a, b] -- A' with mass m which is subject to a conservative" force field through the variation of an associated Lagrange function. Let L: A' x R 3 -- R be given by L(x, v) --+
T
T
(VI V)R3
-
V(x).
Then
rn (t) (cf. Equation (1.2.7))
-y satisfies the
a curve
if and
-gradV1.y(t)
=
only if
d
L(-y(t)
T-r
r, ["
equation
-rh(t), (t)
+
+
Th(t))dt
=
(5.3.13)
0
b]
for all smooth maps h:
[a, b]
-4
R 3 with
h(a)
=
h(b)
=
0. In
fact,
we
have d
Ir
)
L (-y (t) + J,=0
-r
h (t),
(t)
+
-r
h (t)) dt
la,b]
f[a, (M (:Y (t)' f[a,b] (m ( )
h (t)
b]
)
-
R3
dV(h(t)))
dt
d
dt
-
-
m
( (t), h W) R3
dV(h(t)))dt
quickly. Since we will discuss
physical applications which make use of all skipped, a continuation of this guide would lead to a lot of skipping forward and backward. On the other hand, the reader should have by now enough physical motivation in order to read the mathematical sections which we have skipped without getting bogged down. Still, the reader is advised to read on and to skip back only when needed. On these occasions skipped material should probably be read section-wise. Here "conservative" simply means that there is a function V: A R with P -gradV. the mathematics which
10
(t), h(t))R3
we
now
have
---->
=
272
5. General
m
relativity
m (t)
( (t), h(t))R3
+
gradV,-,(t), h(t) ) R, dt.
the first summand vanishes for all choices of h with
Assume, there scalar product
(a, b)
is
a
is
non-degenerate
to
E
m (to) (say
>
0).
Let
h: [a, b]
By continuity there
(m (t)
m (to)
there is
+
an
h(b)
=
=
0.
0. Since the
E
0
any smooth function with
neighbourhobd (t-, t+)
a
h(a) 4
grad Ry(to) with
ho
gradV,7(to), ho W =h
R' be
---*
is
+
with
C
(a, b)
h(to)
=
ho.
of to such
)R3
> 0 for all t E (t-, t+). Finally, let gradVjy(t), h(t) R+ U f01 be a smooth positive function with non-empty sup,0: [a, b] in port (t-, t+). Setting h 7ph the integrand (m (t) + gradVjy(t), h(t) )R3
that
+
--
=
non-negative and strictly positive in an open set. Hence the integral must be positive in contradiction to Equation (5.3.13). This proves m (t) + gradV,.y(t) 0 for all t. is
=
The existence of
Lagrangian formulation is widely seen as fundageneral (classical) physical systems. The physical state of an elementary particle" is described by the section 0: M -- E of an approa
mental for
vector bundle. Its
governing equation should again be determined integral equation whose integrand is built from 0, its derivative, and physical fields which interact with 0. To make this program work one first has to define the derivative of a section in an arbitrary vector bundle. It turns out that one can generalise our treatpriate
the variation of
by
an
ment of derivatives of vector fields
of
general
vector bundles
ble to define
(cf. Sect. 2.6) and define connections (Kobayashi and Nomizu 1963). It is also possi-
notion of curvature for these
generalised connections and one can interpret this general relativity curvature F in terms of physical fields which interact with the given elementary particle. To obtain the complete system of equations one writes down a Lagrange function which depends on 0, its derivative, the curvature F, and perhaps other physical expressions. We denote all these physical inputs collectively by 0: M ---> E and by T -4 0, 0+T a has compact support. The one-parameter family of sections such that equations which have to be satisfied by the physical system are then given by a
analogously
-
to the
case
of
-
=
d
(.Cum)
a--r for all variations
This
the
The mathematical
0, of 0. Lagrange function C.
It is far
0
M
recipe
is referred to
the variation
of
properties of equations
of
as
beyond the scope of this book to explain what this actually is. Our purpose is to vaguely set the following discussion into context. Readers who want to know more are referred to books on gauge theory. only
5.3 Einstein's equation
273
this type are similar to the properties one encounters in the theory of minimal submanifolds (cf. Lemma 4.4.8 and the discussion following this
lemma). We will
now
discuss
an
formulation, the physical
this
example of
system consisting of electrodynarnics and gravity. To keep things simple we will assume that there are no other physical inputs. In particular, there
electromagnetic sources, i.e., there summands,
are no
no
charges.
Our
Lagrange
function will consist of two
L
where
Lgrav
=
Lgrav
+
Lei
-
stands for the contribution from gravity and Lei for the
contribution from
electrodynamics.
Remark 5.3.4. If we had included electromagnetic have to add at least two more terms: -
-
sources we
would also
A term Ckin for the elementary particle which is analogous to the term M ( ) 6 in the mechanical example above, and a
term
Lint which describes the interaction of the elementary particle electromagnetic field.
with the
suspected that the electro-magnetic field generalised connection 2( (Bleecker 1981) The connection Q1 can be identified with a 1-form A which, however, is not invariantly defined. (This corresponds to regarding the Christoffel dA. The first of symbols as tensors.). With this 1-form A we have F then trivial is a Maxwell's equations, dF 0, consequence. According to The reader may already have F is in fact the curvature of a
=
=
the program above we have to vary L with respect to the 1-form A. This 0. Gravity depends will give the second of Maxwell's equations, divF =
on two geometric quantities, the torsion-free connection V and the metric g. We will use the Palatini formalism, i.e., we will independently varyC
independently with respect sense only
variation makes
to V and g. Observe that this
if
we
do not
assume a
independent priori that V is the
Levi-Civita connection. It will turn out that the variation with respect to the connection will fix the Levi-Civita connection and that the variation 12 with respect to the metric will give Einstein's equation. The simplest non-trivial, invariant function L,j which can be defined
is
(modulo
constant factors and modulo the addition of
a
constant
term)
given by -
,ce, (A, v, g) where
(F, F)
=
=
-
167r
F, F)
=
-
-
167r
(dA, dA)
,
gacgbdF,%bFd,
Alternatively, one could assume that the connection is the Levi-Civita cononly vary the metric. We have chosen the Palatini formalism
nection and
because this is
more
akin to the treatment of other gauge theories.
274
General
5.
relativity
Lemma 5.3. 1. Let B
'r," (-A,-'[)
C
and consider the vaxiia-tio- ---
be
terisor
a
4-
,
I
(C.iym)
o
0,
gence
7
where
47r
operator From
Proof.
tbBd
dBbd
I
d
167r
d7-
f
-
g and
(Tiv
is the diver-
we
(dA,, dA,)
get
pm
A M
-L
I (F, dB) -1 IM 9ab9 (Fac)(' bBd I-tm
87r
Am
cd
=
of
t.
tdBb
-
-
-rB, V, g). Then
A M
is the Levi-Civita connection
associated with
with compact support
f ((&F)(B )[tm
=
M
holds,
field
4, 77,
dj,' ;-
-
87r
tdBb)l-tM
gabgcd (Fac)tbBdAM
47r
A M
47r
M
47r
M
(tb (_,abgcd (Fac,)Bd) (F (-, BO) 0)
-
gabgcd tbFacB d) AM
(&F ) (B)) Mm.
The first summand vanishes because of the theorem of GauB and the fact that B has compact support. Since the
respect
fmCal (A, 17, g)lLm
I
does not
depend
on
17 the variation with
to V vanishes.
Lemma 5.3.2. Let h
C
sym(T2(M))
be
support and consider the variation
a
tensor
(A, 17, g,)
field =
with compact
(A, 17, g
+
-rh).
Then d
('COM)
I-F holds,
where Tj is
.
(Tel)abh abltM
0, 2
M
A M
given by Equation (5.2.8)
Proof The equation
0
- 'dT
((g-r) ab (gT )bc) Ir=o
=
habgbc +gab dr d(gr)bc I-r=O
implies d h ac dT
Recall from the
(gr)ac.
proof of Lemma 4.6.20 that the derivative of det(,P,), whereWT 1-parameter family of matrices, is given by (det(W,))* det tr(( W-') W. This implies is
a
5.3 Einstein's
(det(g,))*
=
tr(h) det(g,),
where in this formula tr is the metric trace of
Using d
these two formulas
we
275
equation
covariant
a
(') 2
tensor.
calculate ,
(j
(41
07-) (tLM)
0
A M
d
16,1 -
7r
M
167r
M
dt
-r)
I 7 =0
Iet _(gr) (FacFbd (gr)ab(gr)cd V_-_det(gr))
dxl
A
...
A
d Xn
I-r=o
2FacFbdgcd h ab.",det () g Ad Xn det(g)tr(h))dx (FacFbdgcd (F, F) gab) habMM.
(FF),I.
+
2 1
A
...
1
_
87r
M
4
I
Remark 5.3.5. We have thus obtained the form of the energy momentum tensor
by
variation of the
-(I 67r)
-
1
electro-magnetic simple Lagrange function
(F, F)
with respect to the electro-magnetic potential A. For other matter fields analogous results hold. In this sense it can be said that variational tech-
niques aid For the
in
finding
gravitational
the correct energy momentum tensor. term
we
set
1
Lgrav (A, V, g)
=
167r
(Ricabgab
-
2A),
where Ric is the Ricci tensor with respect to the connection V. This is again the (modulo constant factors and summands) simplest invariant function which
can
be build from the metric g and the connection V.
Recall that for any two torsion-free connections V, the difference is a field K which is symmetric in its covariant indices. In index-
(l)-tensor 2
notation, K fields
V,
is
W. We
tVWa
given by VVWa
simply
Lemma 5.3.3. Let C
write V E
T2"(M)
+ K.
be
a
in its covariant entries and which has
V +,TC and d
d-r
(
0,
=
(A, (V,), g)
(,CgavAM) M
-
0,)
1,=O
we
VcWd for all vector + Ka cd
have
tensor field which is symmetric compact support. Setting (V,)
5. General
276
relativity
161 1M(K
bd
d 6a c
+ K
d
d
cgab
2K acb
-
CcabAM
,7r
where K is
defined by V
'
+ K and
is the Levi-Civita connection
of g. Proof. Since the Ricci we
d
dT Let
tensor is the
only quantity
which involves
(V,)
obtain
(f
Lg,,
(A, (V,), g)
o
A M
x
E
M and consider
(Ric,)aU
=
ac (Fab c
d (Fab
+
+
7Cab) c 7
-
'9b (1-ac c
+
c
c
-Ca b) (Fdc
x
(Ric7-)ab 17.=Ogab4M.
dT
A M
and for
r
=
x.
From
7Cac) c
7Cdc)
+
and the fact that the Christoffel calculation that at
16-7r
IT=O
normal coordinate system centered at
a
d
+
d
(Mm),)
d
(Fac +
-
symbols vanish
d
c
-Ca c) (-Vdb
7
at
0 the derivative of
x
we
Ric,
is
+
see
c
-Cd b)
7
without
given by
d
j,-(Ric-r)abl,r=o,x Since this is
VcCablr=O,x
equation which points of M.
tensor
a
must hold at all
We will
c
=
connection
7
re-express
VbCaclT=O,x-
is
independent of coordinates
c
with respect to the Levi-Civita
c
now
c
-
VeCab-VbCac
and the tensor field K. From the definition of K c
VdCab
tdCab + KdeCab c
=
c
c
e
e
-
KdbCae
e
we
it
get
c
KdaCbe
-
and therefore ab
9
(Vecab
ab
C
-
VbCac
9
( ccab
VbCac ab =
9 +
=
KcceCab
Kb`eCac
+
-
KcebCae
KbecCae
+
-
KceaCbce
KbaCcce)
( CCab tbCac) -
gab (KceCab C
div(tr2,3C +
(Kc gab Ce
-
-
-
KceaCbce
-
KbceCaec
+
Kb'aCcce)
trl,2C) 2
gdb Ked a
b
+ Kf hg
fh6a) Cab, e
trij denotes the (metric) trace over the ith and jth entry. Since the a divergence with respect to the Levi-Civita connection I vanishes integral by the theorem of GauJ3.
where
first summand is its
-
+
5.3 Einstein's
Lemma 5.3.4. Let h E
support and g,
d,r
=
sym(T20(M))
g + Th. For
0,
be
field with compact
tensor
a
(A, V, g,)
we
have
M
I =
-
167r
fm (
Proof. We can split the integral separately, d
=
277
equation
(j
(CgravAM)
0
A M
d 167r
M
A M
d 2A
T-F
cd
gab+ Agab hab Am.
parts which will be considered
into two
I,=O
((RiCab (g,)ab
-
((Ric,)ab ((g,) V
1Riccdg
-
2
d
I
161r
-
d -r
07-)
RiCab
det(g,))
2A)
ab
det(g,)I,=O)dxl
V/---det(g,)) A
dxl
A
...
A
dx n
I-r=O
I-r=O
A d xn.
...
Exactly as in the proof of Lemma 5.3.2 we see that the second summand in the integral equals -Agabhab. For the first summand we calculate d
TT
(gr)ab
-det(g,))
I-r=O
d d-r
((g,)a b) Ir=o
det grT) + 1
-
bd
ac
-g
g
hbd V- det(g)
(g,)ab d7-
det(g,),,=o
-
+ -g 2
ab
cd
g
hed V- det(g)
and therefore
d
RiCab
-
dt
det(g7. ) det(97')) ((g,.)ab,/-
Ir=o
RiCab+
The
following corollary
Corollary be
a
hab V
---det(g).
is the main result of this subsection.
5.3.1. Let A be
a
I-form, V
Lorentzian metric and set F
Einstein's equation magnetical field,
IRiccdgcd 9ab 2
=
be
a
torsion-free connection,
g
dA.
and Maxwell's
equations for
a
source-free
electro-
278
General
5.
relativity
I
Ric2
are
I
Scalg+Ag
(gcd
=
2
equivalent
1
FacFbd
(F,
-
4
F)gab)
dF
7
(41 +,Cgrav)
d-7-
Proof. We
can
Since C is
arbitrary
MAM),
-
Taking 0
=
bd
d ja c
which
add6b c
+ K dd c
gab
2K acb
-
K
ad d
+ nK
ad
+ 2K
d
dcgab
-
d+2K dda
-
+
0
=
2nK ddc
-
a
Ccab
0
-rC, g +,rh) where
and b. This is
2K acb c we
-
4K dcd
a
=
2K bca
=:
equivalent
to
(5.3.14)
0.
get
and b
2(n
separately.
0
2K ad d- 2K dda
the trace with respect to
now
in
symmetric
are
the trace with respect to b arid
=
Taking
c
Cab + K
C
=
consider the variations with respect to A, V, g Lemma 5.3.3 implies at each point x
(K d6a
0
A M
=
bd
=
d
for all variations 0, (A, (V,), g,) (A + -rB, V B, C, h are tensor fields with compact support.
K
divF
0,
to
(
for all tensors
=
we
-
=
(n
-
I)K
ad d-
get
2)K ddc-
These
equations together with the symmetry of Ka bc in b and c: imply 2) that all traces of K vanish. Hence Equation (5.3.14) simplifies K acb -K bca and Kabc is a tensor with the properties
(for to
n
>
=
Kabc We will it is
now
=
-Kbaci
Kabc
=
Kacb-
show that this tensor vanishes. Since it has the property that in two indices and anti-symmetric in two other indices,
symmetric the expressions
sym(K )
and
alt(K5)
0
=
Kabc
+
Kbca
+
Kcab
+
0
=
Kabc
+
Kbca
+
Kcab
-
both vanish. This is equivalent to
Kacb
+
Kbac
+
Kcba
and
These
(Kacb
equations imply Kabc + Kbca + Kcab Kabc + Kbac + Kc.ab symmetries of Kb, 0
=
=
and
we
have V
=
t
Kbac + Kcba)-
+
0 and
=
therefore, using
the
Kcab. Hence K vanishes
5.4 The Einstein
equation
equation dF
The
=
as a
systemof partial differential equations
0 follows
trivially
from the definition F
The second part of Maxwell's equations, divF from Lemma 5.3.1 since B is arbitrary. Since h is
=
0 follows
279
=
dA.
immediately
arbitrary and we know that 17 is in fact the Levi-Civita validity of Einstein's equation follows immediately from
connection the
Lemma 5.3.2 and Lemma 5.3.4.
1
The process to
as
leading to Einstein's equation via Corollary 5.1.1 is referred varying the total Lagrangian 41 + Lgrav with respect to the metric
9.
We have chosen
have
concrete
a
models admit
a
electromagnetism for our matter model in order to example. To my knowledge all fundamental 13 matter Lagrangian formulation such that
d d-r
) f
((Linatter + Lgrav)AM)
for all variations of the metric is
the
=
0
equivalent
to Einstein's
equation for
matter model.
particular Lagrangian
That the
trivial and I have
means
5.4 The Einstein
of
O-r
0
A M
Ir=o
partial
ansatz described in this section works is no
explanation
by
no
for it.
equation as a system equations
differential
Physicists are accustomed to the fact that (classical) physical systems depend on initial conditions and then evolve in a determined manner which is governed by second order hyperbolic differential equations. Since the energy momentum tensor T contains the metric, the Einstein Equation (5.3.11) cannot be simply solved for a given T. Instead, one has to convert the system. of Equations (5.3.11) into a system of partial differential equations for g and some matter quantities. The analogue in relativity would therefore be to fix an (n dimensional Riemannian manifold
(Z, 'g)
which represents
instant of time. This manifold will be isometric to
surface in the solution. Since Einstein's
system
we
specifies
would need to
prescribe
equations symmetric (0) 2
are
functions
or
tensor fields which
our
hypersurface.
represent the initial
an
initial
spacelike hypera
second order
tensor field k which
the normal derivative of the induced metric
the second fundamental form of
at
a
a
_r
g or,
equivalently,
We also need to fix matter distribution
Z, possibly also their normal derivatives. A perfect fluid is a macroscopic concept and the Lagrangian formulation does not work well in this case. See (De Felice and Clarke 1990, chapter
6.5)
for
a
discussion.
280
5.
Geneml rela"'IMI,
The character of this system of partial differential equations will crucially depend on the form of matter assumed. In particular, one can choose unphysical matter models which lead to spacetimes in which it is possible for information to travel faster than light (cf. Corollary 7.4.1). It is also possible to choose unphysical matter models which do not lead to a hyperbolic system of differential equations. Another problem lies in the fact that we have always the freedom to change coordinates. Hence the choice of coordinate system may also have an effect on the kind of system of partial differential equations we
will end up with.
Nevertheless, in most situations of interest, it is possible to obtain a well-posed system of equations. We will show this for the special case that T 0 and A 0. In order to avoid subtleties arising from the theory of partial differential equations we will assume that our initial data 0g, k are analytic and that Z is an analytic manifold. (This restriction allows us to appeal to the relatively elementary theorem of Cauchy-Kowalewskaya.) We will also fix coordinates in which the equations are especially simple. In Chap. 6 we will study the more general case of a perfect fluid. However, we will impose strong symmetry assumptions in order to simthe system of equations plify the problem drastically (cf. Sect. 6.2) will be reduced to a system of ordinary differential equations. Chapter 7 contains an intermediate treatment. We will again consider a perfect fluid but use weaker symmetry assumptions which lead to a system of partial differential with two independent variables. This system of equations is substantially simpler than the general system depending =
=
-
on
4 variables. We will therefore be able to
an
analytic)
existence theorem
(cf.
give
Theorem
a
smooth
(rather
than
7.4.1).
Since the
analogous but considerably simpler discussion in Chap. 6 already exhibits some of the key concepts of the initial value problem for Einstein's equation, the reader may wish to skip the rest of this section on first reading. Let
(M, g)
be
Lorentz manifold Z C M be
smooth, spacelike hypergeodesic - ., with nx. There is a neighbourhood of Z which is foliated by these x(O) geodesics. If this neighbourhood is chosen small enough it is also foliated by spacelike hypersurfaces of the form Zt 1-yx(t) : x E Z1. If one views M as being foliated by spacelike hypersurfaces Zt (ZO a
surface with normal 14
n.
For each
x
E
a
Z consider the
=
=
Z)
with induced metric
fundamental form kt
as
-"'g
then
one
can
the t-derivative of
view the associated second
-I-'_1,g: 2
Lemma 5.4.1. Let Z be
be
a
be
a
14
a spacelike hypersurfaces of (M,g) and JZtj neighbourhood as constructed above. Let x1.... X n-1 coordinate system of Z centred at x E Z.
foliation of
Here
we mean:
a
I
g (n,
n)
=
-
1,
g (n,
v)
=
0 for all
v
G
TZ.
5.4 The Einstein
equation
Then there is
a
xi, j:n-1 i=1 q,j (t,
systemof partial differential equations
as a
neighbourhood
U
of
M such that g
(E
x
-dt2
+
Xn-') dx'dxj.
Moreover, the second fundamental form kt of Zt X
=
281
is
given by kt
c9t z`9
find a neighbourhood U of x such that for every point exactly one point -' E ZnU and one geodesic -y, through: which satisfies , (0) n,. and intersects Z exactly once without leaving U. This gives a chart (U, p) defined by W(y)
Proof. We
can
y E U there is
=
where y -/, (t). It follows from =
given by Ztg
functions. At t
=
0
construction that the induced metric
our
En-1 i,j=
=
X1 I gij (t,
I ....
Xn-1)dx dxi,
where gij
are
on
Et is
suitable
have for each -- E Z
we
n-1 =
gj
-dt2
E
+
Xn-l(.: ))dx dxi
gij
i,j=l
(O)
since
nx _L
=
E
ZO. From
=
=0
tz
0
(:4' axi)
V%% ,axi at,
we
get
(t)
Zt for all
I
V'9X
i,
)
( X' V%axi
+
V
a,
t. This
2
at' Vataxi 0
a,Xi
implies the claim
for the metric compo-
nents.
From Lemma 4.4.6
get
= Vaxiat' axi (Vaiaxi" a",j at gij axi' va"9xj ) at
kt (axi, axj)
0
=
The assertion
we
kt
=
=
-
-LCOtEtg 2
follows
now
*
gij
-
kt (axi ax i ) ,
from the symmetry of
-
kt and
from
V at_,"19) ii
=
V at E19) Oxi. axi) I
=0
'9t -'""gij
-
-,"'gGeataxi, axl)
=0
-
E"g(axi.' xat,9xj)
=
atgij. I
We denote the Levi-Civita connection induced Ricci tensor of
(Zt, Elg) by
El Ric.
on
Zt by EtV and the
282
General
5.
relativity
Lemma 5.4.2. Einstein's equation with vanishing cosmological constant for vacuum is equivalent to the following system of equations.
