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CLASSICAL MEASUREMENTS IN CURVED SPACETIMES
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CLASSICAL MEASUREMENTS IN CURVED SPACETIMES
The theory of relativity describes the laws of physics in a given spacetime. However, a physical theory must provide observational predictions expressed in terms of measurements, which are the outcome of practical experiments and observations. Ideal for researchers with a mathematical background and a basic knowledge of relativity, this book will help in the understanding of the physics behind the mathematical formalism of the theory of relativity. It explores the informative power of the theory of relativity, and shows how it can be used in space physics, astrophysics, and cosmology. Readers are given the tools to pick out from the mathematical formalism the quantities which have physical meaning, which can therefore be the result of a measurement. The book considers the complications that arise through the interpretation of a measurement which is dependent on the observer who performs it. Speciﬁc examples of this are given to highlight the awkwardness of the problem. Fernando de Felice is Professor of Theoretical Physics at the University of Padova, Italy. He has signiﬁcant research experience in the subject of general relativity at institutions in the USA, Canada, the UK, Japan, and Brazil. He was awarded the Volterra Medal by the Accademia dei Lincei in 2005. Donato Bini is a Researcher of the Italian Research Council (CNR) at the Istituto per le Applicazioni del Calcolo “M. Picone” (IAC), Rome. His research interests include spacetime splitting techniques in general relativity.
CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General Editors: P. V. Landshoﬀ, D. R. Nelson, S. Weinberg S. J. Aarseth Gravitational NBody Simulations: Tools and Algorithms J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach A. M. Anile Relativistic Fluids and Magnetoﬂuids: With Applications in Astrophysics and Plasma Physics J. A. de Azc´ arraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O. Babelon, D. Bernard and M. Talon Introduction to Classical Integrable Systems† F. Bastianelli and P. van Nieuwenhuizen Path Integrals and Anomalies in Curved Space V. Belinski and E. Verdaguer Gravitational Solitons J. Bernstein Kinetic Theory in the Expanding Universe G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Space† K. Bolejko, A. Krasi´ nski, C. Hellaby and M. N. C´ el´ erier Structures in the Universe by Exact Methods: Formation, Evolution, Interactions D. M. Brink SemiClassical Methods for NucleusNucleus Scattering† M. Burgess Classical Covariant Fields E. A. Calzetta and B. L. B. Hu Nonequilibrium Quantum Field Theory S. Carlip Quantum Gravity in 2+1 Dimensions† P. Cartier and C. DeWittMorette Functional Integration: Action and Symmetries J. C. Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the OperatorProduct Expansion† P. D. B. Collins An Introduction to Regge Theory and High Energy Physics† M. Creutz Quarks, Gluons and Lattices† P. D. D’Eath Supersymmetric Quantum Cosmology F. de Felice and D. Bini Classical Measurements in Curved SpaceTimes F. de Felice and C. J. S. Clarke Relativity on Curved Manifolds B. DeWitt Supermanifolds 2nd edn† P. G. O. Freund Introduction to Supersymmetry† Y. Frishman and J. Sonnenschein NonPerturbative Field Theory: From Two Dimensional Conformal Field Theory to QCD in Four Dimensions J. A. Fuchs Aﬃne Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory† J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y. Fujii and K. Maeda The ScalarTensor Theory of Gravitation J. A. H. Futterman, F. A. Handler and R. A. Matzner Scattering from Black Holes† A. S. Galperin, E. A. Ivanov, V. I. Orievetsky and E. S. Sokatchev Harmonic Superspace R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity† T. Gannon Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics M. G¨ ockeler and T. Sch¨ ucker Diﬀerential Geometry, Gauge Theories and Gravity† C. G´ omez, M. RuizAltaba and G. Sierra Quantum Groups in TwoDimensional Physics M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 1: Introduction† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 2: Loop Amplitudes, Anomalies and Phenomenology† V. N. Gribov The Theory of Complex Angular Momenta: Gribov Lectures on Theoretical Physics J. B. Griﬃths and J. Podolsk´ y Exact SpaceTimes in Einstein’s General Relativity S. W. Hawking and G. F. R. Ellis The Large Scale Structure of SpaceTime† F. Iachello and A. Arima The Interacting Boson Model F. Iachello and P. van Isacker The Interacting BosonFermion