Otat-"'gij
=
0
=
1
-2EtRiCij Z'Scal
-
Zt,
+
-
4 0
Proof. to Ric Zt
=
(2 atzt'9ijatZt'9k1 19tZt'9ikatZtgil Z"gjl z119 k1, (19tZt'9ij19tZt'9k1 atZt`9ik19t 5,tgil) gii Etgk1
-
_aX,(Etgjkat Zt'gjk)
Ztgjkax., Z tgik
+
Einstein's equation is given by Ric 0 which is equivalent s`1g 2 0. The GauB equation (Proposition 4.4.1) and Ric 0 imply =
-
=
=
Ric (U,
W)
(R (at, U) W, at)
=
+
-
tr (kt) kt (U,
W)
Z"g (kt (U, -) 0, kt (W,
(R(U, at) at, W)
=
(19t -`9ij at Zt'gkl 5'
4
-
at
5t'gik 19t Et,gjl Z1,gj1 zt,gklUiWj
In order to
simplify the term (R(at, U)W,,9t) we may assume that U, W be extended to vector fields U, W which are everywhere tangent to Et and commute with i9t. Using Lemma 4.4.4 we obtain can
=0
1--*1--l
(R(U, at) at,
W
VU Vatat -Vatvuk
(V"tVUat, W) Vqtkt (U, -) 0, W)
W) V",Vatu, W)
I
at
-
2 ==
(.Ca --,.g(U, W))
+
t
X'9tSat-,",g(u, W)
-
2
2
Ot at -"gii
ISa Ety(u, Va W)
2
-C
+ 4
t
t
at -'"t,9 A
+I19t 5t'9ik19tEt9j1g
-Pat -Y,119 (WI -) 0)
k1
) Ui Wj.
4
Hence the spatial components Ricij 0 of Einstein's equation alent to the first system of equations in Lemma 5.4.2. From ==
Rijkl Etgik Etgjl
and the GauB
=
Rijkl (9
=
Scal +
equation
Ric (,9t,
at)
=
we
-
+ at (g
2ffic(Ot, at)
=
Et
Scal +
are
equiv-
t)j1
+ at & a
2(Ric
-
ScIal g) Ot o9t)
get
I(Scal + 2
at)ik(gV
tr(kt )2
_
-
11kt 112)
5.4 The Einstein
Hence Ric
systemof partial
differential
283
equations
=
f El,
Let
as a
0. implies -"I Scal + -14 (tr (-C at Ztg) 2at Ztg 11 2) E,,- 1 1 be an orthonormal frame of Zt and U be a
0
=
equation
field which is tangent to Zt. Then the Codazzi equation
vector
(Proposition
4.4.2) implies Ric (,9t,
U)
1: (R(Ei, Ot) U, Ej)
=
i=1
(V U (kt & at) (Ei, Ej), at)
(7 Ej (kt
0
at) (U, Ej),
at))
n-I
E ( ((V U kt) (Ei, Ej) i9t, at)
=
-
( T Ej
*
kt) (U, Ej) i9t, at
)
j=1 n-1
((-"'VUkt) (Ei, Ej) Ot, Ot) VE, kt) (U, Ej) at, at)' 9
2
Hence
Ricti
equation
=
-"Idiv (kt) (U).
U
tr (kt) +
(
ax, (Zjg'k,9t
0
(i
E
1,
n
Zt
gjk)
+
11)
is
-
Zt, jk
g
a.X.1
Zt'gik) Ui.
equivalent
to the last
system of I
in the statement of the lemma.
The first system of differential equations in Lemma 5.4.2 consists of n (n 1) coupled differential equations for the 2 n (n 1) unknown func-
1
-
-
n tions gij 1, 11). One would expect that these equa, gji (i, tions would uniquely determine g and that therefore Einstein's equation =
-
.
.
.
would be over-determined. Since over-determined systems of differential equations have only very few solutions (if any at all!) and are in general
incompatible with initial value problems, Einstein's equation seems (at first sight) to be very different from other equations in physics. However, the following lemma shows that the over- determinacy of this system is of a very special nature and in fact compatible with a (slightly restricted) initial value problem. Lemma 5.4.3 and Theorem 5.4.2 below hold for initial data which
necessarily analytic. However, we cannot anymore appeal to Cauchy-Kowalewskaya.
not
the
since
the
is then much
Theorem 5.4.1
(Cauchy-Kowalewskaya).
Let F: R 2m- I
Rkand fo: R'-1
---
--->
are
difficult
proof relatively elementary theorem
Rk be analytic
more
maps.
of
Then there
284
5
is
a
f:
U
liahviay
r,
0 and a unique analytic map neighbourhood U c R- of x-R k which satisfies the system of partial differential equations =
ax f=z
719X-.-1f).
rn
f (0, x1.... XM-1
and the initial conditions
Proof (sketch).
fO(X1....
I
I
XM_ 1).
The idea of
proof is to determine the Taylor series of f successive differentiation of the system of partial by jxm 01 differential equations and then to show that this series converges. A formal proof can be found in (Dieudonn6 1971). 1 at
X
E
=
The theorem of
Cauchy-Kowalewskaya rests on the fact that an analytic by its Taylor series and it does not hold when "analytic" is replaced by "smooth". In the non--analytic case
function is determined the word
the structure of the system of partial differential equations matters for both, existence and uniqueness of solutions. This fact indicates that by restricting to the analytic case one may (in general) obtain results which are
misleading
because
Lemma 5.4.3. Let
foliation
as
E
at,9t-'_1,gjj If
at Z
f 1, =
do not
generalise
to the smooth
case.
(M, g)
in Lemma
(i, j
nents gij
they
be a real-analytic spacetime, {ZtltER be a 5.4-1, and assume that the spatial metric compon 11) satisfy -
-
-
.
,
-2z'Ricij
-
(2 atgij'9tgkl
-
19tgik'9tgjlg ji
)9
k1
ZO the "constraint equations"
=
0
hold then
=
'Seal
+
(tr(k) )2
(M, g) satisfies
cosmological constant, Proof. Let
Ric
_
11kJ12,
Einstein's =
0
=
-dtr(k)
vacuum
+
'div(k)
equations with vanishing
0.
Ric be the Ricci tensor associated with g -dt2 + gij dx'dxj. this tensor satisfies Ricij 0 for all spatial compo=
By'assumption, nents. The
2r,'d,cRiCdb)
=
0 implies therefore gab (19a RiCbc identity div (Ric 12 Seal g) 0 which is equivalent to -,9,Scal =
-
-
=
n-1
0
Since
Ricij
=
-,9tRictc
aiRicic
-
0 this is
-
aRictt
linear system of
-
2rdcRiCdbgab
partial differential equaequations are equivalent to Rictt 0 at Z. Hence we have Rictt Ricti 0 everywhere by the uniqueness-part of the theorem of CauchyRicti I Kowalewskaya. =
a
tions for the unknown functions =
=
n
Rictt, Ricti. =
The constraint
=
5.4 The Einstein
equation
as a
systemof partial differential equations
Theorem 5.4.2. Let
(Z, -g)
manifold field which satisfies
and k
Riemannian
0
ZScal
=
(tr(k) )2
+
be G
_
an
(n
-
I)-dimensional real-analytic
sym(T20(Z)) ilk 112,
0
=
285
be
real-analytic
a
-dtr(k)
+
tensor
Zdiv(k).
n-dimensional
real-analytic Lorentzian manifold (M, g) _1'9 and t*ko t*g k, where ko is the second fundamental form of t(Z). Moreover, if (1 1, j) is a second Lorentz manifold with these properties then t(Z) C M and E(Z) C 1 1 have neighbourhoods which are isometric. Then there is
and
an
Fix
Proof.
an
immersion
a
t:
Z
M such that
--
=
=
coordinate system for Z and consider the system of
partial
differential equations
Ot(kt)ij
-[-"1Ricjj](Ztg, ((ht)[kj)k=1,...,n-1 ((ht)[kl])k,1=1,...,n-1)
=
-
Ot-"',gij ,Ot(ht)[k]ij ,9t(ht)[kl]ij where
braic
=
=
=
((kt)ij(kt)kl 2(kt)ik(kt)jlzlgjl) -
Zt k1
g
2(kt)ij, 2,%k(kt)jj, 2a,,ka.,1,(kt)jj,
[ZtRicjj](E1g, ((ht)[k])k=1,...,n-1, ((ht)[kl])k,1=1,...,n-1)
is the
alge-
expression defined by 5'
["7tRicjj](Ztg, (aXk -t9)k=1,...,n-1, (aXkaXI, Ztg)k,1=1,...,n-1)
Z'Ricij.
=
Cauchy-Kowalewskaya implies that for any real-analytic set of initial values 1,-rOgij, ko, ((ho)[k])k=1,...,n-1, ((ho)[kl])k,1=1,...,n-1j7 0 and a unique solution of the system there is a neighbourhood U of t which is defined on U and has these iniof partial differential equations The theorem of
=
tial values. Since
at-tgij
=
the equation
2(kt)ij
at(ht)[k]ij
=
2ax1c(kt)jj
implies 0
=
at (ht) [k] ij
-
a,, k at Z"gij
=
at ((ht) [k] ij
-
aX
k
Z"gij).
of this equation we see that (ht) [k) ij aX -"tgij if that the In same see 19Xk,9X,,-,"gjj 19XkZ"gjj. (ht)[kl]ij waywe (ho)[k]ij if (ho)[kl]ij aXkaXI -Togij. It follows that the solutions Z1,gjj of this
From
integration
an
k
=
=
system of equations also solves
atatzlgij if and
=
only
atE"gij
-2ztRicij
-
(2 19tZ t'9ij19tZt9k1
5'
-
19trtgik at tgjl
t'gjl
Z
19 W.
if the initial conditions
=
2(ko)ij7
(ho)[k]ij
=
aXkZ('gij,
(ho)[kllij
=
aXk19X1,ZOgjj
5.
286
General
relativity
hold. Now the assertion follows
directly
from Lemma 5.4.3.
1
Theorem 9.4.1 is local in character. Also note that the coordinates chosen tend to
develop singularities due geodesics t
and observe that the
A discussion of the smooth
to --+
focusing effects (cf. Proposition 4.6.1) (t, x1, x') are length maximising.
case can
.
.
.
,
be found in
(Hawking
and Ellis
1973, chapter 7). An improved but mathematically more sophisticated theorem is presented in (Hughes, Kato, and Marsden 1977)
6. Robertson-Walker
6.1
cosmology
Homogeneity and isotropy
It is very difficult, and one cannot make with any certainty assertions about the universe as a whole. This is so because we only know a very
small portion of the universe. Hence any cosmological model reflects our own prejudice. Nevertheless, there are certain assumptions which seem to have
high degree of plausibility. After having built
cosmological model Although imperfect, this approach seems to have given us much deeper understanding of the development of the universe than would seem possible at first sight. The first cosmologists placed the earth at the centre of the universe. Copernicus' revolutionary model gave us a much more humble place in the earth was reduced to being just one of its planets. the solar system This model of the universe had such a success that nowadays we not only take it for granted but don't even sincerely doubt that there may be other (more advanced) forms of life in the universe. At Copernicus' times, such a thought would have been considered blasphemous. The a
one can
compare it with the few data
we
a
do have.
-
monk Giordano Bruno
(1548-1600)
truth of such ideas.' Our in
spacetime
is in
places anywhere a cosmology.
was
burned because he asserted the
leads
us to think that our place modesty and that there are no exceptional way exceptional, We will this fundamental idea to build use spacetime. new
no
in
Let x E M be our event in spacetime (M, g) and Ux E TxM be the velocity vector of our world line. If there is not any point (or direction) in spacetime which is special then the universe should be isotropic, i.e., it should not be possible to distinguish any direction in Ux-L by physical measurements. Although a glance at the nocturnal sky indicates that this is at odds with experience, on a sufficiently big scale this assumption coincides very well with observation. The galaxies seem to be randomly distributed, and since at night we mainly see a part of a single galaxy (the Milky Way), our first impression is not very representative.
He is sometimes
styled
and
an
important forerunner of enlightenment. (Bruno 1584) quite unscientific.
must confess that I find his book
M. Kriele: LNPm 59, pp. 287 - 305, 1999 © Springer-Verlag Berlin Heidelberg 1999
I
6. Robertson-Walker
288
cosmology
The mathematical interpretation of the isotropy assumption is that which impossible to construct geometrical objects, using Ux and U-L X
it is
breaking
are
this symmetry. Let E be
(any)
3-dimensional subspace of
(n I)-dimensional space U.- and u, v, w be vectors in E. Since U R(v, U) is a vector, isotropy about Ux and the fact that (R(v, U) U, U) 0 imply that R(v, U)U I-t(x)v for some scalar I_t(x). Consider any 2the
-
=
=:
plane P spanf u, vJ in E. The sectional curvature K(P) should be inof P since otherwise therewould be a plane PO of maximal secdependent tional curvature which in turn defines a distinguished direction POJ- in E. Since by isotropy there should not be any distinguish'ed direction in E we conclude that at x the equation R(u, v)w n(x) ((v, w) u (u, w) v) + =
=
c(u,
v, w) U holds We show -
R(P) lies in
U.J-
=
that
now
((U,V)2
c
=
-
0. The vector
11U11 IIVII)-1/2 R (u, v) U., .
_
and
depends only on P (rather than on the representatives 1 by irp the orthogonal projection U -- P. For each vector 1 let P,, be the plane in E orthogonal to n. Since n E E with g(n, n) 7rp. (R(P,,)) lies in P,, and is therefore orthogonal to n we obtain a vector field 93: n --+ 7rp,, (R(P,,)) on the 2-sphere f n E E : g (n, n) 11. Since the 2-sphere is compact the vector field 93 has constant length since otherwise there would be a distinguished direction of maximal length in violation of isotropy. By Theorem 2.5.11 this is impossible unless the vector field vanishes identically. By isotropy, it should have no component in P since otherwise there would be a distinguished direction in P. It follows that R(P) is orthogonal to P. Further, its length cannot depend on P since otherwise there would be a distinguished plane (and therefore a distinguished vector) in u,
v).
Denote
=
=
E. Let u, v, U1 i U2 E E
W E
we
E.
From,7r,;Panfui,U2J (Rspanf ul, U21)
0 for all vectors
==
obtain
; 02 (R(u, v)U.,, w)
=
(R(u, v
+
w)U-,, w))
=
(R(u, v
+
w) U, v
+
-
(R(u, w)U.,, w))
w)
(R(u, v
+
w) U, v)
;02 1-
This
implies that
-1
(R(u, v) U,,, v))
the tensor field
-
(R(u, w) U, v))
.
(u, v, w) (R (u, v) U, w) is antisym(R(u, v)w, U,,) c(u, v, w), c is a 3form. The first Bianchi identity (Lemma 2.8.2) yields (R(u, v)w, U,') + 0. This motivates the (R(v, w)u, U,,) + R(w, u)u, U,,) 3c(u, v, w) metric. Since
R(u, v)U,,, w)
=
=
following definition.
t-->
=
-
=
6.1
Definition 6. 1. 1. Let
(Uo, U.,) about U.,
if
at
(M, g)
be
a
Lorentzian
spacetime (M, g)
1. The
the curvature tensor
x
Homogeneity
and
isotropy
89
manifold and U., E M, infinitesimally isotropic
is called
satisfies
R(u, v)w
=
n(x) ( v, w)
R(v, U.,)Ux
=
tt(x)v,
u
u, w) v)
-
w E UxL, where y(x), K(X) E R are independent of u, v, w. spacetime (M, g) is called infinitesimally isotropic if there exits a normalised, timelike vector field U such that (M, g) is isotropic about Ux for all x E M. If (M, g) is infinitesimally isotropic, U is called a cosmological observer field.
all u, v,
for
The
Given
infinitesimally spacetime,
an
there may not be
a unique cosmologspacetime all normalised
ical observer field. For instance, in Minkowski
timelike vector fields In the
following
are
cosmological observer
we
will
fields.
that spacetime is
assume
infinitesimally
isotropic. While it can be argued that infinitesimal isotropy about our own velocity vector is backed experimentally fairly well, it is a very questionable extrapolation to assume that spacetime is infinitesimally isotropic. On the other hand, this extrapolation seems to be exactly the
Copernicus. Hence to demand that (M, g) is infinitesimally isotropic appears very plausible to us. (Our acceptance of such a postulate is in striking opposition to the response a medieval scholar lesson learned from
would have
given).
Lemma 6. 1. 1. Let
be
(M, g)
infinitesimally isotropic
about U,,.
Then
R(u, v)U,, for
all u,
v
E
U,,J-
=
0 and
R(U, u)v
=
-I-t(x) (u, v) U.,
and the energy momentum tensor is given
TX
=
0E W
+
Xx)) UX'
(9
U X
+
by
P(x) g,
where
Proof. -
87rc(x)
=
I(n 2
8,7rp(x)
=
(n
Let u, v,
w
E
(R(u, v)w, U.)
tion is
a
obtain
0
2)(n
2)
-
I)r,(x)
(_2 (n
=
-
-
A,
3) n (x)
+ p (x)
The first assertion follows from and
(R(u, v)Ux, Ux)
=
0.
+ A.
(R (u, v) U, w)
The second
(R(Ux,u)v,w) R(v,w)Uu) (R(u, Ux)Ux, v) y(x) (u, v). Taking the
consequence of
(R(Ux, u)v, Ux) we
Uj.
=
-
-
=
=
=
asser-
0 and
trace of R
290
Robertson-Walker
6.
Ric (Ux,
Ux)
Ric (u,
cosmology
(n
=
v)
-
((n
=
-
1) p (x),
2) n (x)
Ric (Ux, /-t (x))
-
v)
0,
=
(u, v)
,
and therefore Ric
((n
=
2)/,t
the
Taking again
n((n-2)n-/_t) tensor is
-
=
(n
+
trace
-
we
2)r,) 0 have
0
0
Seal
+
((n
-
2)K
-(n
=
(-2n+2)jL+(n-1)(n-2)K-
/_t) g.
-
2)(y
-
n)
+
+
The energy momentum
given by
now
I
8-ffT
Ric
=
-
2
(n
=
+
-
Seal g +
2)(/,t
((n
+
K)O
2)K
-
Ag
-
ft
0
&
-
-1 ((-2n + 2),4 + (n 2
=(n-2)(,u+r,)0O0+ (n-2) It follows that the energy
81r(e +p)
=
(n
-
2)(tz
+
r,)
density 8irp
and
E =
3
+ A
9
nn+(n-2)tz+A
g.
-
1)(n
2)r,)
-
-
2
and the pressure p
'(n 2
-
2)(3
-
are
n)r, + (n
-
given by
2)/-t
+ A.
1
Lemma 6.1.2. Let
(M, g) be infinitesimally isotropic and U be the cosmological observer field. Further assume that n > 3 and that E + p : 0. Then Uj- is an integrable distribution. The hypersurfaces perpendicular to U are totally umbilic (cf. Definition 4.4.7) and r, is constant on these hypersurfaces.
Proof. Let X, Y, Z be equation
vector fields which lie in
VUX (analogously
-
(X, vuu
U
=
Uj-
at
x
and
satisfy the
0
Y, Z). This can easily be arranged by considering vechypersurface Z which lie in Uj-. They may then be uniquely extended into a neighbourhood of Z using the above differential equation. From VU (X, U) 0 it X, VUU) U, U) VUX follows that X, Y, Z are everywhere perpendicular to U. Moreover, the derivative VUX is parallel to U since VUX (VUX, U) U (analogously for Y, Z). We will now exploit the second Bianchi identify (cf. tor fields
for
along
a
=
=
-
=
Lemma
2.8.1),
(VUR) (X, Y) Using
-
+
(VXR) (Y, U)
+
(VyR) (U, X)
the formulas in Definition 6.1.1 and the property
==
0.
6.1
Homogeneity
VUX' VUY' VUZ 11 we
and
isotropy
291
U
calculate
(17UR) (X, Y)Z
17U (R(X, Y)Z) R(VUX, Y)Z dr,(U) ((Y, Z) X
=
-
+
K
(VXR) (Y, U)Z
-
(Y' Z)
(X, Z)
-
VU X
VX (R(Y, U)Z) R(Y, VXU)Z
=
R(X, VUY)Z (X, Z) Y)
-
((Y' Z) VUX
0 + P
-
R(X, Y)17UZ
-
/-t
-
((VXU, Z) /-t(- (U, vxz)) Y Y
K
-
dl-t(X) (Y, Z) (K
+
+
(VXY), U)Z
i
R(Y, U)VXZ
-
VX (A (Y' Z) U) -
VUY)
(X, Z) VUY'
tt
R(7ru
-
-
-
-
(Z' VXY) U) (Y' Z) VXU) p Y' VXz) U
-
U
P) ((Y1 Z) VXU
Z' VXU) Y),
-
and
(VyR) (U, X)Z
(VyR) (X, U)Z
=
-
=
-d[t(Y) (X, Z) (r,
-
Inserting these equations 0
=
dti(X) (Y, Z) +
0
=
Since
n
+
> 3
U
-
Z,
Y I 0
=
dy(Y) (X, Z)
K.) ((Y1 Z) VUX
(X, Z) VYU there
[1) ((X, Z) VYU
are
-
into the second Bianchi
dr,(U) ((Y, Z) X -
with X J-
(A
+
U
-
+
-
we
obtain
VUY), + n) ((Y, Z) VXU (6.1.2) (VXU, Z) Y).
(y
-
Vyu' Z) X
identity
U
(X, Z)
X, Z) Y)
Z' VYU) X).
-
pointwise linearly independent vector fields X, Y Equation 6.1.2 implies
Z. For these vector fields
(/,t
+
K)
((VYU, Z) X
-
Vxu' Z) Y).
It follows
immediately that for orthogonal vector u, v the expression. Hence, restricted to Uj-, the bilinear form VO is (u, VvU) 1 a multiple of g restricted to U-L. In particular, VO restricted to U vanishes.
symmetric. We will now show that the tensor field V((n + I-t)U) symmetric in all of T, M. Since U is symmetric on U-L and V ((n + M) U) (d/-. + dy) 0 U + (n + y) V U we only have to show (VX ((n + /-t) U), U) is
is =
=
6. Robertson-Walker
292
VU((n + y)U), X). from
Equation
(1,t
=
+
we
Z and
X 1 Z at
choosing
get
x we
-(p
=
+
n) (X, VUU).
implies
-
-1 2
(n
=
equation with dlt(x)
-
3)dr,(X)
-
have used the formulas
we can
=
n) (VUX, U)
dp(X) and therefore
Y
Setting
6.1.1
dl-t(X) Lemma 5.2.1
cosmology
+
+
dp(X)
provided by
-(M
=
p) X, VUU),
(E
(ft
n) (X, VUU)
+
+
n) X, VUU),
Lemma 6.1.1.
obtain
we
where
Combining
dn(X)
=
this
0. Now
calculate.
(17X((r, + y)U), U) (17U((K + A)U), X)
=
-d(r,
=
(y
It follows that V
((r, + /,t) U)
d((n + M)W) dt (K+p)W.
0.
=
=
to U for each
multiple
of g
By
The
p)(x)
+
is
-dp(X)
=
(p
+
n) X, 17UU),
17UU' X).
K)
+
=
symmetric
in all of T, M and that therefore
the lemma of Poincar6 there is
hypersurface Zi
Ix
=
G
M
a
Tj
t(x)
:
function t with is
orthogonal
f. Its second fundamental form, k(u, v) (u, VvU), is a restricted to Zi which implies that Zi is totally umbilic. I
Theorem 6.1.1. Let
(M,g) be infinitesimally isotropic and U be the cosmological observer field. Assume that e + p0. Then there is an (n 1) -dimensional Riemannian manifold (A , ,) of constant curvature R x M, g-dt2 + a2(t),, where F E f 1, 0, -11 such that (locally) M -
=
e/O
=
Proof.
K
_
I(dn(U)/(K + /_t) 4
2.
It is clear that g (r + A)-2 dt2 + t for some t. Let u, v E Uj- be vectors with 1. Then Equation (6.1.2) gives with v
mannian metric
(v, v) dK(U)
=
=
=
(K + /,t) ( VvU, v) + (VuU, u ).
-
form of the
hypersurfaces perpendicular 1
k
The GauB
=
(u, VVU)
2
X,
u
Y
=
to U is
given by 1
+ /,t
2
OtKg.
equation gives
Rz, (u, v)w
R(u, v)w
(/,z
1
4
+
(
r,
k(u, w)k(v,
+ A
Rie-
(u, U)
I V,
=
Z
Hence the second fundamental
dK(U)g K
t-dependent u
)2)
-
((v, W)
k(v, w)k(u, u
-
(u, W) V)
.
Homogeneity and isotropy
6.1
By the
constant in the
hypersurfaces.
R
a:
R+ is
--->
constant curvature
We will
the
proof
therefore t
=
E
can
I 1, 0, 11
1 n
-
(dr,(U) T4
2
-
-Z
+1z
)
is
words, the hypersurfaces Et have + M)-2 dt2 + a2 (t),, -
l)-dimensional
metric of
-
-
show that the factor
now
dt. To this end
a
e
In other
the factor
therefore write g function and , the (n
constant curvature. We
where
(Proposition 4.3.4)
Lemma of Schur
293
1/(n
+
M)
can
be absorbed into
have to show that M only depends on t (recall from of Lemma 6.1.2 that dn(X) 0 for each X E TZt and that we
=
r,
const,
depends only
so
on
t).
Since
n
at k
=
atK
hypersurfaces
'
t. On the other
on
2 a
-
2
depends only U1
is constant in the
at n. This implies that
is
2
hand,
a
g.
direct calculation gives for
v,w E
k(v, w)
=
(V,'7wU)
=
(r-
+
(/-t
+
V,VwOt)
(K
+
Y)
Ocgat
+
atgca
=
1
=
Comparing the
P)
2
-
depends only we
/-t)gabva rclt wc
+
1 =
2
(N
+
A)'Otgea vawc
r,)aata,. two formulas for k
t. Hence
on
take
(K
19agc't) VaWb
we
obtain that
(a0tK -ata which
=
+
(N(t) +p(t))-ldt
as our new
K
is the differential of
a
function I
time coordinate.
Corollary 6.1.1. If (M,g) is infinitesimally isotropic then it is also spatially homogeneous, i.e. for hypersurfaces Zt orthogonal to U and all x, y E Zt there exists for any pair of orthogonal frames of Y: Zt, TyZt M which maps one of the frames into the other. an isometry M "
-->
Proof This follows is
a
from
Corollary 4.5.1
and Lemma 4.5.5 since
(Eta 2 (t)6 I
Riemannian manifold of constant curvature.
Corollary 6.1.2. Let (M, g) be infinitesimally isotropic and U be the cosmological observer field. Then there is an interval I (t-, t+) and a spaceform (Z, ,) of constant curvature E E I 1, 0, 1 such that the universal cover of (M, g) is isometric to (I x Z, -dt2 + ,) and U at. 4 then there are local coordinates (t, r, 0, p) such that If n -
=
=
g
=
-dt2
+
2 a
(t)
( -1-r2 dr2+ I
r2
(d02 + sin2 (O)d 02)
)
-
(6.1.3)
294
6. Robertson-Walker
Proof The first part
cosmology
of the
obvious since I
corollary is
x
Z is
simply
con-
nected for any spaceform Z. The second part follows immediately from the classification of 3-dimensional Riemannian manifolds with constant curvature
(cf.
Lemma
A spacetime
(M, g)
(6.1.3)
1
which is
finitesimally isotropic spacetime
4.5.5). locally
Robertson- Walker
or
isometric to
Lorentzian manifold is called
cosmology.
a
4-dimensional in-
a
Robertson- Walker
The metric given
by Equation
is called the Robertson- Walker metric.
6.2 The initial value
for
problem infinitesimally isotropic spacetimes
solve Einstein's equations Ric -1 Scal g + Ag 87rT for 2 Robertson-Walker cosmologies. While, in general, Einstein's equations give rise to a system of partial differential equations, in the case at hand In this section
have
we
=
-
shown that the unknown functions
already
depend on only one ordinary instead of a partial system of differential equation. This simplifies the problem greatly. However, even this simple case exhibits typical aspects of Einstein's equation.
we
variable. We will therefore obtain
an
We denote the derivative with respect to t with
Lemma 6.2. 1. Let be tangent to the vature
(M, g)
be
a
hypersurfaces are given by
Robertson- Walker spacetime and u, v, w Et which are orthogonal to U. The cur-
expressions
R(u, v)w
=
( ()2 a2
-'
+
2
) ((v, w)
u
-
(u, w) v),
a"
R(v, U)U
R(v, w) U
=
0,
a"
R(v, U)w
v,
(v, W) U,
a
a
a"
Ric(U, U)
=
-(n
-
1)
Ric(U, v)
,
=
0,
a
(' )) ( W) (n 2) (' )) 6
w)
Ric (v,
(n
+
-
a
Scal
Proof.
=
(n
1)
-
a
(2
2
+
a
V,
2
-,
+
+
-
a2
a
_2
The constant curvature metric 1 9z
=
-
-
dr
-2 61
has the curvature tensor Rz (u,
4.3.3).
2)
2
+r
v) w
2
==
(d02 + sin 2(O)d W2) E
((v, w)
u
-
u, w) v) (Proposition
Hence the formula for R follows from Lemma 4.4.14. The other
formulas
are
direct consequences from Lemma 4.4-15.
1
6.2 The initial value
Corollary
If
6.2.1.
for
problem
the energy momentum tensor is T
(c
=
then Einstein's equation is
p)0
+
equivalent
87rE + A
(n
2
295
infinitesimally isotropic spacetimes
-
1)(n
&
0
+ pg,
2)
( (a/)
given by
to
-
6
(6.2.4)
+
a2
a2
and
87rp
A
-
=
-(n
-
3
n
(
2)
+
E
+
(6.2.5)
2
a2
2
a
((a?
Proof. Recall from Lemma 6.2.1 that the Ricci tensor Ric restricted to spatial subspace U,,-L is a multiple of the metric g restricted to this 1 Scal g + Ag 87rT subspace. It follows that Einstein's equation Ric 2 restricted to this subspace,
the
=
-
a// +
(n
-
2)
LaL)2
e
+
a2
a2
a
n
2 2
is
equivalent
+
(n
-
2)
(a? +
a2
a
+
2
A)
g
8-7Tpg
Equation (6.2.5). From Lemma 6.2.1 we get Ric(v, U) v J_ U which implies that the only other non-trivial Einstein's equation is given by evaluating it on the pair of
to
0 for all vectors
component of vectors
(U,,, U,,). at/
(n
-
1)
n-1 +
a
which is
We obtain
2
equivalent
to
(2
at' +
(n
-
2)
a
Equation (6.2.4).
(a' )2 +
a2
a2
)) -A)gjE
=
87rEgIE 1
equation is not a well posed system of differenhave only two equations for three unknowns, we Instead, equations. of the function a appear in our system derivatives Moreover, only a,,E, p. of equations. The first problem has a direct physical resolution. Just specifying a perfect fluid is not enough to specify a matter-model completely. Rather, perfect fluids give a framework which is fitting for many different matter models. In particular, vacuum is a (very degenerate) perfect fluid, and so is dust. In order to arrive at a determined system of equations we therefore have to specify an additional relation between the energy density c, the pressure p, and the metric described by a. In the following we make a rather simplistic assumption, namely that there is a given equation of state: p f (c) for some smooth function f : R --> R. Having made this assumption, we still have the problem that we have Observe that Einstein's
tial
=
6. Robertson-Walker
296
cosmology
equations for a rather than a system of two differential equation for a and E. We can resolve this problem by replacing one of our equations with an equation of motion (Lemma 5.2.1). two differential
R Corollary 6.2.2. Assume that there is a smooth function f: R with f (e) > F_ for all 6 E R. Let ao G R+ \ f01 and 6o c R and assume -+
-
2 (ao )2 + 2) (81rEo Tn-_(n 1)
that
-
-
A)
(a, c) of Einstein's equations a,
Then there exists
> 0.
e
-
such
a(O)
e(O)
and
ao
=
unique solution
a
=
E0.
The
functions
satisfy
E
(i) (n
-
2)
a'
-8ir
a
(f(c) n-3.) +
2
+
n-1
A,
n-1
a
(ii) (n
E+P
a
and
a'(0) Proof. Assume
2(ao)2_ (8-7rEo
(n
1)(n
-
-
+
2)
A)
-
solution of Einstein's equation.
EquaEquations (6.2.4) and (6.2.5). Equation (ii) follows from Lemma 5.2.1. Finally, the equation for a'(0) follows immediately from the equation for E in Corollary 6.2. 1.
tion
(i)
is
first that a,
linear combination of
a
For the
converse
unique solution a, [--2(ao)2
a'(0)
=-
is also
E are a
E
notice first that for
initial conditions there is
our
which satisfies the system of equations
(i), (ii)
a
and
We have to show that this solution
(87rEo A) -(n2) solution to the system of equations given in Corollary 6.2-1. It +
V7
-
n
a
is clear that this
system of equations is satisfied
at t
=
0. We will show
that the first equation is satisfied for all t. Since the second equation is a linear combination of the first equation and equation (i), it must then also be satisfied for all t.
Defining
87rc + A
-
2
(n
1)(n
-
have to show that 0 vanishes for using equations (i), (ii) gives
87rE'
-
2
(n
1)(n
-
-
-
Taking
2aa'
2)
-
8-7r(n
(ii) a'
-
2
the derivative of
+
6)
-
(N
-
1)(N
-
2) a
(a/ )2
a'/
(a
+6
a2
a"
87r(n
-
I)E
+
(n
-
1)(n
-
2)
+ a
a
-(N-1)(N-2)a' a
0
a4 a'
1)a(E +p)
a
((a')2
a/
(i) =
all t.
2aa"a2
1 =
2
a
we
0'
2)
-
a'/
(a
(a/
)22+6 a
87r(n
-
3)E
+ 2A
and
6.3 Geodesics and redshift
297
2a/ a
Since
0(0)
=
0 and 0 is
a
solution of the differential equation 22a 0, ordinary differential equations implies that
the fundamental theorem for
0
must vanish for all t.
I
This solution of Einstein's equation is is often
for the are
not
that
typical in two aspects. Firstly, it exchange part of the original set of equations 0. Secondly, Einstein's equations motion, div(T)
advantageous
to
equations of a free system of differential equations but =
we were
not free to choose
equation for
a.
In other
a'(0)
even
words, only
constrained. Recall
are
though
had
we
a
second order
restricted set of initial values
a
had the chance to lead to solutions of Einstein's equation. The system equations which was solved was derived from Einstein's
of differential
equation but
not identical to it. We had therefore to show that the
solutions to this system are also solutions to Einstein's equation. We did so by deriving an additional linear differential equation and used our
constraint
this equation
the choice of
(i.e.,
a'(0))
implies that the original
phenomenon has
to show that the solution of
set of
equations is satisfied. This settings where we
direct counterpart in more general have to deal with systems of differential equations. a
6.3 Geodesics and redshift In 1929 Hubble made
galaxies
are
moving
a
cosmological discovery which implies
that distant
(and each other)
proportional
away from
us
at
a
rate
to their distance. This astronomical fact shattered the
long cherished idea physical processes take place. 2 It is instructive to describe Hubble's discovery in slightly more detail: Each star has a spectrum of light which contains characteristic gaps due to absorption of light of certain frequencies in the atmosphere of the star. Since we have physical explanations for these absorptions, we can calibrate these patterns and thereby obtain information about the chemical composition of the star's atmosphere. Hubble discovered that for stars in galaxies which are not too close 3 these gaps are shifted towards smaller frequencies. Moreover, this shift is proportional to the distance of the galaxy. From his observation it was then that
our
universe
was an
eternal
arena
in which the
2Einstein introduced his cosmological
constant a decade earlier because he wanted to have static solutions in accordance with the prejudice of his time. Had he not done so, there would have been another striking prediction by
general relativity. very nearby galaxies
3For
overshadows this effect.
the
(local)
movement of the
galaxy relative
to
us
298
Robertson-Walker
6.
cosmology
concluded that all
galaxies are moving a-way from each other. (Everyone analogous effect: If a fast car is approaching one has the impression that the noise of the engine is higher pitched than when it is moving away: In other words, if the source and the detector of a sound move away from each other, the frequency of the sound appears is familiar with
to be
an
smaller).
In this section
we
will show that Hubble's
discovery can be undercosmology (cf. Corolof the great successes of general relativity
stood within the framework of Robertson-Walker
lary
6.3.2
below).
This is
one
and the
isotropy assumption. Recall from Sect. 1.4.3 that we can describe photons by null geodesics. The energy of a photon -y measured by an observer u is given by E hv (u, ). Here v denotes the frequency of the photon (as measured by u) and h denotes Planck's constant. In Robertson-Walker spacetime we have a natural unit vector field U which is approximately tangent to the world lines of the galaxies. We will therefore define the energy of a photon using this distinguished observer. In this section we will always refer to this energy. Let -y be a photon which moves from x e M to y c M. In general, it is possible that its energy is not constant along the world line of the photon. This is traditionally expressed using the fractional increase z of the world lines of
=
the associated
being
=
11v
detected at y is
(t, Y), (t, Y)
I
Z
-
hIE: z
of
a
photon originating
at
x
given by A (Y)
AW A(X)
*1 Y) If
=
The redshift factor
Definition 6.3.1. and
A
wavelength
=
-
the events
occupied by two galaxies, then the a (t) dz (Y, Y), given by d ((t, i), (t, Y)) where dz (Y, Y) is the distance of Y and Y in (Z, ,). We will show that there is a constant H such that we have approximately z(x, y) Hd((t, Y), (t, Y)) for galaxies which are distant enough for Hubble's disE
x
are
distance of these events at time t is
==
==
covery to hold but still
this end
we
close that it is sensible to linearise
so
must first calculate the null
geodesics
z.
To
in Robertson-Walker
spacetimes. Lemma 6.3.1. Let
(M, g)
finitesimally isotropic
8
is
a
aeodesic if and
::=
x Z, -dt2 + a 2(t) manifold. The curve
((t-, t+)
Lorentzian
F-+
-Y(s)
=
(t(s), 1(s))
only if
d2t
a' dt +
d 82
'Y,
a(t)a'
=
0,
+ 2 a
a,
0
,)
be
an
in-
6.3 Geodesics and redshift
hold,
where
(t(s), I(s))
t is
denotes the induced covariant derivative a
null
geodesic then the
conservation
on
Z.
299
If y(s)
equation
dt
a(t(s))
const
holds.
Proof. The first part follows immediately from the corresponding formugeneral warped products (Corollary 4.4. 1). Assume now that 7
las for is
a
null
geodesic.
equation and
(4, -4)
dt
dt
2
From the first
a
ly
(dt/ds)2
=
we
obtain
d
TS_ (a (t(s))
)2
a'(t)
a(t)ddS22t
+
0.
=
Corollary 6.3.1. Let (M,g) ((t-,t+) x Z,-dt2 + a 2(t)g,) be infinitesimally isotropic Lorentzian manifold. The curve s 1-4 -y(s) (t(s), I(s)) is a null geodesic if =
(i) -F (ii) and
is
1-4
unit
a
there is
a
speed geodesic
constant
c
(Z,
in
such that t (s)
8
=
C
ft.
a
an
(i) di,
1(s)
di
(ft,,
ly
a(i)
Proof Assume that 'y' is a unit speed geodesic and that the integral equations (ii) hold. The curve -y(s) (t(s), I(s)) is a null curve because .
=
of
-(dt/ds)2 =
=
ds
26
d
( J_T
1 +
d
z,
2 (dt/ds)2
dT
,
integral equation
equation d
0
a
d
(dt/ds)2
It follows from the first
Hence the first
+
d-r in
d
v
Y7
(ii)
that
dt
a
))
=
0.
adt/ds
is constant.
in Lemma 6.3.1 follows from
dt
(a (t(s)) a-s )= a'(t) ( TS_ )2
and the fact that -y is
Z,
V/ r
null
curve.
+
a(t)ddS22t
The second equation follows also
direct calculation:
t ,=
I
-
Y
c2a2
t7
d
='
a7r'y
(
I
d
a2 d-F
)
-2 da d
c2a5 d-r d-r
z,
-2 a
da.:, ds
by
300
6.
where To
Robertson-Walker
have used
see
that all null
to notice that
where
d' ds
we
e
is
a
(up
to
1 d' ads
ale-*
tange,,-iat
from
(M, g) c
x
be described this way it is sufficient any null vector can be realised as 9t + e
can
niultiple)
6.3.1. Let moves
I -
geodesics
a,
unit -vector
Proposition photon which
cosmology
to 'S4,
be
infinitesimally isotropic
M to y
and 7 be
a
M. Then
c
1.
Z
a(t(x)) Proof -
A
From Lemma 6.3.1
.
jd;t ds
(U, =
we
implies that A/a
ka in the definition of
a(t) ds!
get that
h/(a(t) jdt,) ds
=
is constant. Hence E
k is constant.
=:
Inserting
proves the claim.
z
Corollary 6.3.2. Let xo (to, Yo) E M. Then the frequencies emitted by nearby galaxies (situated at y (t,:V) E M) appear to be red-shifted at xo by z Hto d((to, Yo), (to,: o-)), =
=
where
Hto
=
a'(to)/a(to)
Proof. We
assume
Hence
obtain
we
is the Hubble "constant" at xo.
that dz (
Fo, go) < 1 which in turn implies It a(to) + (t to)a'(to) and therefore
a(t) a(to)
1
z
-
-
+
I
;:z
t
3.
(M, g)
=
If
(i) there is a to E (t-, t+) with Hto > 0 (ii) E, p are continuous on (t-, t+), (iii) e +A/ (87r) > 0, (iv) there exist constants c such that E n
-
3
p
n
then t- >
limt,t-
a
-oo.
In
2(t)(6(t)
-
I
addition, +
+
c
we
A/(81r))
-
:5
< c_
=
and p
Al (81r) AI(8-x)
limt-t_ a(t)
have oc.
For t+ there
lim.
a'(t)
satisfy
< -
=
0 and
are
the
limt-t_ a'(t)
following possi-
bilities.
1
8
lim.
t+, t
-1
1
(t),
lim.
t -t+
a'(t)(E(t)
+
A/(8?r))
t-t+
00
1
0
00
00
0
0
finite
0
-00
00
00
0
a
t+
Proof If c can be extended beyond t- or t+ as a bounded function so call p. For given E, p, Corollary 6.2.2 (i) can be viewed as a linear differential equation of second order for a. Consequently, if t is finite, a could be extended
as a
C2- function if it is
(Dieudonn6 1960, limsupt-t, a(t)
not infinite
or implies liminft,t, a(t) 10.4.6). (M' g) is C2- -maximally extended by assumption.
This
(iv) imply
=
that 8,7r
(P
+
n-3. n-1
)
2A/(n
0
-
1)
is
=
Conditions
positive
Remark 00
since
(iii)
and
Hence a" (t) < 0 -
by Corollary 6.2.2 (i). The function a' is therefore monotone and we can replace lim inf (and lim sup) by lim for a and a'. First we investigate what happens near t-. The inequality Hto > 0 implies the inequality a(to) > 0. Since a"(t) :5 0 for all t, the graph of a lies below the graph of the map t -4 a(to) + a'(to)(t to). This linear for all t
-
graph
(a 0)-axis a(to)/a'(to), to).
intersects the
is finite: t- E
there is
a
[to
-
6 > 0 with
=
at
a(t) < to which implies that tto a, (to) Since p A/(87r) > c- (,E + A/(87r)), -
-
6. Robertson-Walker
302
cosmology
J
3
(I
+p
E
c-)(f, + A/(87r))
+
)(E
>
+
A/ (87r))
n
2 + J
A/ (87r)).
+
Corollary 6.2.2 (ii) implies (E + A/(87r))' -(n 1)(E +p)a/a < -(2+J)(e+A/(87r))a'/a and therefore ((E+A/(81r))a2+6)' < 0. The equa0 implies now immediately that limt,t- O(E(t) + tion limt-t, a(t) oo. From the equation for the energy density c in Corollary A/(87r)) Hence
-
==
=
6.2.1
(a/)2 diverges
infer that
we
For t
t+ there
also.
possibilities. If a has no maximum oo. Since ((E + implies that t+ A/(87r))a2+J)' < 0, the function (e + A/(87))a2+6 is decreasing which 0. The equation for the energy implies limt-t+ (E(t) + +A/(8ir))a2(t) density E in Corollary 6.2.1 implies now that < 0. The assertions about -->
limt,t+ a(t)
then
are
oo.
=
several
a"
0
3 and
as-
following
statements
-
+
-
-
6)
161rma2-n
=
+ 2A.
0 (n 1) is equivalent to (n 1) + Proof. The equation 6 obtain this anda' 0. we 6.2.2 equation Integrating : Corollary Ean m for some constant m. The equivalence of (ii) and (iii) is clear =
-
-
n
a
e
because of =
from the formula for
In the
case n
explicitly
=
4, A
1
in Lemma 6.2.1.
c
0 the
=
and the solutions
resulting differential equations given by
can
be solved
are
t
E
1-4
a
(t)
-
C. 'F_j + (1 + t/C)2 0 (4C2)1/4VFt CVj_ (I t/C)2 1.
-1
-
-
where
c:
is
a
constant of
We will make the of matter is 6.5.1 and
integration.
assumption that the
negligible.