Model C. Itzykson and J. M. Drouﬀe Statistical Field Theory Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J. M. Drouﬀe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems† C. V. Johnson DBranes† P. S. Joshi Gravitational Collapse and Spacetime Singularities J. I. Kapusta and C. Gale FiniteTemperature Field Theory: Principles and Applications, 2nd edn V. E. Korepin, N. M. Bogoliubov and A. G. Izergin Quantum Inverse Scattering Method and Correlation Functions† M. Le Bellac Thermal Field Theory† Y. Makeenko Methods of Contemporary Gauge Theory N. Manton and P. Sutcliﬀe Topological Solitons† N. H. March Liquid Metals: Concepts and Theory I. Montvay and G. M¨ unster Quantum Fields on a Lattice†
L. O’Raifeartaigh Group Structure of Gauge Theories† T. Ort´ın Gravity and Strings† A. M. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† L. Parker and D. J. Toms Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity R. Penrose and W. Rindler Spinors and SpaceTime Volume 1: TwoSpinor Calculus and Relativistic Fields† R. Penrose and W. Rindler Spinors and SpaceTime Volume 2: Spinor and Twistor Methods in SpaceTime Geometry† S. Pokorski Gauge Field Theories, 2nd edn† J. Polchinski String Theory Volume 1: An Introduction to the Bosonic String J. Polchinski String Theory Volume 2: Superstring Theory and Beyond V. N. Popov Functional Integrals and Collective Excitations† R. J. Rivers Path Integral Methods in Quantum Field Theory† R. G. Roberts The Structure of the Proton: Deep Inelastic Scattering† C. Rovelli Quantum Gravity† W. C. Saslaw Gravitational Physics of Stellar and Galactic Systems† R. N. Sen Causality, Measurement Theory and the Diﬀerentiable Structure of SpaceTime M. Shifman and A. Yung Supersymmetric Solitons H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, 2nd edn J. Stewart Advanced General Relativity† T. Thiemann Modern Canonical Quantum General Relativity D. J. Toms The Schwinger Action Principle and Eﬀective Action A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects† R. S. Ward and R. O. Wells, Jr Twistor Geometry and Field Theory† J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics †
Issued as a paperback
Classical Measurements in Curved SpaceTimes FERNANDO DE FELICE Physics Department, University of Padova, Italy
DONATO BINI Istituto per le Applicazioni del Calcolo “M. Picone,” CNR, Rome, Italy
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521889308 © F. de Felice and D. Bini 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN13
9780511776694
eBook (NetLibrary)
ISBN13
9780521889308
Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or thirdparty internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface Notation 1 Introduction 1.1 Observers and physical measurements 1.2 Interpretation of physical measurements 1.3 Clock synchronization and relativity of time
page xi xii 1 1 2 2
2 The theory of relativity: a mathematical overview 2.1 The spacetime 2.2 Derivatives on a manifold 2.3 Killing symmetries 2.4 Petrov classiﬁcation 2.5 Einstein’s equations 2.6 Exact solutions
11 11 17 27 28 28 29
3 Spacetime splitting 3.1 Orthogonal decompositions 3.2 Threedimensional notation 3.3 Kinematics of the observer’s congruence 3.4 Adapted frames 3.5 Comparing families of observers 3.6 Splitting of derivatives along a timelike curve
34 34 39 41 41 48 53
4 Special frames 4.1 Orthonormal frames 4.2 Null frames
59 59 69
5 The world function 5.1 The connector 5.2 Mathematical properties of the world function 5.3 The world function in Fermi coordinates 5.4 The world function in de Sitter spacetime 5.5 The world function in G¨ odel spacetime 5.6 The world function of a weak gravitational wave
74 75 77 80 82 84 87
viii 5.7
6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14
Contents Applications of the world function: GPS or emission coordinates Local measurements Measurements of time intervals and space distances Measurements of angles Measurements of spatial velocities Composition of velocities Measurements of energy and momentum Measurements of frequencies Measurements of acceleration Acceleration change under observer transformations Kinematical tensor change under observer transformations Measurements of electric and magnetic ﬁelds Local properties of an electromagnetic ﬁeld Timeplusspace (1 + 3) form of Maxwell’s equations Gravitoelectromagnetism Physical properties of ﬂuids
88 91 91 95 96 97 99 99 102 108 108 111 113 115 115 116