6.5.2, radiation dominates
dominates at late times imation to
use
interaction between both
Observe that due to the formulas
((a
>>
1).
at
early
Hence it
the radiation model for the
model for the present universe.
times
seems
early
(a
to be
(ii) < a
types
in Lemmas
1)
and dust
good
approx-
universe and the dust
7.
Spherical symmetry
chapter serves two purposes. Firstly, a large isometry group simpliproblem of solving Einstein's equation considerably. Virtually all explicitly known solutions of Einstein's equations for physically plausible matter fields have a high degree of symmetry. Secondly, a spherically symmetric spacetimes are very good descriptions of non-rotating, isolated stars and therefore of astrophysical interest. (If the star rotates
This
fies the
the rotation axis breaks the
symmetry).
In Sect. 7.2
we
will
see
that
unique 1-parameter family of spherically symmetric solutions to Einsteins's equation for vacuum with vanishing cosmological constant. The parameter can be interpreted as the mass of the isolated star. If the mass of the sun is chosen, one obtains an excellent model of the gravitational field in our solar system. Some aspects of this model have been there is
a
verified
experimentally.
These solutions also form the basis for much of
intuition of black holes.
our
In this
chapter
we
will also discuss the initial value
that the energy momentum tensor represents gether with a non-interacting electric field (cf. Sect. case
more
general
less arrive at
case we a
will not obtain
explicit
a
problem for the perfect fluid to-
7.4).
solutions
we
While in this will neverthe-
non-trivial existence theorem for the considered class of
spacetimes. This section will hardly be of primary interest to a geometrically oriented reader. Since the discussion uses elements of the theory of systems of hyperbolic partial differential equations even physically oriented readers may wish to skip the proofs on first reading. The validity of the physical conclusions from this (and also the following) chapter depends very much on the question of whether the corresponding properties of our explicit solutions are stable under perturbations. We know only very little about the stability of Einstein's equations. Since they are highly non-linear it is well possible that these properties have little to do with our actual universe which has only "approximate isometries".
M. Kriele: LNPm 59, pp. 307 - 355, 1999 © Springer-Verlag Berlin Heidelberg 1999
308
7.
Sphericai sprimetry
7.1 Pseudo-Riemannian manifolds with
spherical symmetry
A property in R 3 is spherically symmetric if it is invariant under rotations about the origin. The rotational isometries defined below form a group
which is our
(locally)
isometric to the rotation group main justification of the following definition.
SO(3).
We take this
as
Definition 7.1.1. A
pseudo-Riemannian manifold (M, g) is called spherically symmetric if it has a dense open subset M' such that (MO, g) can locally be written as a warped product (Z x S', gz + r'dS?'), where dQ' is the metric of the 2-dimensional unit sphere, (Z, gz) an (n 2) dimensional pseudo-Riemannian manifold, and r: Z R a positive function. The sets f xJ X S2 (x E Z) are called spheres of symmetry and those isometries which map all spheres of symmetry into themselves are called -
-
---+
rotational isometries. The set C
Lemma 7.1-1. Let
(Z, gz) res
(M,g)
be
M
=
\
M' is the. centre of symmetry.
spherically symmetric spacetime and manifold orthogonal to the sphe-
a
be the 2-dimensional Lorentzian
of symmetry. For each frame f U, QJ of (Z, gz)
(U'U)=-1' there
are
g
and U
=
U'Q)=O'
(Q'Q)=I'
adapted coordinates with respect =
-
e2v(t,q)dt2
e-',9t, Q
=
e2,\(t,q)dq 2
+
to which
r2 (t, q) (d02
+
with
sin20 dW2)
+
e-'\'9q-
In these coordinates the energy momentum tensor T is 1
87rT (U,
U)
(1
=
r2 +
(U
+
2(-Q
*
r)
*
Q*
2
(Q
_
r
r) 2)
,
(U* A)(U* r))
+
given by
-
A,
+
A,
r
87rT(U, Q)
=
-
2(U
Qr
-
(Q
-
-
OW
-
r)),
r
1
87rT(Q, Q)
=
-
+
72 (1
+
2(-U
(U
*
r)2
U
*
r
e
-
+
(Q
(Q
9
9
r)2)
v)(Q
e
r))
r
87rT(V, X)
81rT(I'22, r
(911 r
)
=
=
0
for
81rT(aI(0) r
=
all X
Q,, Q +
sin
0
V
+
U
*
E TZ
and V E
TZ-L,
a, 1
r
sin (0)
(Q U
*
*
r
V)2 +
_
Q
*
U
Q
U,, A
r
-
_
(Q
(U e
9
A)2
v)(Q
*
r)
7.1 Pseudo-Riemannian manifolds with
(U
-
87rT('90, r
01* r sin(O)
A)(U
o
+
spherical symmetry
309
A,
0.
Proof. The existence of the adapted coordinates (t, q, 0, o) is clear from 1 Ric Seal g + Corollary 2.4.2. Hence we only need to calculate 8,7rT 2 Ag. By Lemma 4.4.15 we have for X, Y tangent to Z and V, W orthogonal =
-
to E
1
Ric(X, Y)
=
Ric(X, V)
=
Ric(V, W)
=
2
-Scalz (X, Y) 0
I(V, W) r2
Since
ScalE
=
VVr(X, Y)
=
+
X
VVr(U, U)
=
U
9
U
*
r
VVr(U, Q)
=
U
o
Q
o
r
VVr(Q, Q)
=
Q
o
Q
9 r
which in turn
A) (U
9
r).
-
-
impliesAr
Hence
we
r
9
-
I
(grad(r), grad(r))
+
r2
r
4
_2Y
e
Ar
(
-
2
Seal
-VVr(X, Y),
-
2
2
'Ar
_2 (grad(r), grad(r))
-
r
VXY
-
*
v)(Q
o
r),
(Q
*
v)(U
o
r)
(U 9,\)(U
o
r)
-U*
obtain
r we
*
(Q
=
=
Q
Uor+Q oQ
o
U
9 r
or+
-
(U *,\)(Q
(Q*v)(Q or)
2
-
-
72
2
4(-U*Uor+Q*Qor+(Qov)(Qor)2
r))
0
72 (_(U
-
0
02
+
(Q
or
)2))
-
A
I
(I + (U r)2 0
(Q
_
0
r)2)
2
_r(Q*Qor-(U*A)(Uor)) 87rT(U, Q)
=
-
2(U
Q
o
o r
(Q
-
-A,
v)(U r)),
o
o
r
87rT(Q, Q)
=
IScalz -2(Q*Qor-(UoA)(Uor))
2
r
2
+
-
_2
4(-U
o
U
or
+Q
o
Q
*r+
I -
2
(Scal-v
(Q ov)(Q or)-
r
2
(U
(U
(ScalE
r
(U 0'\)(U 2
r),
1 +
r
2 +
9
get
IScal_r -2(U*Uor-(Qov)(Qor))
87rT(U, U)
(V, W),
o
A)(U
9
+ (U 1(1 2
r))
9
r)2
-
-
-2(-(Uor )2 +(Qor )2)) (Q
o
r)2)
+ A
9
310
7.
Spherical symmetry 2
(U Let X
9
r
(Q
-
v)(Q
9
r))
o
+ A.
0 and (X, V) 0 (TZ) 1. Since Ric(V, X) 0. Since there is an isometry which maps 87rT(X, V) a" aP we have 87rT(00,'90) 81rT( sin (0) sin ). The Ricci (0)
TZ and V c
c
it is clear that '90
U
9
a,
into
=
=
=
=
T
r
sin(O)
r
r
1
r
r
tensor restricted to (T.,Z)J- is a multiple of the metric which implies '91" 0. Finally, we the last component of T which needs to ) r r sin (0) be calculated is given by
T(-'91-,
=
1
-(I(-U*Uor+Q*Qor+(Qov)(Qor)
87rT( L11,21-1)= 2 r
r
r
I -
(U
9
A)(U
9
r))
1 2
(Q
9
v)(Q
2-(-(U
72
(-
+
o
r)2
9
po
+
r(-Uo Uor+ Q
-
72
o
(U
2
r)2)
4
2
(Scalz
-
+
+
r)
r)2
(U
-
(Q
+
A)(U
9
r)2))
0
9
Q
or
r))
o
+A
U*U*?-+Q*Qor
r
+ _
(Q
o
v)(Q
(U
0
U
Scalz
2(U
=
9
r)
A +
0
where in the last equation
9
we
-
(U
(U 0
A)(U
o
A)2
_
Q
r))
9
0
Q
0
V
_
po
V)2)
+
A,
have used
U *,\ +
(U
0
A)2
_
Q
0
Q
0
1',
_
(Q
0
,,)2)
(cf. Proposition 4.3-5). We will if
now
re-arrange these
in
form which is
more practical complete generality but the following genericity assumption on the matter model.
equations
a
wants to solve them. We will not do this in
one
rather
assume
Recall that for any normalised timelike vector v the number T(v, v) represents the energy density measured by u. This number should be
positive. For null Given
a
pect T(N, N) to
vectors N
obtain then
T(N, N)
> 0
by continuity. plausible to ex0 for all null vectors N. In this case it is always possible and T simultaneously (Greub 1981, Chapter IX 3). we
non-extreme matter distribution it is therefore
diagonalise
>
g
Lemma 7.1.2. Let
(M, g)
be
spherically symmetric Lorentzian mantensor field which is also spherically ifold symmetric (0) 2 T Assume and that symmetric. g can be simultaneously diagonalised. and T be
Then there exist coordinates g
==
-e
a
a
2v(t,q)dt2
+
(t, q, 0, W)
such that
e2,\(t,q)dq2+ r 2(t, q) (d02
+
sin20 d(P2)'
(7.1.1)
7.1 Pseudo-Riemannian manifolds with
T
=
E0
0
0
+
Q
PradQ
+ Psphr
spherical symmetry
+ sin 20
2(t, q) (d02
d(P2),
311
(7.1.2)
:= e-',Ot and Q e-AOq are invariantly defined if E '7 -Prad87rT is equivaMoreover, Einstein's equation, Ric Scal/2 g + Ag, lent to the system of differential equations
where U
=
-
(U
1 +
Q*Q*r= (Uer)(U* A)+
9
r)2
-
(Q
*
r)2
2r
'A
47rr(,E Q*Uor
(7.1.3)
+
87r
(Q*r)(U* A),
=
(7.1.4) (U r)2
1 +
e
(Qor)(Qov)-
U*U*r=
_
(Q
0
r)2
2r
A
41rr(Prad
-
-
-
8,7r
),
(7.1.5)
_(U*,\)2+Q*Q.V+(Q.1.,)2+
UOUOI\=
47r (E
-
and the equation
-
Prad +
2p,ph)
1+(Uor)2
=
0,
is
equivalent
2(E
+
Psph)
U
U
+
Prad)U
A
0
(Q
*
r)2
(7.1.6)
7
of motion, div(T)
-
r2
-
e
to
r
(7.1.7) r
Q
0
(E
Prad
+
Prad) Q
* v
-
2 (Prad
-
Psph)
Q*r
(7.1.8)
r
Lemma 7.1.1
Proof. -
-
0 T(X, V) and T (V, V)
for all X E T
=
Since
implies that
=
T (W,
by assumption
W)
Z,
T satisfies V E
there is
a
Q,
V, W
E
(-- Z)
frame which
mentum tensor T and the metric g
fields
(TM)'
for all unit vectors
diagonalises the simultaneously there must
U tangent to Z and functions r, E, Prad i Psph: Z g
T
=
_U5
=
EW
(j5
W
+
+
Q
(&
Pra&
-->
energy
mo-
exist vector
R such that
Qb +r2dQ2, (D
Q
+ Psphr 2dp2.
The existence of
adapted coordinates follows now from Corollary 2.4.2. equations are equivalent to Equations (7.1.3)-(7.1.6) follows immediately from the definitions Of E7PradiPsph and Lemma 7.1.1. Observe that for any vector fields X,Y tangent to Z the decomposition VXY -"VXY and that for any vector fields V, W tangent to the of spheres symmetry the decomposition VVW (V7 W) grad(r) + That Einstein's
=
S2
VVW
div(T)
=
holds.
(dE(U)
Using Proposition +
4.3.5
we
ediv(U))W +,E(VUU)
obtain
+
(dPrad(Q)
+
Praddiv(Q))Q'
312
7.
Spherical symmetry
+
+ Psph
+ Psph
=
I
Prad(VQQ)5 + (Psphdiv(
(de(U)
( (
i9o)) r dO r
I
V
r
ao r ao
r
sin(o)
I
V
+
I
+ Psph div r
sin (0)
a ,)r sin (O)d(p
aw
aw
E((U A)
2(U
+
9
+ Prad (U
*
A)) U
5
r
2
(dPrad (Q)
((Q V)
+ Prad
(Q Or)) +,E(Q. v))Q5
+
0
r
cos(O) Psph
dO
1dr + 0 + o
-
-
r
=
(dE(U)
+
E
((U A)
2(U
+
0
Idr
-
sin(O)
r
*
dO) U
r))
+ Prad (U
*
A)
+
r
+
-
(dPrad (Q) 2p.,ph r) Q5,
+ Prad
r
2Psph
U5
r
((Q V) 0
2(Q
+
e
r)) +e(Qe v)
r
r
where
we
have used
S2
div(,9o)
Proposition
4.3.5 to calculate
dQ2(S2,V ao 190,ao
+
dQ2(S2,7
9W sin(O)
ao
sin(O)
cos(O) sm(0) S2
div(
aw sin(O)
The functions U
dQ2(S2,Vao sin(O)
ao)
=
0.
A and
Q * v are well defined invariants since the Q given by [U, Q] (Q * 1/) U (U . A) Q. For spherically symmetric spacetimes we can define an invariant tion which can be interpreted as a mass. mutator of U and
is
Definition 7.1.2. Let Then its
mass
function
=
(M, g) m
is
The term "mass function"
sUmptions that there that it is
exists
be
com-
-
func-
spherically symmetric spacetime. defined by m :=:= 112 (1 (grad(r), grad(r))) a
-
can a
be motivated under the additional
centre of
symmetry, that Q
* r
as-
>_ 0, and
possible to diagonalise, g and T simultaneously. Recall from special relativity that mass and energy are equivalent concepts and that the energy of a point particle measured by an observer depends on this observer.
7.1 Pseudo-Riemannian manifolds with
In order to determine the in
of
mass
a
would first fix
313
spherical symmetry
object consisting of spacelike hypersurface repeach particle which intersects material
spacetime particles resenting an instant of time. The mass of this hypersurface would be measured by the infinitesimal observer represented by the normal of the hypersurface (cf. Sect. 1.4-3). The sum over all these numbers is then the mass of the material object with respect to the chosen hypersurface. Since we assume that we can simultaneously diagonalise g and T there must be a timelike eigenvector of the linear map Tab 9bc. Observe that we
a
it is orthogonal to the spacelike const the hypersurfaces t words, hypersurfaces of this should that This indicates use we defined. family are invariantly it is
unique if
-Prad. In this
E
case
const. In other
t
=
hypersurfaces in order to define mass. Multiplying Equation (7.1.3) with Qer and inserting Equation (7.1.4) * r obtain we (Q * r)2)) =2Q*m Q e (r(l + (U * r)2 ft(c + A)r2Q 81r =
-
and therefore r(t
m(to, qo)
=
47r
qo)
in
A C+ 87r
)
r2dr
where t
to is fixed.
=
(7.1.9) This
integral
can
also be written
A
as a
volume
integral,
) sin(O) ),r2 sin(O)(Q r)e- 'dq (6+ ) (Q r) (U 1-tB)
JB (
m(to, qo)
C+
r2
_Ir
dr A dO A
dW
A
9
A
dO A
d o
A
J
0
8?r
Ix
where B is the ball
E
M
:
t(x)
=
to, &)
< qo
1. Equations Q
*
r
=
I
the energy
0 would imply we integral 2 in view of E can also be interpreted as a me density c which 0 then T represents a smooth mass density. If, in addition, Prad Psph of freely falling particles and we would obtain the 3-parameter family smooth analogue to the motivation using individual particles above. In general, however, Q * r = 1. This reflects that one also has,to take into account the energy contribution of the gravitational field.
and A
had just
that
=
over
an
-
,
Lemma 7.1-3. Let
U
m
be the
* m
=
mass
-47rr
function of (M, g). Then
2(U r) 9
Q*m=47rr2(Q*r)
(Prad A)
A
-
E+
8,7r
87r
314
7.
Proof.
r)
-
Spherical symmetry
We have Us
(U
o
A) (Q
QOM
Q. +
Qer
r). By
o
r
=
Q*Uor+[UQjor= Q*Uor+(Q*v)(U*
Lemma 7.1.2
(r2 (I + (U r)2 o
(yQ
U
o
Qor +
m
o r
-
(Q
Q
r)2))
o
r
o
r
(Q
-
calculate
we can
o
r) Q
9
Q
a
e
A
r((U r)(Q r)U o
o
r)
(Q
-
r)(U
o
o
r)U
o
r)
A
r
A
-
(Q
47r(Q
o
r)Tn
r)r
o
+
47rr(Q
r)
o
81r)
+
e
(6+ A)
2
8,7r
and Uor U
0
M
+
m
=
r
((U r)U o
o
U
o
r
(Q
-
o
r)U
Q
9
r
Uor
(Uor)(Qor)Qov-(Uor)m,
m+r
=
r
-
-
47rr(U (Q
e
9
r)
r)(U
o
A
(Prad
87r
r)Q
+
o v
(Q
(Q
9
r)Q
o
m,
+
r
-
m(U
9
r)
-
47rr(U
o
r)
r
=
While
-47rr'(U r) o
(A:
orthogonal coordinates
nates introduced below
are
U
o
r
r)2(U O'\))
Uor =
o
(Prad
A -
87r
A ad
-
87r
are
often very useful double null coordiadapted to the geometry of
much better
spherically symmetric spacetimes.
(double null coordinates). Let (M,g) be a 4-dimenspherically symmetric Lorentz manifold. Then there exist local coordinates (u, v, 0, o) and functions G: (u, v) --+ F(u, v) E R, r: (u, v)
Lemma 7.1.4
sional
r(u, v)
E R such that
g
The v
i--*
function
b(v)
=
G(u, v)dudv
G is
unique
+
2 r
up to
and
In these
Fu'
=
(U, V)
(dO'
+
sin2(O)d W2)
interchanging of coordinates, the Christoffel symbols
9,, (In r),
the
transformations of
form
u
-4
ft(u)
the coordinates.
I'&
are
given by 1
=
FufV
==
9u (In G),
FouO
u
==
sin
2
(0)
rWp
2rc9vr G
7.2 The Schwarzschild solution
F,v,
I
Since in
Proof.
point exactly
I
Fv00
Ov (In r),
=
r, '.
=
0, (In G),
=
=
sin
I"
Wo F11W
=
ow
r.V,
sin(O)
I
2
(0)
315
2r8,,r
FVW V
G
sin(O) cos(O)
two-dimensional Lorentz manifold there exist for each
a
linearly independent, lightlike directions, the existence immediately from Definition 7.1.1 and Corollary 2.4.2.
two
assertion follows
ft, f) be coordinates and 6(fi, 0), (ft, f)) be coordinates with G(fi,,b)dftdb + 2 (,a, f)) (d02 + sin2(O)d W2). Since the warped product invariantly defined we have G(u, v)dudv G(ft, ))dftdb. At each point Let
g
=
is
of
=
a
2-dimensional Lorentzian manifold there
tions whence
we can assume
functions fu,
fv with a,
(without and
fuau
=
of GauBian vector fields vanishes 0
and therefore
VU au fV 19,11
=
avfu
=
we
af,
0. This
=
=
are
exactly
generality')
f,,9,.
two null direc-
that there exist
Since the commutator
obtain
fu A fV) 191,
=
I
a,,f,
loss of
-
implies
fl, (C')V fu) au
f fu(u)du
fi
and b
f f,(v)du. It is
formula
to calculate the Christoffel
straightforward
Fbac
I =
2
gad (abgdc
+
Remark 7.1.1. The function
acgbd
-
M
r:
adgbc)
-4
R
symbols using the I
-
gives the
area
of the orbits S., via
the equation Area(S.,) 41rr 2and is therefore invariantly defined. That G is almost an invariant is one of the two main reasons why double null =
(u, v)
coordinates
are a
very
practical choice. The other reason is that in (M, g) is explicitly described.
these coordinates the causal structure of
7.2 The Schwarzschild solution In this section
we
ically symmetric
will solve Einstein's equation for the case of a spherspacetime. These solutions describe the grav-
vacuum
itational field caused
empty
space. As the
by
sun
a
single non-rotating star which is situated in slowly and space is almost empty,
rotates rather
these solutions describe gravitation in the solar system very well. Theorem 7.2.1 uum
(Birkhoff).
spacetime. Then either
given by
m
=
11(2VA-)
or
M' C M such that each
with local coordinates '
Otherwise
we
x
Let 1
r
=
E M'
(t, r, W, 0)
exchange
A
there is
ft and f;.
a
(M, g)
be
a
spherically symmetric vacmass function is
is constant and the constant mo and
admits which
a
a
dense, open subset neighbourhood
local coordinate
satisfy
7.
316
Spherical symmetry 2
2mo
r
9=
A
3
r
Proof. Lemma
dr2
dt2+
_
7.1.3
2mo
r2A
r
3
+
r2
* M
9
( 3)
=
U
in
a
and
Q
*
m
=
Q
6
constant ?no such that
m
mo +
=
open set then from the definition of
an
2(0)d(P2)
Ar
( 3)
0
6
Hence there is
+ sin
implies Ar
U
(d02
m we
A3
If
.
6
get
(Qor)2 mo +
2
(Uor)2
=
Ar3
which
6
const. Since T 0 we can choose coordinates implies that r which simultaneously diagonalise g and T. Equation (7.1.3) implies then
in turn
r2=
=
1/A
Let
and
obtain
we
m
that
us now assume
orthogonal
are
=
coordinates
r/2
=
=
1/(2v/A).
gO (dr, dr) -(Ugr)2 + (Q * r)2 : 0. (4, T) such that r 4. Since T 0 this
There
=
=
=
choice
of coordinates
trivially diagonalises g and T simultaneously and we can generality that in Lemma 7.1.2 we have t t, 0. Hence Equation 4. We immediately obtain U * r e-'atq q (7.1.4) yields U e A 0 and Equation (7.1.3) implies without loss of
assume
=
=
=
=
=
m
Q*Q*r= Since
Q
=
e-Aar
this equation is
e-
and
can
be
_2_
integrated e-
to
2A
mo
Ar
2
r2
3
equivalent
2Y-
Ar
Mo
arA
_2
+ 3
2mo
r2A
2m =
-
3
r
where
to
give
A(t)
=
2A
Ar
A(t)
-
r
A(t) is an integration constant. Equation (7.1.5) implies e 2A arV A-- and therefore e -2AarV -e 2X,9rA which in turn yields v -
=
-
-
=
=
3
r
B(t) (1
-
r
_
A. After
(Q,,,r)2)
the assertion is
a =
re-parameterisation of L 2
(1
-
e
-2A)
L(l 2
-
t
A +
we
can
choose B
=
2m/r) implies A(t)
0. =
Tn
1 and
proved.
Observe that any spherically symmetric vaccum spacetime is automatistatic in the region gO (dr, dr) > 0. In the region gO (dr, dr) < 0 it is not static but has a fourth spacelike Killing vector field.
cally
spacetime has (for A 0) first been obtained by Schwarzschild static, spherically symmetric vacuum equation. (1916) Birkhoff then showed that staticity was not needed as an assumption. This
who solved the
=
7.2 The Schwarzschild solution
Definition 7.2.1. A
ishing cosmological The coordinates
spherically symmetric
constant
(t, r, W, 0)
In the rest of this section
The
regions
r
these coordinates
r
at
2m. Below
we
will
A
assume
0.
=
be matched
> 2m cannot
geometrically determine
null coordinates of the solution. This will show that
and that there exists
singularity spherically symmetric
Fig.
vacuum
7.2.1. Schwarzschild
Proposition
F:
R+ \
naively using
101
a
-+
r
=
singularity
useful double
2m is
a
spurious
unique, inextensible solution of the
equation.
spacetime
in Schwarzschild coordinates
r -->
more
structure),
With
7.2.1.
f : R+ \ f Of
van-
a
7.2.1 which represents the causal time it has been believed that there is a physical
some
=
will
is called
(cf. Fig.
and for r
we
and
< 2m
spacetime with
vacuum
Schwarzschild spacetime 2 called Schwarzschild coordinates.
(M, g)
are
317
(-oo, 1),
r
F-->
f (r)
=
-
2m
(
2M
Ir
32m 3 e-'l
R+ \ fol
r
F-4
F(r)
=
)
e
r/(2m)
(2m)I
r
let B ,chw
(X, Y)
E R
Then the Lorentzian
(R
2
X
0. satisfies Ric related through =
2 :
XY < I I and
S2, 9B,;cl,,
+
(f-I(Xy))2 (d02
The coordinates
ln
( X) Y
In the literature this
F
of
(XY)
dXdY.
manifold
(t, r) of
+ sin
2(0)d(P2
Theorem 7.2.1 and
(X, Y)
are
t
XY
2m'
=
f (r).
name is usually reserved for a subset of the maximally extended Schwarzschild spacetime, the shaded region in Fig. 7.2.2.
318
7.
Proof.
We
Spherical symmetry
restrict to the base manifold
can
field N which is,
!, yic',r,,s
B,,I,w. For
any null vector
of synr).nnetry the
equations
2rn/r) (N') 2+ (I 2m/r) 1 (Nr) 2 0 holds, whence we have Nt 2m/r) 'Nr. Double null coordinates can now be obtained through integration of these two vector fields. Since fo (1 2m/r)--ldr
(I (1
-
-
=
-
-
-
r
an
r
-
+ 2mIn
=t+
r
(2.
1)
-
we
define
coordinates
our
by
r
(r+2mln (2m 1))
t
-
,
-
(r
This gives ddk (dt + (I 2m/r)-ldr)(dt d t2_(1_2m -2 dr 2 and therefore 9B,,,,,,, (1 2(r + 2rn In( lm - 1)) we obtain =
-
=
+ 2mln
r _
2m
1))
(I 2m/r)-ldr) d dk From Y
-
-
(
-
.
-
2m
e(,k- ')/(4m) which
-=
er/(2m)
r
r
(2m
2m
e/ (2m)
f (r)
2m
r
implies
9B,_j_
We set X
=
=
(
2m 1
-
2me -r/(2m)e(
ddk
r
e-'k/(4m)
,
(4m) d,dk.
r
Y
=
-ekl (4m )
and
finally
obtain
32M3 e-'/(2m) 9B,,cllw
Furthermore, Ox and Oy
Y).
are
f (r)
both future oriented
=
XY.
(this
has been the
the minus sign in the coordinate transformation for It remains to show that the inverse of f exists for all r > 0. But this
reason
follows
for
dXdY,
-_
choosing
immediately
from
f(r)
r =
_
-ZM-2 e
-"--
2-
< 0.
1
The coordinates
provided by Proposition 7.2.1 are called Kruskalcorresponding spacetime is often called Kruskal-Szekeres-spacetime. This spacetime is locally isometric to the metric given in Theorem 7.2.1 but the global structure is different from the global structure obtained by using Schwarschild coordinates (cf. Fig. 7.2.2). Nevertheless, in this book we will refer to the inextensible spacetime given in Proposition 7.2.1 as Schwarzschild spacetime. Szekeres- coordinates and the
Remark 7.2. 1. The motivation for the Schwarzschild spacetime is to dea non-rotating star. During the lifetime of the star
scribe the exterior of
the radius may change (typically, it may shrink and 0). If we denote the r-component of the star at t will need for each t
Fig.
only the part r region can be
7.2.2. The white
purposes.
perhaps even reach by rstar(t) then we > rtar (t) of the shaded region in completely discarded for physical
7.2 The Schwarzschild solution
319
Fig. 7.2.2. Schwarzschild spacetime. Radial null geodesics are the const. The region covered by const and Y straight lines X =
=
Schwarzschild coordinates is shaded
We will
glimpse
investigate this solution
now
at what is known
as a
singular
will show that this spacetime is need to calculate its
geodesics.
Lemma 7.2.1. Let
(M,g)
....
be
in
black hole: In
detail and get a first Proposition 7.2.3 below we
more
but inextensible. But first
pseudo-Riemannian manifold and diagonal,
a
we
(xi,
be coordinates such that gab is
Xn)
n
gabdxadXb
=
1: gadxadxa. a=1
Then the
geodesics
^1(s) of (M, g)
s
are
given by
n
d
i-s (ga
0
7(s), a(s))
=
2
1: 19agb
2
0
'Y (S)
( b (S)) (no
summation
over
a).
b=1
Proof. Then
We we
suspend
have gab
the summation convention if the
=
repeated index
and obtain ga ja b d
V
aa
7,
aa
aa
ds
(ga a)
_
b c C)b,
d
,s (ga a)
c_ (a.-ga jab+ 19agbe 2
-
19bga ja) c
is
a.
320
7.
Spherical synimetry d
Proposition
d
1- b
TsS (ga a)
caagbc
2
7.2-2. Let
s
(s)
-->
be
-=
a
1
jS (ga a) -
geodesic
( b)
-
2
2
49a 9b
in Schwarzschild space-
time
(R2 with
S2, 9B
X
f-2 (XY) (d02
_I_ +
+
sin2 (O)d
77 E f 1, 0, 11 and assume that y-(O) Then there exists a rotational isometry 0 such that 7 =
fx
-
dt
2m
2
E, E2
=
_
r o
(x) is
2rn
=
given by
7r
0
L,
2
(dra-s )2 (I 2m) (_,, +
:
dW ds
r
02))
+
L2 /r 2),
r
where
Proof.
E,L
constants.
are
We will
the metric is
the coordinates
use
diagonal
(t, 0, p)-components
we
can
provided by Theorem 7.2.1. Since apply Lemma 7.2.1 and obtain for the
of
d
dt
((I -r 2m) js- ) (0) ds) a-s ( ( is- )
=
0,
=
0,
=
r
_
is'
d
r2
sin
dW
2
d
2
dO
r
2
ds
There is
rotational isometry 0: x 0. Then 0 o -y last equation. The first two equations
sin(O) cos(O)
dW ds
such that 0
o 0 o %0) ir/2 unique solution of the can immediately be integrated. To derive the fourth equation in the assertion of the proposition it is more convenient to use the conservation property ( , - ) 71 than to use the r-component of the geodesic equation in Lemma 7.2.1. In fact, it follows directly from
and
an
dO(O.,y(O))
=
=
ir/2
=
is the
=
2m) (dt )2 ds
r
after
+
2m)-1 (dr )2 r
ds
2
+
r2
(d o ds
inserting the equations for dtlds and dWIds.
Lemma 7.2.2. Let Then there is
a
null
(M, g)
geodesic
be in
spacetime which is locally extensible. (M, g) which is incomplete and extensible. a
321
7.2 The Schwarzschild solution
(M, g)
Let
Proof.
be
(M, g),
local extension of
a
future
x
time
M
\ M, oriented) E
and y E M.
broken null
(not necessarily in geodesic from x to y. This broken geodesic y must intersect OM C M that of loss assume Without can we a point z generality 'Y (t) 7 (0). M this geodesic is incomplete for t < 0, Itl sufficiently small. Since z Then there is
a
or
=
I
and extensible.
Proposition 7.2.3. Schwarzschild spacetime is inextensible and geodesically incomplete. A future directed null geodesic is incomplete if and only 0 and the if it enters the region f 1 2m/r < 01. It then approaches r 48,tn2 Rabed scalar Kretschmann Rabcd /r6 diverges along this given by =
-
curve.
-2', 71 Ropop Rabcd
implies Rrero
Lemma 4.4.14
Proof.
==
=
Rrtrt
2m-r
r), and 2(0) Rotot 'T (2m _-i_-_!1__(OJ Rptwt components which are not related to these com-
2rmsin
0 for all other
i- 1703 Rwrw 1
-
,
ponents by the general symmetries of the Riemann tensor. It follows Rabcd Rabcd 48M2/,r6 and therefore that any curve -y(s) with (7Bo,,_h,7'YS2) in B chw X S2 r o - (s) --+ 0 is inextensible. A curve -y is extensible if and Only if 'YB.,,:I,w is extensible in B ,ch, and 'YS2 is extensible in S2 By Lemma 7.2.2 we only have to study null geodesics in order
that
=
=
.
to prove that
(M, g)
is inextensible. If 'YB.,,h, is extensible then dr/ds 7.2.2 and r 74 0. By Corollary 4.4.1 'YS2 is a
is bounded
by Proposition
pregeodesic
with bounded acceleration in
fore also extensible. Hence
we can
in this 2-dimensional
null
geodesics (X, Y) these geodesics
are
the reflection isometries
only
(Bschw gB_11w) and study spacetime. In Kruskal coordinates
given by X
(X, Y)
1-4
compact manifold and there-
a
restrict to
=
7
const
or
and
(Y, X)
Y
const. Because of
=
(X, Y)
--+
(-X, -Y)
Y > 0. The
region
r/2m
(i)
Y > 0 is the >
r/2m,
(ii)
1,
=
=
r/2m
(iii)
1,
each of them being invariant under future directed null
const, Y
we
const, geodesics of the form X disjoint union of three different subsets,
need to consider future directed
0.
We have to estimate the affine parameter of our null geodesics. If -Y is a null geodesic given by X const, Y > 0 then there is a function =
Y
h(Y)
--4
with
h(Y)Vayi9y)
==
=
h(Y)ay
h(Y)(h'(Y)
and
'7h(Y),9y(h(Y),9y)
h(Y)FyYy)ay. 1, where c: (gxy) +
From
=
h(Y)(h'(Y),9y +
Fyyy
=
9y ln(gxy),
-
we
obt ain therefore h (Y) In
-y(s)
=
region
h(Y)
=
c
is
a
const ant.
(ii) gxy is constant which implies that -y satisfies const and is therefore future complete.
=
Now consider
regions (i), (iii). Since
dr/ds
=
VE2
-
L2/r2 (I
-
2m/r)
-+
JEJ
> 0
1-Y ds
o
322
for
7.
r --
In
for all
Spherical symmetry
the parameter
oo
(i)
case
s.
we
have
(I
diverges if and only if r diverges. 2m/r) < 0 and the square root is well
s
-
r
dY/dr
defined
The equation
h(y) dsldr
=
32m
3e'/ (2m) (E2
-
L 2/r 2
(1
-
2m/r))
-1/2
diverges 3. Hence s diverges for Y --> oo and complete. In case (iii) it is clear from XY f (r) m-) er/(2m) that 2m (1 our future directed null geodesics X const, Y > 0 are approaching 0 and are therefore inextensible and incomplete. r I implies that geodesic
diverges
r
the
if Y
-y must be future
-
r
=
=
The
region
r
black hole is
< 2m, X > 0 is the simplest model loosely characterised by the fact that
of
a
black hole. A
light ray which enters it cannot leave it any more but instead reaches the edge of the universe before the affine parameter of the corresponding null geodesic has reached the value oo. 4 Since a black hole does not emit a single light ray one is tempted to say that it is black, whence the name coined by J. A. Wheeler. However, this name is slightly misleading, since the black hole is not in the past of any observer who is situated outside this region. Rather than appearing black it is simply invisible. An observer who enters the region does not have a very low life expectancy. The longest timelike curve within the black hole region is given by X Y, X E [0, 1]. In Schwarzschild coordinates this corresponds to the path t 0, r E (0, 2m). Hence the observer's life is bounded by a
=
==
0
As
7.2.1
=
f2
0
9B _ lw
((9r, ar) dr
m
Experimental
In this section
=f2m
V"2-m/r1-1 dr -
-
=
7rm.
tests for the Schwarzschild solution
will
investigate the region 2m/r < I which may be a non-rotating, spherically symmetric star of mass m. The discussion applies in particular to the gravitational field produced by the sun which was Schwarzschild's motivation for solving Einstein's equation in this special case. considered
as
we
the exterior of
3This property could also have been seen geometrically: The lines X const < 0, Y > 0 intersect all the hyperbolas r const > 2m. 4 A widely accepted general definition of black holes does not exist. The definition we have just given has the disadvantage that any RobertsonWalker solution which satisfies the assumptions of Theorem 6.4.1 and e I is a giant black hole. In this special case one would have to replace the condition that the null geodesics in the black hole don't reach the affine parameter oo by the condition that they don't end in the cosmological future singularity given by t t+. =
=
=
7.2 The Schwarzschild solution
Since the exterior
onal to the
Ro9t E) (,9t)
contains the timelike
Killing
field
09t orthog-
spheres of symmetry it admits a natural infinitesimal split of spacetime into space and time. Moreover, the distribution
integrable whence we obtain geometrically defined hypersurfaces constant time. These hypersurfaces are given by
(,Ot)' of
1
region
323
is
Zt,,
I (t, r, 0, p)
=
t
:
=
to, r
1.
> 2m
pullback of the metric to Et does identify all Et through projection along the t coordinate. A timelike curve in spacetime corresponds to a curve in the Riemannian manifold (Zo, (I 2Tn/r) -'dr' + r 2(d02 + sin 2(O)d W2)) at
Since
is
depend
not
a
Killing t and
on
vector field the
we can
-
which represents space. In
our
case,
we
may
imagine the non-rotating located in the centre
> 2m. It is
radius
system but the Schwarzschild solution is of
r
star to be the
sun
with
0 of the coordinate
=
only
course
valid for
>
r
2m.' It follows that the region r < 2m can be excluded from introduced above. our discussion and we can utilise the spacetime split It is natural to identify this spacetime split with the infinitesimal splits defined by our own world lines. "Space" has then its intuitive meaning. While in general timelike geodesics represent freely falling particles, in asteroids and our case they should be interpreted as planets (or perhaps
rsu,,
>
satellites)
-
Proposition 7.2.2 we can assume that light ray is contained in the plane 0
Because of
single planet
or
Lemma 7.2.3. Let
-y(s)
=
(t(s), r(s), O(s), W(s))
be
the movement of =
a
a
7r/2.
geodesic. Then
we
have
(
1
r2
dW) 2+ (-77L2
I
dr
+
2m)
2
p(s)
=
7.2. 1. Let -y (s)
dr/ds
and
d olds
in
1
be
a
geodesic.
Then
11r(s) satisfies d
-77m
dW2 Proof. Substituting p(s)
(-,qlL 2+02)(1
-
2mg)
=
=
+
+ g
L2
11r(s)
E 2IL 2.
3me2.
in Lemma 7.2.3 gives (do/d(p)2 + Differentiating this equation implies
1
the assertion.
5
for
(t (s), r (s), 0 (s), W (s))
=
L2
r
Proof. This follows by dividing the equations Proposition 7.2.2.
Corollary
E2 =
Hence it does not matter that the Schwarzschild metric is not defined at
the centre
r
=:
0 where the
sun
is located.
7.
324
Spherical symmetry
Bending of light rays. Since (null) geodesics are influenced by curvature, according to general relativity, light rays should appear bent near regions where gravity is large. In particular a light ray passing the sun at a short distance should appear to be slightly bent. The experimental verification of this effect was one of the first tests of the theory. To describe this effect we need to determine the angle a under which central a object appears to an observer in Schwarzschild spacetime. This angle can then be compared with the corresponding angle determined by the background metric dr2 + r 2dS?2 of space (cf. Fig. 7.2.3)
ly
Fig.
7.2.3. The size of
a
central star in Schwarzschild
spacetime
Lemma 7.2.4. Let -y be a lightlike geodesic and -y(O) Then a (ILIIE) /(-Or, (O)) satisfies ro sin(a)
VF1
=
=
=
-
(to, ro, Oo, wo). 2m/r.
"
Proof. Since (s) dtlds,9t + ly the null condition (I 2m/r) Jdt1dsJ' + 0 implies 1-412 2m/r) -1/2 E. Since (I V1 2m/rdt/ds ly the is I a a, angle given by sin(a) IL/rl/((l ld ods,9,1111(s)j aw ==
=
-
-
=
=
2Tn/r)
-1/2
-
-
-
I
E).
Corollary 7.2.2. Let 7 be a lightlike geodesic with past endpoint -Y(O) (to, ro, Oo, Wo). If -y) passes the boundary of a centred star of radius r,, 3m, then the angle a,, defined in Fig. 7.2.3 satisfies
sin(a.) Proof. Assume
(dr/ds) ,
=
r.
I
ro
I
-
== -
=
>
2m/ro 2m/r*
that -y passes the boundary of the star at s s*. Then r has a minimum there. Proposition 7.2.2 implies =
0 since
r*EIILI
=
V1
-
2m/r*
7.2 The Schwarzschild solution
and the assertion follows
by inserting this equation
325
into Lemma 7.2.4.
1
angle a. of the star is larger than one would expect physics where sin(a,,) would just be given by r,,/ro. This effect has been verified by a British team lead by Arthur Eddington (1882-1944) which measured the bending of light rays close to the sun during the total eclipse in 1919. They used the limiting behaviour given Observe that the
in non-relativistic
below. 7.2.4. Let IY be
Proposition
region 2m/r > minfr o -y(s) : s E RI along respect to the flat metric gflat and intersect at
an
A
angle
inextensible null
an
not enter the
Then there is
1. -y. =
+
which does =
Furthermore, dr2 + r2dQ2 which
4m/ro
=
geodesic
minimal radius ro there are two lines with a
are
asymptotes of -y
o( 77) ro
r o -y(so) exists since -y is inextensible Proof. The minimal value ro and r o -y > 2m by assumption. We may choose our spherical coordinates 0, o so that the light ray lies in the plane 0 7r/2 and the equation oo the coor0 holds. Proposition 7.2.2 implies that for s (p oy(so) =
=
-4
=
dinate W converges to limits V. The lines (with respect to gflat) which pas's through the origin under these angles W are therefore parallel to asymptotes of -y. The differential equation provided by Corollary 7.2.1 can
be solved
and has the solution
exactly 9
(P (Q)
1
d
=
_ 2 We
are
angle
+ 2
M0 3
+
(ro)-2
interested in situations where the ratio
A
=
2
lime-o o(p)
vanishes when
with respect to the parameter x Differentiating the function m/ro
=
m/ro
F-
W
C9(mo/ro)
mo(ro)-3
m/ro
is small. Since the
0
will linearise
=
we
3 (_ 2
+ 2
MJ3
0.
3 -
ro
ro
11,ro
W(g)
and then take the limit
gives
0
aw
m,
2
-
+
(ro)-2
2
-
mo(ro)-3)3/2
dp
and therefore
19 W
,9(mo/ro) Hence
we
have zA
0
),o=0,m0/r0=0 =
4mo/ro
+
=
ro
JI/ro ((ro)
o(mo/ro).
3
-
ro-
-2 -
3
2+)3/2
-4
2
I
326
7.
Spherical symmetry
The
perihelion precession of Mercury. Mercury moves around the sun describing an orbit whicli i.,., *1 to an ellipse but not closed. 6 the in 19th Already century one has ineasured the angle between consecutive. local minima of the distance between
Mercury and the sun and explain this angle within the Newtonian theory of gravity. 7 While such a "precession" occurs if one takes into account the gravitational fields caused by the other planets, this does not give a quantitative explanation of the measured value. The first outstanding success of general relativity was Einstein's demonstration that his theory could explain this discrepancy. 8 In order to calculate the "missing angle" we have to compare the Newtonian solution of the two-body problem (the sun, Mercury) with timelike geodesics (Mercury) in the Schwarzschild solution which dehas tried to
scribes the
sun.
In Newtonian
gravitation,
spherically symmetric
star
particle in according
a
moves
the
gravitational field
to the
ordinary
of
a
differential
equation
d21(8)
M
_ (S) 111(s)II,
-
ds2 Lemma 7.2.5. Let
ential equation
o
is
1(s) -
(1/9N, W)
are
and
Le
polar
R
:
Then the
equivalent
ds where
> 0
(7.2.10).
Equation (7.2.10)
dW
m
-->
curve
R 3 be y
(7.2.10)
-
a
solution
contained in
is
of the differa plane and
to
d 2ON
2
+ ON
,
coordinates
:--:
m/L
of this plane
2
(7.2.11)
.
and L is
a
constant.
"
Proof Equation (7.2. 10) implies that (s) x ds (" x being the vector product in R 3) is constant with respect to s. Hence I is contained in the plane spanned by -!L (O) and (O). If (r, W) are polar coordinates ds of this plane (x r cos W, x2 r sin W), Equation (7.2. 10) is equivalent cross
=
to 2
d 2,r(s)
r(s)
-
dP
(
ds
)
M
r(s)
r2 (8)
The second equation implies that 1 /r, the first equation is therefore
dV(s)
2 r
==
ds
equivalent
d2 0(s) -
+2
dS2
dr(s) =
ds
L is constant.