7 Nonlocal measurements 7.1 Measurement of the spacetime curvature 7.2 Vacuum Einstein’s equations in 1+3 form 7.3 Divergence of the Weyl tensor in 1+3 form 7.4 Electric and magnetic parts of the Weyl tensor 7.5 The BelRobinson tensor 7.6 Measurement of the electric part of the Riemann tensor 7.7 Measurement of the magnetic part of the Riemann tensor 7.8 Curvature contributions to spatial velocity 7.9 Curvature contributions to the measurements of angles
126 126 129 130 131 132 135 141 143 149
8 Observers in physically relevant spacetimes 8.1 Schwarzschild spacetime 8.2 Special observers in Schwarzschild spacetime 8.3 Kerr spacetime 8.4 Special observers in Kerr spacetime 8.5 Gravitational planewave spacetime
152 152 156 165 169 184
9 Measurements in physically relevant spacetimes 9.1 Measurements in Schwarzschild spacetime 9.2 The problem of space navigation 9.3 Measurements in Kerr spacetime 9.4 Relativistic thrust anomaly 9.5 Measurements of blackhole parameters
186 186 194 199 206 212
Contents 9.6 9.7 9.8 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8
Gravitationally induced time delay Raytracing in Kerr spacetime Highprecision astrometry Measurements of spinning bodies Behavior of spin in general spacetimes Motion of a test gyroscope in a weak gravitational ﬁeld Motion of a test gyroscope in Schwarzschild spacetime Motion of a spinning body in Schwarzschild spacetime Motion of a test gyroscope in Kerr spacetime Motion of a spinning body in Kerr spacetime Gravitational waves and the compass of inertia Motion of an extended body in a gravitational wave spacetime
Epilogue Exercises References Index
ix 215 218 225 231 231 237 239 241 247 248 255 261 265 267 299 305
Preface
A physical measurement is meaningful only if one identiﬁes in a nonambiguous way who is the observer and what is being observed. The same observable can be the target of more than one observer so we need a suitable algorithm to compare their measurements. This is the task of the theory of measurement which we develop here in the framework of general relativity. Before tackling the formal aspects of the theory, we shall deﬁne what we mean by observer and measurement and illustrate in more detail the concept which most aﬀected, at the beginning of the twentieth century, our common way of thinking, namely the relativity of time. We then continue on our task with a review of the entire mathematical machinery of the theory of relativity. Indeed, the richness and complexity of that machinery are essential to deﬁne a measurement consistently with the geometrical and physical environment of the system under consideration. Most of the material contained in this book is spread throughout the literature and the topic is so vast that we had to consider only a minor part of it, concentrating on the general method rather than single applications. These have been extensively analyzed in Cliﬀord Will’s book (Will, 1981), which remains an essential milestone in the ﬁeld of experimental gravity. Nevertheless we apologize for all the references that would have been pertinent but were overlooked. We acknowledge ﬁnancial support by the Istituto Nazionale di Fisica Nucleare, the International Center for Relativistic Astrophysics Network, the Gruppo Nazionale per la Fisica Matematica of Istituto Nazionale di Alta Matematica, and the Ministero della Pubblica Istruzione of Italy. Thanks are due to Christian Cherubini, Andrea Geralico, Giovanni Preti, and Oldrich Semer´ ak for helpful discussions. Particular thanks go to Robert Jantzen for promoting interest in this ﬁeld. All the blame for any inconsistencies and errors contained in the book should be addressed to us only. Fernando de Felice Physics Department, University of Padova, Italy Donato Bini Istituto per le Applicazioni del Calcolo “M. Picone,” CNR, Rome, Italy
Notation
: The real line. 4 : The space of the quadruplets of real numbers. {xα }α=0,1,2,3 : A quadruplet of local coordinates. {eα }: A ﬁeld of bases (frames) for the tangent space. {ω α }: A ﬁeld of dual bases (dual frames) ω α (eβ ) = δ α β . g = gαβ ω α ⊗ ω β : The metric tensor. g−1 : Inverse metric. g: Determinant of the metric. X # : A tangent vector ﬁeld (with contravariant components). X : The 1form (with covariant components) gisomorphic to X. (left contraction): Contraction of the rightmost contravariant index of the ﬁrst tensor with the leftmost covariant index of the second tensor, that is [S T ]··· ··· = S ···α Tα··· . (right contraction): Contraction of the rightmost covariant index of the ﬁrst tensor with the leftmost contravariant index of the second tensor, that is [S T ]··· ··· = S···α T α··· . p
(left pcontraction): Contraction of the rightmost p contravariant indices of the ﬁrst tensor with the leftmost p covariant indices of the second tensor, i.e. S p T ≡ S α...β1 ...βp Tβ1 ...βp ... .