0.
ds
Setting
2 to d ON /d (p2 + ON
=
ON
m/L
2
1
6This
true for all
is
but in the
case actually planets, Mercury especially pronounced. 7In order to do so, Astronomers have assumed the existence of a further planet. However, this planet has never been seen. 8 He did this using the equation Ric 87rT before he arrived at his final theScal/2 g 87rT. This was possible since for these calculations ory with Ric only the vacuum equation is needed. =
-
=
of
the effect is
7.2 The Schwarzschild solution
Equation (7.2.11)
is
an
inhomogeneous
constant coefficients. It is easy to
see
327
linear differential equation with (in the generic case L = 0)
that
' there exist constants C1, C2 such that ON (W) + cl sin(W) + C2 COS (W) L2 Our polar coordinates are only fixed up to a rotation in the plane. Hence =
-
without loss of
we can assume
generality that
there is
constant
a
e
> 0
such that M
ON (
This solution is
periodic.
O)
f2 (1
=
We could
+
now
COS(W)).
e
attempt solve the corresponding and to calculate the difference of
in the Schwarzschild
solution, equation angle W (modulo 27r) between two consecutive minima of the coordinate radius as a Taylor polynomial inmo/ro. However, it is difficult to use this strategy in practice because it would involve integrals which are quite complicated. We will therefore employ a different method and obtain approximate solutions from approximate differential equations. Observe that Equation (7.2.11) is the Newtonian analogue to Corollary 7.2.1 and that both 2 2 equations differ only by the quadratic term mg m/r which is very small. The idea is now to view the Newtonian solution as an approximation to the relativistic equation. Inserting the Newtonian solution into the quadratic term gives a third equation =
d2Oapprox dW2 which is also
linear
a
2
Tn
M
+ Papprox
+ 3Tn
=
1-2
( -L-2 (1
+
e
COS(W))
inhomogeneous differential equation
with constant
coefficients. It appears to be a better approximation than the first differential equation since the term 0 has been replaced by the term Un(pN )2
approximation for 3mg 2. While this argument justification appears to be too complicated to be
which should be
a
better
only heuristic
a
real
is
worthwhile in
our
context.
This third equation gives
3Tn3
'M
Papprox
=
1-2 (1
+
To calculate the
angle
g(W).
of the function
d0approx dW
e COS
(W))
+
at the
perihelion
Un 3
sin (W) +
-
y-2
e2
e2
2
6
+
we
cos(2W)
+ eW
sin(W)
have to calculate the minima
The equation
Tne =
L4
(I
!
_L4
(! sin(2W)
+
3
sin(W)
+ W
cos(W))
0 as was to be expected. gives that Qappmx has a perihelion at Wo A comparison with ON indicates that the next perihelion should be at =
27r + 6 where J is small. Hence
with respect to
(27r
+
6) cos(27r
+
6)
we can
-
neglect
and obtain
f 3
sin(2J)
+
sin(6)
7.
328
SpIlericai symmetry
3M3e
me
0
L2 With
J and
tan(6)
s,n(6)
(27r
+
L2
neglecting
6) cos(6).
+
J with respect to 27r, this equation
implies 67rM2 J
L2 which gives
a
correction to the Newtonian value in very
good agreement
with observation.
7.3
in
Quasi-linear hyperbolic systems two independent variables In this section
we
of
equations
prove a theorem about hyperbolic systems of partial in two independent variables which will be applied
differential equation in Sect.
7.4.
The material is very technical and of a different mathematical topic than the rest of this book. The reader may wish to skip this section
For the
on
first reading.
following
call j'(f ): R1 projection R2
theorem
R1
x
Rk,
R, (t, q)
we
_,
X
q is
i-->
The system
Otf
+ A
o
some
notation. If
.
C1(R2
Definition 7.3.1. Let h E
Rk,Lin(Rk,Rk)).
f : R1 -4 Rk we (x, f (x)) the O-jet of f The canonical denoted by pr2-
need
Rk,Rk)
x
and let A
(E
C1(R2
X
of differential equations
jo(f),Oqf
=
h
o
jo(f)
quasi-linear system of hyperbolic equations in two variables if for O-jet (t, q, F) E R2 x Rk the linear map A(t, q, F) has k linearly independent left eigenvectors. The directions R(at + Ap9q) where Ai are the left eigenvalues of A are called characteristic directions. The (unparameterised) integral curves of the characteristic directions9 are called the characteristics of the system of differential equations (and the given is
a
every
solution). The aim of this section is to prove the following fundamental existence and uniqueness theorem for quasi-linear systems of hyperbolic equations in two variables.
C' (R2
Theorem 7.3. 1. Let h E
Rk, Lin(Rk, Rk))
atf Here
we
mean
x
Rk, Rk)
and let A
E
C' (R2
X
such that
integral
characteristic directions
+ A
o
jo(f)aqf
curves
=
h
o
jo(f)
of vector fields which
are
tangent
to the
7.3
in two
Quasi-lineay hyperbolic systerns
independent variables
329
quasi-linear system of hyperbolic equations in two variables. For any function fo E COO ([a, bj, Rk) there is an open neighbourhood U of 101 x (a, b) C R2 and a unique smooth solution f : U --> Rk of the system fo (q) for all q E (a, b). of differential equations such that f (0, q)
is
a
=
proof of Theorem
The main part of the
7.3.1 is contained in the
following
lemma. Lemma 7.3. 1. Let
h,
/\ E C' (R2
R2
x Rk, Rk) of the
at least two
and
a
0 de
is
The characteristic directions
7.4.1.
Corollary
of the Einstein equation
are
U, z=U+ It
Proof AB.F ev
d2
follows that for =
> 1
FLdpcPQ,
information
h2H2(nr)4/(6+p)2 de2
V dir: Q). The second dp
U
= U-
can
=
FLdPEPL
travel
faster than light.
e-2ve2A 2 implies dE
assertion follows from
0tv1'A__BFaq
Corollary
7.3.1. 1
Remark 7.4.1. The system of differential equations is especially simple 0. Then it reduces to the following system of case of dust: p(E)
in the
=
ordinary
differential equations. U
0 r
Y, 2 + Y 2- F
U
62 +
2r3
t. The
depend
decouples and
be calculated from U
7.5 Static
on
o E
6
energy
Q9*Y -
density
6
2c Y. r
perfect
Most stars do not
fluid stars
change
therefore reasonable to
very much
assume
tions should be
an even
better
all of their fuel. In this section
over
long
time spans. It appears
that their interior
static, spherically symmetric solution
ity has
2
equation for the
where F does not can
rA
+ 2r
to Einstein's
description
once
be described
by
equation. Static solu-
these stars have burned
will show that the
we
can
assumption of static-
an absolute upper bound for the symmetric star. Further, this bound is so
consequence: there is
striking static, spherically small that it is exceeded by a multitude of known stars, which indicates that many of these stars will collapse into singularities once their fuel is mass
a
of
a
exhausted. ,
In this section
we
will model
a
non-rotating
star
by
a
spherically
sym-
metric, perfect fluid spacetime. Under the assumption of staticity, Einstein's equation for a perfect fluid reduces to the following ordinary static
differential equation.
7.5 Static
be
(M, g)
Theorem 7.5.1. Let
perfect fluid
stars
349
spherically symmetric 4-dimensional
a
spacetime which is C' and piecewise smooth, and assume that there exists a timelike Killing vector field U such that the energy momentum tensor
(,E +p) 0 0 U5 +p g, where E, p are given, smooth function. given by T 0 then p has a well defined centre of symmetry r solution the If the Tolman-Oppenheimer-Volkoff equation satisfies is
=
=
dp
m
(p +
dr where
m(r)
=
4,7r
87r
-
f''or
Conversely, let
A
3 (r) + 47rr (P r(r 2m(r))
+
c:
R+
2 &.
911
R+,
-+
R+
p:
-->
R+ given continuous func-
tions such that
(i) ( and p can be extended to R- as smooth, even functions, (ii) E and p vanish for r > ro, (iii) E(r) + p(r) > 0 for r < ro, (iv) E and p are smooth for r < ro, (v) E and p satisfy the Tolman- Oppenheimer- Volkoff equation, :
m(r)
41r
=
A
jr (,,(,)
+
7r
0
)
where
2
a unique 4-dimensional, spherically symmetric LorentzC' -manifold (M, g) which is piecewise smooth and satisfies
Then there exists ian
(a) (b)
Ric
There is a
-
U is an
a
-!Scalg 2
8-x(E
=
timelike
mass
mo +
o
r)U'
vector
field.
+p
r
mo > 0 such that
spherically symmetric
and
o
Killing
0
U5
(p o r) g,
+
for r > ro this spacetime is isometric to spacetime with cosmological constant A
vacuum
: 6-3'
Proof. Equations (7.1.8) and (7.1.5) imply
(Q
-
r)
Q-P +P
E
Since
by
(
definition of m,
Q
*
r
Oppenheimer-Vo,lkoff equation For the
converse we
Volkoff equation
==
A
m
_2 +
F F_ 2M, 1
I
-
follows from
81r
the
d2
10m
Q
9 r
validity
1Q r
r
Tolman-Oppenheimer-
exist functions
0
(
is
r(I
v(r), A(r)
trivially
(Q 2
of the Tolman-
!QR:2 Qer'
=
r
isfy Equations (7.1.3)-(7.1.6). Observe first that Equation (7.1.4) (7.1.3) is equivalent to
ar
-
have to check that the
implies that there
r
47rr(p
0
r)2
which sat-
satisfied. Equation
7.
350
which
A(t)
Spherical symmetry
implies m(r)
is
L(I
=
constant of
a
(Q sr)')
-
-P
A(t)
e-
2
n. Fro-in nz f in-Itc-gratio 9) g. .1
1
") + A (t),
p-,, et A
-,,m
(t)
where
=
0 and
therefore e
2m
2A(r)
r
)_
(7.5.26)
The 4-dimensional solution should be smooth at tation
by
is
7r
an
isometry
under the transformation
being
even
functions of
r
r
-->
But this is
-r.
0. Since
equivalent
We also need that A satisfies
r.
=
a ro-
it is clear that the metric must be invariant
A(O)
to A and
V
0 because
=
for any other value we would get a conical singularity. In fact, consider a centred sphere with area A(r) and (geodesic) radius R(r). In the limit 0
-->
r
obtain
we
47rr2
A(r) W2 (?-)
lim R-0
3
lim
=
47r
2
3
2(Q 9,r)-2 jr=0
4-Tr -
-e
=
(Q
r-O
&)2
2A (0)
3
which reduces to the Euclidean relation in the tangent space at the centre of symmetry if and only if A(O) 0. 12 ==
To
see
of g to
r
=
2 x
=
that
r cos
our
are
0, x3 2
=
e
=:
r
e2A
'dt2 +
Observe first that there =
xi
extendibility
=
r cos
0
cos
sin 0. Then the metric is of the form 3 -
1
are
3
7-
3
Ei= I (X i)2
(XI, X2, X3)
sufficient for the smooth
0 consider the coordinate transformation
0 sin
g
conditions
i x
xjdx'dxj
E(dx
+
i,j=l
)2
i=1
smooth functions
Fj,
in
a
neighbourhood
of
0 with
3
F,(XI, X2, X3)
==
V(
3
E(Xi)2),
(Xl' x2,x3)
E(Xi)2)
if and
only if v and A are even functions. Assume now that A is even. 2A I is a series in the variable y3 Taylor series of e _i=l (x T. The equation A(O) 0 implies therefore that the quotient
Then the
_
=
E3 (Xi)2 i= I
e2A
I
(Xi) well defined at
(x 1 ,x2,x 3)
12
would get the
If A(O) > 0 can
by
we
=
0 and smooth.
analogue
of the
be visualised in the 2-dimensional
a
circle.
2
case
tip of a 3-dimensional cone. This with the sphere being replaced
7.5 Static
A(O)
We obtain r
that the
integrand
=
0 from
=
Equation (7.5.26)
0. The function A is
order at
Tt
of
m
is
Equations (7.1.5) simplifies
to
c
+
-
2A
e
1
=
_
We
therefore determine
can
that
v
is
(P- ')
47rr
-
Q
vanishes to third
(7.5.26)
and the fact
=0
8,7r
is
m
m
351
stars
even.
r2
2m/r,
-
since
because of
even
M
2A,9' V
e
which, using
81r
perfect fluid
+
equivalent
A
47rr3
(p
r(r
2m)
up to
-
to
-
87r
constant of
a
integration. Observe
since the function
even
A
m(r) + 47rr 3 P r(r 2m(r))
87r
-
integrated is uneven. The Tolman-Oppenheimer-Volkov equation implies now the equation of motion (7.1.8). Since this equation is independent from Equations (7.1.3)-(7.1.5) but Equation (7.1.8) is a consequence of Equations (7.1.3)-(7.1.6) we can derive Equation (7.1.6) from the system of Equations (7.1.3)-(7.1.5), (7.1.8). Let to be
=
we
extend A
-
6 2mo
using e2A(r) (1 It is C1 if and only if
beyond
then A is continuous.
M(r)
ro,r 0.
equation of state p: c i--4p(E) and let an m, E R which only depends on the equation low for ene7gy densities, p: [0, F-,] --+ R such that m(r) :5 m, E
R+.
equivalent to Q 9 Q * v + (Q 9 v)' (together with Equation (7.1.8)) implies
Proof. Equation (7-1.6)
p)
which
e'Q
Q
an
Then there is
of state for all r
47r(c
-
Assume that there is
(ii)
+
associated with the radius r, satisfies 5r,/9. If, in addition, we assume that p stringent inequality m(r,) :5 4r,/9 holds. mass
e'(rr-) (Q
*
V) I
r
for
r
Q
From
< r,
o
v
'9Q ar
=
obtain
o r we
rev(rc:)
Oev >
wr
-
>ev('-) (Q
e'(0)
0
0
We re-express Q 9 r in terms of m using the resulting equation. This gives
ev(r')
(Q
-
TA r)
0
V),
m(I 2
_
(Q r) 2)
and
0
integrate
r,;
Or,
f V-1--L rM(1)
(7.5.27)
dr.
2m(r)
0
r
In order to estimate this
for all
(0, r,). Comparing
E
r
integral
m(r)
show first that
we
!
m(rc')
the derivative of both functions
we
Tr
obtain
that the function
d
f(r)
(m(r)
jr-
r2(4,7re(r) satisfies df (r)
=
dr
with
(0, r,) m(O)
the existence of
47rr2 dc(r)
r
G
(ri, r,)
implies
e'(0)
rc using the known equation of state. Clearly, m(r) depends continuously on the data r, rn(rc). This implies that mc c
sup f m (r)
:
(r, m (rc))
E
C I is finite.
Theorem 7.5.2
gives
larities in
universe. In Sect. 7.2
our
an
important indication for the we
have
seen
existence of
that all
singunon-trivial,
7.5 Static
maximally extended, with non-constant
r
non-flat
perfect fluid
spherically symmetric
355
stars
vacuum
of Einstein's equation fail to be static in
a
solutions subset of
spacetime and contain a region where curvature diverges. We saw that it is possible to enter this region but impossible to leave it. Moreover, once having entered the region any observer will fall into the singularity
(where
curvature is
is the Schwarzschild
the
vacuum
infinite) mass.
solution also
within the finite time span 7rm where m question arises whether this property of
The
occurs
for real stars which have
energy momentum tensor. Since in Newtonian
central
singularity
in the
vacuum case
which
gravity
can
tempting
artifact of the
vacuum
also have
a
be avoided if the mat-
ter of the star is not assumed to be concentrated in to argue that the
non-vanishing
we
a
single point,
it is
property of the Schwarzschild solution is
an
equation.
Theorem 7.5.2 indicates that this is not the
case.
We have
proved that
upper limit for the concentration of matter in static, spherically symmetric perfect fluid stars, m/r < 5/9, if the energy density of the star decreases outward and is positive. These physical assumptions
there is
are we
an
very weak and seem to be satisfied for all known objects. Moreover, have seen that for any star which is governed by an equation of state
there is
an
absolute
mass
limit. What is more, this
mass
limit
can
be
estimated using only the equation of state for low energies. This means that we get bounds even if we do not know the physical configuration of
extremely dense stellar cores. It has been shown13 that there
are
stars which exceed the
mass
limits
in this section. This indicates that these stars will
collapse into given black holes once they have exhausted their nuclear fuel. In Chap. 9 we will give a very general argument to the same extent which does not rely on spherical symmetry. It should be noted however, that all these arguments in favour of the existence of black holes have loop holes. In this section, we heavily rely on spherical symmetry and the assumption of a perfect fluid. Moreover, it is conceivable that there are non-singular solutions which fail to be static. There are other loop holes in Chap. 9 which
we
will address then.
The argument uses input from book, cf. (Hartle 1978)
physics
which is
beyond
the scope of this
8.
Causality
In this
chapter we link the concept of causality to the conformal strucby the metric and present some elementary causal properties and their interpretation. In Minkowski spacetime, causality is trivial since lightlike geodesics are straight lines. Lemma 3.1.4 shows that the local causal structure of arbitrary Lorentzian manifolds is the same as the causal structure of Minkowski spacetime. All non-trivial aspects of causality are therefore global in character. In this chapter we will also discuss in detail the possibility of "causality violation" due to the global geometry of spacetime. Chapter 8 requires Sect. 3.1 and develops rather specialised mathematical techniques. It contains a number of technical results which are needed in Chap. 9 where the existence of singularities in generic spacetimes is proved. We will restrict to those results which are necessary to prove and interpret these singularity theorems. For a more comprehensive mathematical treatment of causality see (Beem and Ehrlich 1981; Hawking and Ellis 1973; Penrose 1972). For more examples which exhibit the subtleties of causality and singularity theorems see ture induced
(Senovilla 1998).
According we assume
Cx
cone
(at
=
least
from
a
experience no signal is faster than light (photons). As photons move along null geodesics, the integrated light 0, vx future orientedl should expx (Jvx G Tx M : g (vx, vx)
to
our
that
=
locally)
given
determine which events
event
x.
It is therefore
can
plausible
in to
principle be identify the
influenced conformal
structure Q of spacetime with the causal structure of the universe. Postulate 8.0.1 E M
can
objects
can
x
A proper theory of this it is
(Causality
and conformal structure
causally influence y E M if and only if y reach y from x if and only if y E 1+(x).
justification of Postulate 8.0.1 would physical particles and fields. This is
E
coincide).
J+(X).
Material
corresponding beyond the scope of book. While Postulate 8.0.1 will not be important for our theorems, crucial for their physical interpretation. require
a
far
Remark 8. 0. 1. For
arbitrary matter models, Einstein's equation does not cone structure and causality. For inspherically symmetric fluid with equation of state c p(,E)
respect the link between the light stance,
a
M. Kriele: LNPm 59, pp. 357 - 382, 1999 © Springer-Verlag Berlin Heidelberg 1999
i-
8.
358
Causality
satisfying dp/dc > 1 has spacelike characteristics (cf. Corollary 7.4.1). Consequently, the characteristics of the initial value problem associated with Einstein's equation is spacelike whence perturbations of the initial data propagate faster than light. For this reason one usually regards these matter models as unphysical. In fact, to date all classical (i.e.,
non-quantum)
matter models which describe real matter have causal
characteristics. The local causal structure of any Lorentzian manifold is trivial, i.e. the same as in Minkowski spacetime. This follows immediately from Lemma 3.1.4 which is fundamental to this
lary
chapter. The following technical
corol-
will also be useful.
Corollary 8.0.1. Let (M,g) be a Lorentzian manifold neighbourhood of x E C. Let K C C compact and -y be a
and C causal
a convex curve
in
K. Then -y is extensible.
Proof. Let curve
-y
[a, b)
-/:
--*
C be
a
future directed causal
be future extended if
can
liMt-b -y(t)
curve
in K. The
exists. In order to
see
that
this limit exists, let
{'Y(ti)bEN7 fY(Sj)IjEN be convergent sequences with b and x, y be their limit points. For any i limi-oo tj limj,,,, sj there is a j > i with y(tj) E J+(7(sj),C) and for any j there is an i > j with -y(si) (2 J+ (-y(tj), C). Hence we obtain x G J+ (y, C) and y G J+ (x, C). Hence by Lemma 3.1.4 (i) there are two future directed causal vectors v, w with x expx (w). Traversing the expy (v) and y =
=
=
geodesics
t
expy(tv)
i-->
past directed is
a
vector
we see
that at
x
there is also
a
causal
with expx (u) y. Since the exponential map expx of an open set C C TM to C we must have w u.
u
=
diffeomorphsim only possible
But this is
8.1
=
backwards
=
if both vectors vanish.
Causality conditions mathematically construct a spacetime with closed timelike first glance one is tempted to rule out such spacetimes since it seems possible to perform experiments in them which lead to logical contradictions. In this section we will investigate this issue in some detail. We will also define a slightly stronger "causality condition" which will Play an important r6le in subsequent sections.
It is easy to curves.
In
a
At
general
Lorentzian
manifold,
it is
possible
for closed timelike
curves
to exist.
Definition 8. 1. 1. Let
M We say that causality (resp., chronology) only if there exists a closed, non-trivial causal (resp., timelike) curve from x to x. The chronology violating set is given by
is violated at
x
if
and
X G
-
8.1
JX and the
causality violating
Ix
E
M
:
3
a
E
M
set
by
X
:
(=-
Causality conditions
359
1+(X)l
non-trivial causal
curve
-y
from,
x
xJ.
to
A Lorentzian
manifold (M, g) is causal (resp. chronological) if the causality violating (resp., chronology violating set) is empty. If (M, g) is chronological (resp. causal), we sometimes say that the chronology condition (resp. causality condition) holds. set
The term
'causality
closed causal
curves
matical arguments
not
violation' is somewhat is not
contradictory
misleading:
the
itself and there
possibility
are no
of
mathe-
against causality violation.
The idea that there may be closed timelike curves in our universe is new: The concept of cyclic time was a widespread idea in ancient
Greek
philosophy (Kanitscheider 1984, p. 45). These Greek philosophers accepted our fundamental experience of local linearity of time but they compactified the time line to a time circle. Its circumference was identified with the time of
model of
one
revolution of the universe
(according
to their
planetary just according 'arbitrary' laying down'). length of this period is sufficient to explain why nobody of us motion
or
her/his
own
to
The sheer ever
has reentered
We
can
easily
obtain
a
past.
spacetime whose causal
structure is
to the causal structure of this ancient Greek model. Just take
a
analogous horizontal
strip of 2-dimensional Minkowski space and identify the upper and the lower boundary (cf. Figure 8.1 .12) Another very instructive example is .
closed timelike
curve
identify
Fig.
8.1.1. A
dimensional space
strip of twoMinkowski
where
future
boundaries identified.
past
and are
the Lorentzian manifold '
2
Plato, The are
for
instance, chose 10,000 years (Kanitscheider 1984, in this and other figure indicate how both sides
arrows
oriented
p.
55)
to be identified
8.
360
Causality
(R
x
S1, 2dwdt
given in (Misner 1967) (cf. Fig. 8.1.2). given by f (t, W) : t < 01.
first is
+ td
closed timelike
Fig.
8.1.2. Misner's
spacetime
(S'
W2)
The
chronology violating
set
curve
x
R, 2dtdW
+
td(P2)
compactification" arising in these examples is trivial in the examples there is a locally isometric Lorentzian manifold which satisfies the chronology condition. In Lorentzian geometry however, there also exist non-trivial examples where causality violation arises geometrically and not merely topologically. An example which will also be of importance in Chap. 9 is the G,5del solution (G6del 1949).
The "time sense
that in both
Example 8. 1. 1. The G6del solution describes a solutions of Einstein's + Ag Ig equation with dust and postive cosmological constant, Ric 2 8,7reu 0 u where u is a timelike unit vector field and F_ A/(4?r). The metric is given by -
=:=
2 g
=
dt2
-
+
dr2
+
A
(sinh2 (r)
-
sinh4(r))&P2 +
2,v/2 sinh 2 (r)
d odt)
+ dz
2
have r > 0 and identify W with W + 21r. The vector field 9, integral curves and it is spacelike for r < arsinh(l). For r arsinh(l) the integral curves of a. are lightlike (but not null geodesics) and for r > arsinh(l) they are timelike. Since sinh 2(r) is an even function 0 we have only the usual coordinate singularity of r it follows that at r associated with polar coordinates. Hence spacetime has the topology R4 and is in particular simply connected. It follows that chronological where
we
has closed
=
=
violation is
an
inherent property of the solution.
8.1
Causality conditions
361
A
physically interesting solution of Einstein's vacuum equation with vanishing cosmological constant is the Kerr solution. For details cf. (O'Neill
1995), (Wald 1984), (Hawking Despite
and Ellis
the existence of these
1973).
examples
most
physicists regard causalcausality
ity violation as 'unphysical'. The reason for this rejection of violation is the following thought experiment:
Suppose,
you
are
travelling
in
spacetime and reach
a
point
in
past before your departure. Now you decide not to your travel after all and instead to stay home. Contradiction. own
At
glance, the possibility of "free will" seems to be at the centre of However, following (Wheeler and Feynman 1949) Clarke (1977) has re-formulated the thought experiment in terms of a simple machine and has argued that the thought experiment is fallacious: Assume that a
first
the issue.
gun directed at
there is
a
with
shutter
a
which,
if
a
target in spacetime. This target is connected
closed, blocks off the path between the
gun
and the target: If the gun is triggered, the bullet will hit the target which in turn will cause the shutter to fall. A second shot will now be blocked Now
by the
assume
shutter and therefore cannot hit the target (c.f. Fig. 8.1.3). configuration is located in a region with causality
that the
violation such that the shutter falls
along
a
closed timelike
curve so
that
,gers shutter
Fig.
A gedanken experidisprove causality viola-
8.1.3.
ment to
tion
it blocks the bullet to arrive at
a
before
the gun had been
triggered. Again
contradiction: If the shutter is open the bullet
we seem
can
hit the
362
8.
Causality
target. But the target closes the shutter which
in turn blocks the
path
of the bullet.
This
may be resolved
as follows. The angle a, under which by the shutter depends continuously on the shutter's position x at the time the bullet passes the shutter. For simplicity we assume that the shutter will descend with constant velocity v. This velocity is continuously related to the angle a. If the length of the closed causal curve is T we obtain the relation x Tv(a(x)). This equation
paradox
the bullet is deflected
=
has at least In
physical
solution xo which leads to a contradiction-free situation. terms this can be explained as follows: The original contraone
diction is due to the fact that the shutter is
thought to be either up or However, the position of the shutter depends continuously on the parameters of the system. What happens is that while the shutter descends it grazes the bullet and thereby deflects it so that the mechanism works only imperfectly. As a consequence, the shutter is released rather late and not yet in place when the bullet hits it again due to causality violation. Hence it grazes the bullet and we are in a paradox-free time loop. This scenario appears to be highly non-generic but Clarke argues that exactly this is the effect of causality violation: It picks out those non-generic data which are in accordance with the causal anomaly. The gist of the argument rests on the assumption that physical processes are continuous, an assumption which does not hold for quantum mechanical systems. These systems may be in discrete pure states such as spin up or spin down. However, if one tries to set up a quantum thought experiment one is faced with the fact that all predictions are probabilistic which invalidates the whole thought experiment from the outset. There are also arguments against Clarke's resolution of the paradox. Instead of releasing the shutter directly when the target is hit we may have a device which automatically releases the shutter a certain time after the impact. This can be achieved with an electronic switch rather down.
than
a
mechanical connection between target and shutter. It seems now probable that this device always releases the shutter such that
much less
it grazes the bullet when coming down. For Clarke's argument to work the bullet must comes out of the gun so slowly that it just touches the
target but does not really hit it. Otherwise it cannot be explained that the second device is not successful in time which would lead to
a
releasing
the shutter at the pre-set
contradiction.
Whether Clarke's argument is correct
or
not
-
we are
only
able to
conduct local experiments. But causality violation is a global effect, and so the lack of experience cannot give evidence of its absence. Any objec-
against causality violation every-day experience.
tion
of
rests
on
an
(unjustified) extrapolatiom
8.1
Causality conditions
363
There is another point which should be addressed. Causality vioseems to constrain free will. While this is not really a physical
lation
problem, such an effect would have some bearing on philosophical and questions. But an almost trivial observation resolves any possible argument concerning free will at once: If we want to incorporate the notion of free will into a physical description we have to view it at least as a quantum effect (or caused by another yet undiscovered 'mechanism'), but certainly not as something fitting into the framework of classical physics. We only can expect that general relativity is a classical limit of such a theory, It is therefore quite possible that 'free will' is something like a second order effect and that the classical "limit-spacetime" of our world contains closed timelike curves even though we still enjoy free will. With this discussion in mind we should always be very watchful if in order to obtain physical results the seemingly innocent assumption of chronology has to be made. moral
Lemma 8.1.1.
The chronology (resp., causality) violating set consists of connected components of the form I+ (xi) n I- (xi) (resp., J- (xi) n J- (Xi)) 0 1, ). =
-
-
-
Proof. We only show the lemma for chronology violation. The proof for causality violation is completely analogous. Let C be a connected component of the chronology violating set and x E C. Since C is connected there is for each pair of points Jx, yJ C C a (not necessarily causal) curve -/ c C which connects
neighbourhood
of
and y. Since for all z G C the set _T+ (z) is a curve -y is compact, there are finitely many
x
and the
z
zi e -y such that zi+1 E 1+(zi) and the neighborhoods 1+(zi) cover -Y. It follows that there is a timelike curve from x to y. By the same argument
there is
timelike
a
form y to
curve
assertion follows since 1+ (x) n I-
The
following proposition chronological.
Proposition 8.1.1. If of M is non-empty.
Proof.
We
k). a
can cover
(x)
Hence C c I+ (x) n I-
shows that
a
I
with x, E
Ui=1I1+(x,(i)) X, E
a
compact spacetime
M is compact then the
M with
finitely
C
I+
there is
we
1+ (xi)
set
I
an
(i
E
would have
(U JT+ (XI(O)
in contradiction to the definition of xi. If xi
argument
cannot be
chronology violating
many sets of the form
=
i=1
same
and the
1+(xi) f2,..., kJ and 1+ (x,(1)), xi V Ui-=', i+ (x,(i)). This implies
since otherwise
I+ (XO,(1))
(x)
is connected.
If x, is not contained in
permutation
X,
x.
U 1+(X,(i)) i=1
1+ (xi)
to x1 instead of x1. Since there
are
we can apply t he only finitely many
8.
364
one
xi,
Causality
of the xi must be in its own future for otherwise none of the 1+(xi).
we
would have
I
that X.(k) is in
Proposition 8.1.1 is often taken as a reason for dismissing compact spaceas unphysical. While the chronology condition and the causality condition are very intuitive, from a technical point of view, a slightly stronger condition is advantageous:
times
Definition 8.1.2.
neighbourhood
any
any causal
curve
The strong causality condition holds at x E M if for of x there is a neighbourhood U C V of x such that
V
intersects U at most
once.
40Z
(remo,
remove
7-
7
iden
rp.mnvp
ify
8.1.4.
Fig.
A
spacetime which strongly
is causal but fails to be
causal
words, if the strong causality condition does not hold at x, there curves starting at x which come arbitrarily close to x after leaving a giving convex neighbourhood. Hence the chronology condition
In other
timelike
are
is almost violated. In the next section
will
we
see
causality condition. Finally, we wish to introduce global hyperbolicity 3,
the importance of this the strongest
causality
condition which is often assumed. Its relevance stems from the fact that in
a
globally hyperbolic spacetime
given -points properties in the
two
is
compact in
is
the set of causal
natural
topology.
curves
We will
connecting use
related
next section.
Definition 8.1.3. A subset
A is
a
strongly causal
of A
C M is
said to be
globally hyperbolic if
and for any two points x, y e A the set J+ (x) n J-
(y)
compact.
This
name
partial
has been coined
by Leray (1953)
differential equations.
in connection with
systems of
8.2 Cluster and limit
365
curves
a ove
remove
remove
P
identify
remove
Fig. 8.1.5. A spacetime which is strongly causal. An infinitesimally small perturbation of the metric results in a spacetime with chronology violation
8.2 Cluster and limit In this section
based
(Beem
on
practical
It is
to continuous
study
we
sequences
section will be
of this
The results
curves
and Ehrlich
to
generalise
1981). the concept of
every
point
:
y
point on
x
timelike
or
causal
curve
x
7 n C
can
be
-y is called causal
curve
(resp.,
time-
neighbourhood C such that any connected by a causal (resp., timelike) C'
-y has
on
a
convex
which is contained in C.
curve
this definition coincides for Cl-curves with
Clearly,
Lemma 8.2.1. Let
of x,
causal
our
previous Defini-
(iii).
tion 3.1.3
of C
a
curves.
Definition 8.2.1. A continuous
like) if
of causal curves and there limits. fundamental to what follows. It is
x
E
M. There is
constant k >
a
curves
-y in C
can
a convex
0, and coordinates be
coordinate
(xO,
parameterised by
.
.
,
neighbourhood
xn- 1) such that all
xO and the coordinate
t
inequality n
E (,ya(t) 7a(S))2 -
0 =
exists
a
respect
to the flat metric
I
-
kodt2 + Eni=1 (dx' )2.
In
particular,
p satisfies
366
8.
Causality
nst
Fig.
8.2.1. The
proof of Lemma
8.2.1
ko
==
En-I(Ai)2.
ko (itO)2
f eo,..., en-11
and write
If
we
11va ea 112
=
denote the standard basis of Rn
VEa-=01 (va)2,
we
by
obtain t
a
(t)ea
-
,a (S) ea 11 2 =11,a M ea
io. The
V
of -
curve
in M there exists
an
-y is called the limit
io
E N
curve
such that -yj C V
of the
sequence
for
f -Yi jiErq
-
terminology is slightly at odds with the traditional definition of general relativity but closer to generic mathematical terminology. Often, not limits of curves (with respect to a natural topology) but curves which are better thought of as a set of pointwise accumulation points are called "limit curves". We will reserve the term "cluster Our
"limit curve" in
curve" for such accumulation
curves:
Definition 8.2.3. Let -y: [a, b] -y is said to be a cluster curve
-M, of the
-yi:
[a, b]
sequence
M (i E N) be curves. 17ifiEN if there exists a
-4
8.2 Cluster and limit
8.2.2. A limit
Fig. of
a
367
curves
sequence of
curve
curves
-y
-yn
subsequence f -yij jj,2N such that for all x E -y each neighbourhood of x intersects all but finitely many of the curves -yjJ. Following Beem and Ehrlich (1981) we will say that the sequence f7ijbEN distinguishes the cluster
curve
-y.
It will turn out that for curves are
essentially
the
strongly causal spacetimes same
(cf.
from
curve
for
a
An
8.2.3.
Fig.
x
cluster and limit
below. There is
8.2.2)
Theorem
example where
to y is not
sequence of
a
cluster
a
points from
x
no
limit curve
to y
logical relation between limit curves and cluster curves. Consider in cylinder (S' x R, d odt) (cf. Fig. 8.2.3) the sequence of identical const. The curve curves 7,, which connect x with y and satisfy W t(x) and then connects x with y satisfying -y which first traverses t const is a limit curve of -y, but it is not a cluster curve. In Fig. 8.2.4 a
strict
the flat
=
=
=
we a
have
limit
may
a
cluster
even
be
curve
-/ of
a
sequence of causal
general, a cluster curve spacelike (cf. Fig. 8.2.5
curve.
In
of
a
Proposition 8.2.1. If (M, g) is strongly causal and -y is of a sequence f-yjjjEN of causal curves, then -y is causal. Proof.
Since
(M, g)
is
strongly causal,
bourhoods Ci such that
no
causal
we
may
curve can
which is not
curves
sequence of causal
cover
-y
by
a
curves
cluster
convex
enter any of these
curve
neighneigh-
8.
368
causabity
identify
Fig.
A spacetime,
8.2.4.
which is causal but fails to be strongly causal identi
Fig.
8.2.5. Assume that
b1c is
rational and
a/b is ir
rational. Then the
.
c/a
......
Ar
b
R2
from
depicted a-
projecslope
tion of the line with
identify
to the torus
in the
dense
curve
curve
is
a
figure
is
a
7. Hence every
cluster
curve
of
ly
bourhoods twice. Consider
one
such
neighbourhood
and denote it
by
C.
Let x, y be points in -y n C and denote by f 7ij JjEN a subsequence which distinguishes 7. Then there are sequences xj, yj E 7j., n C with xj --> x and yj -+ y. Since yi., E J+ (xi, , C) Lemma 3.1.4 implies the existence of a
causal vector vj with
expx,, (vj)
=
yi.,. These vectors have
an accumu-
lation point v with exp,(v) y. The vector v must be causal since the set of causal vectors is closed. But this implies y G J+ (x, C). If x and y are arbitrary points on -y, we can find finitely many neighbourhoods Ci such that the segment from x to y is covered by Uj Ci. We can now apply the preceding argument finitely often to conclude that x and y are
causally
related.
I
The basic Lemma 8.2.4 below is which
we
an
application
of the theorem of Ascoli
will present first.
Let A C
Rk be
a
compact
CO(A,Rl)
=
ff:
set. The space of continuous functions
A
--
R
k :
f
is
continuousl
8.2 Cluster and limit
then be
can
regarded
normed vector space in
a
as
a
369
curves
natural way. Just
SUP., C A I I f W I I Moreover, this norm is complete, i. e., evCauchy sequence Ifil c C'(A, R1) converges to a function f E ery CO (A, R1) . (f can be constructed pointwise using the completeness of
I I f I lo
set
=
.
R1.) C(A, R1)
Lemma 8.2.2. Let B C > 0
e
every
radius
E
there
and 13 C
are
finitely
Uj(') B'(xj). i=1
Proof If the lemm 'a covering B such that .
6
be
closed set and
a
many balls B
''(xi),...,Bj,()(xj(,))
for
with
Then B is compact.
is not true then there no
that
assume
are
finite subset
open sets
(t
U,
E
I)
B. Let
covers
JBI'(xi),...' B3,(1)(xj(j)) cover B. By our assumption by finitely many U,. (Otherwise we would obtain a finite cover of B by finitely many sets which are in turn finitely covered by sets V,). We denote this ball by B0. Assume that we
be
a
one
finite set of balls of radius 1 which
of these balls cannot be covered
have constructed balls
jBjjj=o,...'k-1
such that
(i) Any two consecutive balls intersect, (ii) Each ball Bi has radius 2', (iii) None of these balls can be covered by finitely There
are
fore also
balls
(xi), B1, ?
.
.
Bk-1. Since Bk-1
must exist at least
one
.
,
many
k(2-k) B-k (Xk(2-k))which 2
cannot be covered
B2"! 2- (XII)which
U,.
cover
B and there
U, there Bk-1 and cannot be by Bk we have induc-
by finitely
many
intersects
k
covered by finitely many U,. Denoting B2"lk(x,) 2tively defined a sequence f BiliENUO of balls which satisfy (i)-(iii). Denote the centres of these balls by yi. For any natural numbers m < n we obtain n
n
JYn
-
YmIl
Y_
---
x] of -y n C between y and x would intersect A then we would obtain a contradiction to the achronality of A from y E (1+ (A) U A) \ D+ (A) c I+ (A) and
(-y
n
C) [y
x] _t_LLy).
--->
Let U C C be
a
neighbourhood
of
y which does not intersect D+ (A) and consider a point z E (I- (y, U) n 1+ (A)) \ D+ (A). Then the concatenation A of a timelike curve from z to y in U and the
segement (-y n C) [x -- y] does not intersect A. Since H+ (A) and A is closed a slight deformation of A results in a causal curve from z to some point in -: E D+ (A) which does not intersect A.
x
(=-
We could
this
prolong
now
curve
to the
past of
z
E M
\
D+ (A)
thereby
obtaining past inextensible curve which does not intersect A. This gives a contradiction to , E D+ (A), whence we have proved (-y n C) C D+ (A). a
Assume that there is
D+ (A). Let A be
a
y E
an
timelike
(7 n C) n I- (D+ (A))
curve
from y to
This
z.
and let
e
z
1+ (y)
n
cannot intersect
curve
A to the future of y because of y E 1+ (A) U A and the achronality of A. If y V A then there is a neighbourhood U of y with U C I- (z) which does not intersect A. Since -y is a cluster curve Of jYijiEN there is an i and a point from E -yj n U. It follows that there is a causal curve G -yj to z
which does not intersect A. If y cz A then
x
A
E
\ edge(A)
by the construction of C. The point y has a convex neighbourhood of U c C n I- (z) which is intersected by infinitely many -yi. Let i E N with -/i n U :A 0, Consider a timelike
E -yj n
be
U, and
a
timelike
from
curve
9
to
z.
which is contained in C. curve p from I- (x) n C to The concatenation of 1-t and the part of -yj to the future of intersects A because of
and xi
V edge(A)
x
C-
1+ (x). The equation 7i
A
n
=
0 implies
cannot intersect A by the achronality of A. that M intersects A. Hence We have shown that in either case, y V A and y G A, the concatenation of is past inextensible the past endpiece of -yj with future endpoint and
and does not intersect A. This gives a contradiction to 0 Consequently, we have shown (-y n U) n I- (D+ (A)) We have 7 n C C D+ (A)
\
I-
(D+ (A))
curve
-yj C
inductively x
endpoint
H+(A) we
obtain
which has either
Since
H+(A)
n
&. The concatenation of y n C and
with future
no
causal
a
endpoint curve
past endpoint
is achronal this
curve
x.
-yc,,, C or
E
D+ (A).
H+ (A). The past endpoint
=
of 7 n C lies in H+(A) since H+(A) is closed. If can repeat the construction thereby obtaining a curve
with future
z
Repeating this
edge(A) we d C H+ (A) n d gives a construction
H+ (A) with future endpoint
has past endpoint in an achronal null
must be
edge(A). geodesic. I
9.
Singularity
theorems
chapter we prove and investigate "singularity theorems". These are usually interpreted as an indication that black holes exist and that there has been a big bang or at least that there are regions in spacetime where general relativity breaks down. They are one of the main motivations for attempting to quantise general relativity. While there is a lot of evidence in favour of this interpretation we will see that there are also open problems which have to be addressed in order to justify this interpretation.
In this
theorems
-
Chaps. 7 and 6 we have seen that spacetimes describing a single, non-rotating star and the simplest cosmological models of our universe contain regions where the curvature diverges. One may think that these singularities are only an artifact of our high symmetry assumptions, but in this section, we will give an indication that a physically realistic spacetime must contain such singularities. More precisely, we will show that there exist causal, inextensible geodesics which are incomplete. Recall that a freely falling particle is represented by a timelike geodesic. If the geodesic cannot be extended to a complete one (i.e. if its future endless continuation or its past endless continuation is of finite length), then either the particle suddenly ceases to exist or the particle suddenly springs into existence'. In either case this can only happen if spacetime admits a "singularity" at the end (or beginning) of the history of the particle. This singularity may be a curvature singularity, there may be a topological obstruction, or spacetime may simply cease to be sufficiently smooth. However, the Schwarzschild and Robertson Walker solutions indicate that these singularities are accompanied with diverging curvature. (But cf. Sect. 9.5.1 below where we present a spacetime which indicates that such singularities are very mild). We will prove a singularity theorem which only establishes the existence of incomplete causal geodesics rather than incomplete timelike geodesics. While the innocent looking extension to null geodesics is necessary for the proof, the name singularity theorem is in this case somewhat misleading, because there exist In
This should not be confused with pair creation or pair annihilation of particles and anti particles, because during these processes nothing really ceases or
starts to exist.
changes
These quantum mechanical
of state.
M. Kriele: LNPm 59, pp. 383 - 424, 1999 © Springer-Verlag Berlin Heidelberg 1999
phenomenons
are
merely
384
9,
Singularity theorerns
perfectly regular spacetimes
with
incomplete
null
geodesics contained
compact subsets. On the other hand, it has been argued that such
ples this
are
very
and that in
special
phenomenon does
not
occur
in
exam-
stable, physically realistic spacetimes
(cf. (Hawking
and Ellis
1973)).
Energy conditions
9.1 In
general, a maximally extended Lorentzian manifold need not contain incomplete causal geodesics. In order to prove a singularity theorem, we will have to make some physical assumptions. There are two sorts of fundamental physical experience which come to mind. Firstly, energy density as measured by the energy momentum tensor is positive. Secondly, gravitation is attractive. Recall that the energy density measured by an observer -y (with g( , ) -1) is given by E T( , ). We feel that this energy density should be positive. Recall also that in the motivation of the energy momentum tensor (cf. Sect. 5.1) we have obtained the energy density E T(U, U) as an average of a positive mass distribution. For our purpose this should be enough of a motivation of the following definition =
=
=
Definition 9.1.1. M
We say that the weak energy condition holds at
T(u, u) For
x
E
if
a
> 0
for all causal
vectors
u
E
TXM.
physical verification of the weak energy condition one would have physical matter models. This is beyond the scope
to consider all realistic
of this book but
far the available evidence points to the fact that the
so
weak energy condition does hold. Gravity is attractive if and only if any two nearby freely falling observers will be forced to approach each other under the influence of the
underlying spacetime geometry. This can be formulated infinitesimally in a rigorous manner. A freely falling observer is modelled by a timelike M. Let f : (-J, J) x [a, b] --> M a geodesic variageodesic -y: [a, b] tion of -y and J f,(0, -) be the variation vector field. Observe that J is a Jacobi vector field. From Taylor's theorem we get with respect to any coordinate
ordinate
expression
can
i
f (0, t) + 8 ji (t) + 0 (S2). This cobe interpreted in Newtonian terms as follows.
system f '(s, t)
=
The observers 7 and f (s, -) have (up to first order) the same rest space and are separated by the space vector sP. Hence up to first order it
speak of the (Newtonian) force F with which -Y acts on -msj' where m is approximately given by P the mass of the observer f (s, .). (The minus sign is inserted because the force vector points from f (s, t) to -y (t) and J points into the opposite direction). Clearly there cannot be a direct translation of Newtonian makes
f (s, .).
sense
to
This force is
=
9.1
(ft (s, t))'
'(t)
for
s
V7
7_)
-4
:
03 (01):
(7-,
04 (a):
(f,
:
,
1
=
(9E,
6
=::
vector fields
+ a, x,
+
, -)
a)_: _, (I
+
7,
+ a,
-)
(f, _: j
_; +
ce).
vector fields
corresponding Killing
sic, (6j,
I
W _ --
j
+
a) , ;E)
are
0 ,
63
4
give four constants Writing -y(-r)
of motion
along
ci is constant.
the
geode-
i(-r))
we
obtain
C,
=
-t
Y WX
This is
a
x 2
yx
2wJU
-
-
-
TW 2,;,
-
-'
linear system of equations for
W2X2
(fx,y-,z-),
C3
=
Z-
and its solution is
given by The isometry group is five-dimensional, but the additional group of isometries is not important for the argument.
1-parameter
400
t= cl
If
Singularity theorems
9.
-
2w -:
C3)
we rename
2w
X
(Cl, C2, C3
2
(C2
C3)i
d,
V
our
without loss of
Values for
projection of
V)
-
geodesics
we
d Z=
-7
: 1)2
+
-y to the
have the
(9.4.3)
--
'
C
generality that
and radius
cle with centre
causal
obtain
C
i into this
/)2 Hence the
C4
X
C
Inserting
: -C3)'
V2-w-i-
2 j;
v/2-C v/2_wJ (.'r
can assume
we
T
v2C!
t
We
+
:
C
,
20)2;,
Y
equation
-
d
-
2
we
V
2 ' -
2
traverses
d2
1
qC2.
+
2
(x, y)-plane
1, 0,
71 E
obtain
+,qC2.
an
arc
of
a
cir-
Observe that for
inequality < -
which
R+
that the circle is
implies
\ 101
the
R. In
2
wholly contained in a compact subset of coordinates: , in remain bounded along
particular, is bounded imply that Since these coordinates are global, it follows x
9.4.1
Equations
,y.
must range from
-oo
The G6del solution is see
this,
we
g
=
coordinate system.
I
oo.
axially symmetric
2(
There is
9.4.2.
E R
2w-
our
that the affine parameter
with respect to any
point. To
will introduce different coordinates.
Proposition
(8, r, 0,,i)
to
in
x
-
R+
d82
\ f01 +
dr2
x
a
S'
dense open set N C M and coordinates R such that 91N is given by
x
+ sinh
2r(I
-
sinh
2
r)d W2
+ d'
2
-
2V2-sinh 2
rd ods).
Proof. Observe that the metric is a direct product, (M, g) (R 3 -dt2 + dX2 + Ie 2Nf2"d y2 2eV2"dtdy. Hence it R, h + dz 2) with h =
=
-
2
sufficient to show that there is
(s, r, o)
hll
=
a
dense open set
9
C
X
is
R 3 and coordinates
such that
2L,)2
(-dS2
+
dr2
+ sinh
2r(i
-
sinh 2r)d
02
-
2N 2 sinh 2
rd ods)
.
9.4
Singularities and causality violations
The assertion follows then with i
12
=
V2w
via the
(s, r, W)
=
cosh(2r)
=
sin(W) sinh(2r)
=
e-
12wx wye
I
(2 ((p
We define the coordinates
equations
ev2wx
tan
z.
+ wt
V2-s)
-
2r
+
cos(W) sinh(2r)
(21W)
tan
(9.4.4)
(9.4.5) (9.4.6)
-
To show that h has the desired form in these coordinates is
ward" but very cumbersome. 6 We first differentiate Equations
vl2weV2xdx
0
+
-
2(sinh(2r)
-
1
0
2
-
sin(W) cosh(2r)dr
cos(W) sinh(2r)dW
(1 (tan(W/2 112wt (1'+ (tan(W/2 wt12 +
-
+
I +
2
(
-2r -
e
(tan(W/2
(I
+
+
)2) s/v/2)) 2) V2
112V2-s)
-
2
+
to obtain
sin(W) sinh(r)dW
+
2
(9.4.4)-(9.4.6)
"straightfor-
cos( p) cosh(2r))dr
+
V/2_W2ev/2-w xydx + evf2-w xwdy
0
401
_ds
(tan(W/2) )2
wt12
_
S
/ N/2-)
wdt
+ 2e
-2r
tan(W/2)dr
+ 1
) 2dW.
V2wx system of equations we can eliminate y and e using Equations The and then be considered can as a linear system (9.4.5). system (9.4.4) for dt, dx, dy which only depends on s, r, W, ds, dr, dW. Solving this linear system gives (after some simplifications)
In this
2
V2-ds dt
-4 W
w
sm(W/2) cos(W/2)e rdr 2 (Or COS2 (W12) +sin (W12)
(- sin 2((p12)
+
4r
2r e
-
e
+ W
v2 dx
1/2
(Or COS2 (W12) 4r
(2e COS2 ( o12) (e 4r COS2 (W12)
w
In
+
COS2 (W12)) dW
(sin(W/2)
)2)
sin2(W12)) dr + sin (W12))
-
2
2
fact, in his original paper, G6del chose to derive this form independently geometric assumptions which led him to the metric.
of the
402
9.
Singularity
theorems
V2_sin(112 o) cos(W/2) (I e4r ) dw (Or C092 (W12) + sin 2 (w12)) -
+1/2
w
dy
sin( o/2) cos(W/2)e 4'dr
4
=
(Or COS2( p/2) + sin 2 (V12) )2 (sin2(W12) + es' COS2 (W12) e4r ) dW +1/2 w (e4r COS2 (W12) + sin%o/2) )2 w
-
We
simply calculate
can now
h
tion
=
e-
2r
-
2
(s, r, p) using (9.4.4) (which is equivalent
ev"2wx
-dt2 + dX2 + Ie 2/2x dy2
=
in the coordinates
(e
4r
our
expressions for
2eV2"dtdy
dt, dx, dy and Equa-
to
COS2 (W12)
(sin( p/2)
+
)2)).
This gives with
A(r, p) B(r,W)
sin2( p12)
=
-sin
=
+
2(W12)
e8 COS2 (W12) r
2
+
e
-
e4
e4 rCos2(W/2)
r -
and using trigonometric identities
ds2 g
-2
C2
-4r
+
v/2-e -2 rA(r, W)
(
+
(8e (e4
W2
(e4
r
+ 2B (r,
COS2 (W/ 2) +
sin2(W12))
2(W12) COS2 (W12) (e 4r)2 COS2 (W/2) + sin2( p12) )2
r
(sin2(W12) e4 COS2 (W12) )2 dr 2 W2 (e4 COS2 (W12) + sin2 (W12)) A (r, W) (B (r, 0))2 + B(r, o)e r
)
-
r
-2
+
W2 1
9
(e4
-4r e
r
COS2 (W12) +
(A(r, W))2 W
2 w-
dW ds
sin
W2
+2
W) V2_
2(
-
2
d '32 + dr
(e4 2
r
I
'r
(sin( o/2) )2
COS2(W/2) sin2(W12) (I
_
COS2 (W/ 2) +
+ sinh
)2
2 r
(1
-
-
sin2( p12) )2
sinh
2
r) d W2
e4r)2
d p2
+d,z2
2V2 sinh 2rdWds).
We will
now
const,
=
s
dates for
show that there
are spacelike submanifolds given by r vanishing expansions. They are our canditrapped surfaces in a suitably perturbed metric.
const which have
strictly closed
9.4
Let
Proposition 9.4.3. 2 surface given by s
t such that the
there is
=
a t+ segment of leaves V and then re-enters this set.
between
V of
-1(t)
-y(t)
and
All other properties of
our example are unchanged since the new spacelocally isometric to the old one. In order to state a theorem which justifies the claim that the only impediment to a version of Theorem 9.3.1 in the presence of causality violation is the possible existence of almost closed causal curves in f + (T) we need the following technical definition.
time is
Definition 9.4.3. Let -y be a curve and choose any Riemannian metTic h on M. Let ft: (a, b) be a reparameterisation of -y which satisfies
h(A, A) JA(t) : exists
=
t E
a
1.
(a,
We call -y almost closed if there exists a vector u E that such for every neighbourhood it of u in TM there b) I
deformation A of 1,t
satisfies X(t)
E
WTM(it)
-,:*
in
(t)
7rTM(5-0 E
which
yields
a
closed
3A.
Observe that this definition is independent of the choice of h.
curve
and
ISItrength
9.5
of
Theorem 9.4. 1. A spacetime
singularities and
(M, g)
cosmic
409
censorship
is not causal
geodesically complete
the timelike convergence condition and the
genericity condition
if
(i)
hold,
-
(ii) there exists at least one of the following: (a) a (locally spacelike) strictly closed trapped surface T, (b) a compact achronal set T without edge, (c) a point x such that along every past (or every future)
inexten-
sible null
geodesic from x the expansion of the null geodesics starting at x becomes negative, (iii) neither f + (T) (respectively, f + (f xJ) nor any f (D), where D is a compact topological submanifold (possibly with boundary) with D n T = 0 (respectively, x E D) contains any almost closed causal curve that is a cluster curve of a sequence of closed timelike curves. -
This is
a
proper
(iii) just
tion
generalisation
of Theorem -93.1. The technical condi-
states the situation which
we have already anticipated by example. The proof of Theorem 9.4.1 is far too technical to be reproduced here. It basically consists of a cutting and pasting procedure (Kriele 1990).
Newman's
analyzing
The closed
trapped
surface in Newman's counter
example has the topol-
ogy of a torus. In a physically realistic collapse scenario of a star one would rather expect that there exists a closed trapped surface of topol-
S' surrounding the collapsing conjecture: ogy
star. This motivates the
Conjecture 9.4. 1. A 4-dimensional spacetime (M, g) cally complete if
(i)
the timelike convergence condition and the
following
is not causal
geodesi-
genericity condition
hold,
(ii)
there exists
a
strictly
closed
trapped surface
of
topology S'.
In Newman's is
example the generalised future focal set of T Jn((I+V 3_)/2) generated by closed null curves. This is impossible if T has topology
S2 In
open
spite of this small piece of evidence and the importance of Conjecour interpretation of singularity theorems it is completely whether this conjecture is true or not.
9.5
Strength of singularities
ture 9.4.1 for
In this section
dicted
we
will
by Theorem
and cosmic
censorship
investigate the character of the singularities pregive an example (cf. Sect. 9.5. 1)
9. 3. 1. We will also
410
Singularity theorems
9.
which shows that the theorem
of Hawking and Penrose may only imof "singularities" which are so weak that the energy exists in a distributional sense. Our example is not very physical for a start, it is 3-di'mensional rather than 4-dimensional. On the other hand, it is a good test case for the mechanism behind the singularity theorems.
ply the density
existence
-
The existence of
incomplete causal geodesic does not imply that there singularity. This is the reason why "singularity theorems" are often referred to as "incompleteness theorems". The standard counter example in general relativity is the Taub-NUT spacetime (cf. (Hawking and Ellis 1973, chapter 5.8). The following two-dimensional example is especially simple.
is
a
2 (Clifford-Pohl torus Let (M, g) (R 2\f0J, _UTTV_T dudv). Then the curve -y(t) is an and the incomplete geodesic 0) map 0: (u, v) (2u, 2v) is an isometry. Defining
Example
9. 5. 1
=
.
-4
x
we
obtain
a
-
y : 0.
neighbourhood U
some
of -y such that
=
curvature invariant
the Riemann
=
homogeneity condition is important because otherwise we could appropriate power of a weakly diverging curvature invariant in 1 order to obtain a diverging integral: While fo, -1--dx is finite, the integral _X The
take
fo
an
716dx
not.
is
Hawking & of such will
a
give
Ellis state that while
conjecture' they a
may not be true in the
as a
are
convinced of the
validity
black
present form.
singularity in our universe, we would like to interpret hole, i.e., we would hope that it is invisible just as the
If there is it
they
unable to prove it. In Sect. 9.5.1 below we 3-dimensional example which indicates that Conjecture 9.5.2 are
a
-
singularity in the Schwarzschild spacetime. Otherwise we would not have a chance to globally solve Einstein's equation as a Cauchy problem since the singularity (whose data are unknown) would influence the geometry spacetime to its future. There are also important theorems for our interpretation of black holes which need a assumption similar to cosmic censorship. The prime example is the "area theorem" due to Hawking which states that the area of black holes can only increase 8) (Wald 1984, of
theorem
12.2.6).
Since it is easy to find which contain visible (or
examples
of inextensible Lorentzian manifolds
additional assumptions spacetime must be made in any conjecture which "censors" naked singularities. The following conjecture is due to Penrose.
"naked") singularities,
on
Conjecture
9.5.3
((strong)
If (M, g) is qualitatively physically reasonable then no future causal geodesic -/ lies in the past of any
cosmic
censorship).
stable and its matter model T is
incomplete,
future inextensible
XEM.
7They 8
state their
This is
only
conjecture with respect to a different singularity theorem. general relativity without taking quantum
true in classical
effects into account
412
Singularity theorems
9.
It is not sufficient just to demand
"physically reasonable" solution,
a
matter model
because of the Reissner-Nordstr6m
27n 9
+
-
e2)
-
dt2
r2
r
+
27n
(I
-
+
e2) _1d
-
+r
The energy momentum tensor T e
T
Ub
U5
=
_
87rr 4
where
81r
Qb
2 r
r2
r
(Ric
2
(d02 + sin2Od W2)
(Scal/2)g)
-
Qb +r2 (d02
is
given by
+ sin 2OdW
2))
have set
we
2m
(I
U
e2)
2
19t
;72-
2m
e2
r
r2
Q
I
'21
)
ar
This
spherically symmetric spacetime satisfies Einstein's equations for electromagnetic field (cf. Lemma 7.4-1) which is certainly "physically reasonable". On the other hand, unlike in the case of the Schwarzschild solution (e const < I are space0) where the hypersurfaces r const < I in the Reissner-Nordstr6m solike, the hypersurfaces r an
=
=
=
lution
are
such that
timelike. It is easy to
1+(x)
see
that there exist
points
x, y G M
1-(y) contains timelike future inextensible curves which approach r 0. (For a more thorough discussion of the ReissnerNordstr6m spacetime including its global properties cf. (Hawking and Ellis 1973, chapter 5.5)). This Reissner-Nordstr6m spacetime therefore vin
=
olates
Conjecture
son
and Penrose
the
case
for
an
9.5.3 if it is
(1973)
qualitatively stable.
and McNamara
intuitive notion of
stability.
Calculations
by Simp-
indicate that this is not
(1978)
It is
generally believed
that
generic, physically acceptable perturbation of the Reissner-Nordstr6m spacetime results in a spacetime which is qualitatively more similar to the Schwarzschild spacetime, even though the Reissner-Nordstr6m spacetime itself can be thought of as a perturbation of the Schwarzschild spacetime. a
9.5.1 A
Let
simple, 3-dimensional example
(M, g)
be
a
3-dimensional spacetime and assume that the energy cUb (9 0 where U be the spacetime given by T
momentum tensor is
=
velocity of the dust particles and
e
their energy
density.
We
are
seeking
solutions of Einstein's equation Ric
-
1Scalg
=
81rEU
(9
U5,
2 c: M -4 R is a function. In general, this is still t ,oo difficult even 3. Assuming that there is a foliation of spacelike though we assume n hypersurfaces orthogonal to U simplifies the problem dramatically.
where
=
9.5
Lemma 9.5. 1.
only if
the
Strength
of
singularities
The vector
field Pfaffian system f 0 1
and cosmic
(i.e., dO
U is irrotational is
censorship
=
413
0) if
and
integrable.
Proof. By Lemma 2.5.8 the integrability of f U51 is equivalent to the 0. Recall from Lemma 5.2.1 that U satisfies the equation dO A 0 0. Let X be any vector field. Then we have geodesic equation VUU dW (X, U) VXW (U) VUW (X) g(VXU, U) g(VUU, X) =
=
=
0
-
0 since
=
-
g(U, U)
=
-1. It follows that
=
-
dO
completely determined
is
by evaluating it on vectors orthogonal to U. Let orthogonal to U. Then we have dO A 0(v, w, U) equivalence follows.
w
v,
=
be two vectors
-dO(v, w)
and the
I
(M, g) be an irrotational, 3-dimensional dust spacehypersurface which is orthogonal to U. If at p c Z the second fundamental form of Z is not a multiple of the metric, then p has -dt2 + V2 dX2 + a neighbourhood with coordinates (t, x, y) such that g W2 d y2' where V, W are functions of t, x, y and T 6(t, x, y)dt 0 dt. These coordinates are unique up to transformations of the type x -4 X(x), y -4 Y(y), t i--> t + const. and interchanging of x and y. Lemma 9.5.2. Let time and let Z be
a
=
=
Proof. Since f 0 1 satisfies dW We
can
write g
=
the bilinear form
-dt2 +
0 there is
=
a
function t with dt
=
0.
(2)g,j (t, Xl' X2)dx'dxj, where for each t E2j=l i,
(2)g(t,
is
a
Riemannian 2-metric and Z is
given by
to. Sinceat(2)g is not umbilic at p, there exists a frame lei, e2j Of Z in a neighbourhood of p such that (2)g andat(2)g are both diagonal
t
=
1
jw W 2 I be the dual frame. It follows (x, y) such that at t to both (2)g and 19t (2)g are diagonal with respect to a, a.. In fact, we only need to show that there exist multiples ale,, a2e-2 of el, e2 such that [alel, 012e2] 0. This is equivalent to dai(ei+l mod 2) + 0 e2l) (no summation over i) which is a system of ordinary differential equations and can be solved by Theorem 2.4. 1. With respect to the coordinates (t, x, y) the 0 (i, j E f x, yj) imply equations Tij with respect to this frame. Let
,
that there exist coordinates
=
==
=
=
atat(2)g,j- tr(atat (2)j) (2)g,j
_
1tr (19t(2)g)at (2)gij I
+
4
Since at t
that the has
a
=
to the bilinear forms
right hand side
is also
up to
(2)gk1at (2)gik at (2)g jj
( (tr (at g) )2 -31 (2)
(2)gij
and
diagonal
unique solution and there exists
simultaneously diagonal, (2)g ',9t (2)g have existence. For
+
2
a
19t(2)gii at t
=
are
at Mg
12) (2)gj.
diagonal,
it follows
to. Since the system
solution when (2)g, 19t(2)g
Must be
diagonal
for all t. Hence
are we
uniqueness observe that the frame lei, e2j is unique multiples and permutation, and that the coordinate t is already
414
9.
Singularity theorems
chosen
so
that it is
coordinates
X,
9,y I I Oy
and
or
unique
Y with the
up to same
additive constant. Thus any other properties must satisfy either o9-- 11 Ox an
9., 11,9y and i9y 11 9x. This
I
proves the lemma.
Corollary 9.5.1. Let (Z, Mg) be a 2-dimensional Riemannian manitensor field which is not proportional to fold and k be a symmetric (0) 2 (2)g. Then the initial value problem for irrotational, 3-dimensional dust spacetimes with initial data (Z' (2)g, k) reduces of ordinary differential equations.
to
a
constrained system
we will summarise properties of generic, irrotational, 3-dimensional dust spacetimes using standard differential geometric terminology. Consider a 2-dimensional Riemannian manifold (Z, gz) and denote the set of unoriented lines in TZ by PZ. Then there exists
In Theorem 9.5.1 below
a
natural map -1: PE
-4
PZ which maps an unoriented line 1 E PZ to a section 1 of PE nowhere geodesic if
to it. We call
the line
orthogonal local, non-vanishing vector field L with L(p) cz l(p) Vp we have 1 gz(L 7LL) : - 0 Vp. This condition does not depend on the chosen representative L. It is a local but not necessarily a global genericity condition on 1. Locally, this condition is slightly stronger than demanding that 1 does not have any local integral curve which is a geodesic. Let (z, g_') be a 2-dimensional, spacelike submanifold of a Lorentzian 3-manifold for any
(M, g) with future directed normal n and second fundamental form k(X, Y) -g(17XY, n). We denote the bilinear form associated with the the of corresponding matrix by k2' i.e., (V), (g_r)1mkjjkj,,,. square =
=
eigenvalues kj, k2 of k with respect to gz the principal curZ and the (unoriented) lines spanned by the eigenvectors the
We call the
of principal directions of Z. vatures
Theorem 9.5. 1. Let
(R2, (2)g)
be
a
Riemannian
2-manifold
and 1: R2
which maps each point p (2 R2 into an unoriented line l(p) C and assume that 1 is nowhere geodesic. Let C C R2 be a smooth
TR2
TpR2 curve
which divides R2 into two disconnected regions such that 1 and 1--L intersect TpC transversely at each p E C. Finally, let KI, K2: C --* R be smooth
junctions.
the set of points p E R2 such that the integral l' through p intersect C. There exists an irrotational, 3-dimensional dust spacetime (M, g) and an isometric embedding L: (Z (C' 1, 11), (2)g) ---> Z C M such that the second fundamental form k of Z in M satisfies (a) the principal directions of Z are given by t.1, tlj-, K1, (b) along C the submanifold Z has principal curvatures k, K2 k2 R2. Then (M, g) is inextenC can be chosen such that 0 (C, 1, 1 -L)
(i)
Let
curves
O(C, 1, 1 of 1
and
=
:--
-
=
sible
if (R2,
(2)g)
is
So.
For any p
(ii)
k-
(p)
c Z
and cosmic
singularities
censorship
415
let
(p), k2 (p))
min (ki
=
of
Strength
9.5
and
k+ (p)
max(ki (p), k2 (P))
=
The world line
of the dust particle through p ends in a curvature 1 1 finite proper times k (p) k+ (p) if k+ (p) > 0 > k- (p), -
singularity at
finite
if k+(p)
in
-1
proper time
There
< 0.
-
,
k
(p)
ifk-(p) other
are no
>
0,
at
finite
proper time
singularities are weak in the sense that for all volume, vol(U) fu -, /_det(g,,b)dtdxdy spacetime average of the energy density, All
(iii)
U with bounded
=
I
fu
vol(U)
e
k+ (p)
singularities. open sets oo, the