p
(right pcontraction): Contraction of the rightmost p covariant indices of the ﬁrst tensor with the leftmost p contravariant indices of the second tensor, i.e. S p T ≡ S α... β1 ...βp T β1 ...βp ... . r s tensor: A tensor rtimes contravariant and stimes covariant. [α1 . . . αp ]: Antisymmetrization of the p indices. (α1 . . . αp ): Symmetrization of the p indices.
Notation
xiii
[ALT S]α1 ...αp = S[α1 ...αp ] . [SYM S]α1 ...αp = S(α1 ...αp ) . eγ (·): γcomponent of a frame derivative. ∇: Covariant derivative. ∇eα : αcomponent of the covariant derivative relative to the frame {eσ }. α1 ...α4 = [α1 ...α4 ] : LeviCivita alternating symbol. ηα1 ...α4 = g 1/2 α1 ...α4 ; η α1 ...α4 = −g −1/2 α1 ...α4 : The unit volume 4form. 4 δ α1β...α = α1 ...α4 β1 ...β4 = −η α1 ...α4 ηβ1 ...β4 : Generalized Kronecker delta. 1 ...β4
[∗S]αp+1 ...α4 =
1 p!
Sα1 ...αp η α1 ...αp αp+1 ...α4 : Hodge dual of Sα1 ...αp .
“·”: Scalar gproduct, i.e. u · v = g(u, v) = gαβ uα v β for any pair of vectors (u, v). ∧: The exterior or wedge product, i.e. u ∧ v = u ⊗ v − v ⊗ u for any pair (u, v) of vectors or 1forms. {eαˆ }: An orthonormal frame (tetrad). D ds :
Absolute derivative along a curve γ with parameter s, i.e. D/ds = ∇γ˙ .
a(u): Acceleration vector of the world line with tangent vector ﬁeld u, i.e. a(u) = ∇u u. D(fw,u) ds :
The FermiWalker derivative along the curve with parameter s. For any D X vector ﬁeld X: (fw,u) = DX ds ds ± [a(u)(u · X) − u(a(u) · X)].
CX : The congruence of curves with tangent ﬁeld X. ω(X): The vorticity tensor of the congruence CX (the same symbol also denotes the vorticity vector). θ(X): The expansion tensor of the congruence CX . Θ(X) = Tr θ(X): The trace of the expansion tensor of the congruence CX . £X : Lie derivative along the congruence CX . C γ αβ : Structure functions of a given frame. ω α1 ...αp = p!ω [α1 ⊗ · · · ⊗ ω αp ] : The dual basis tensor of a space of pforms. δT = ∗ d[∗ T ]: Divergence of a pform T . Δ(dR) = δd + dδ: de Rham operator. LRSu : Local rest space of u. P (u): uspatial projector operator which generates LRSu .
xiv
Notation
T (u): utemporal projector operator which generates the time axis of u. [P (u)S] ≡ S(u): Total uspatial projection of a tensor S such that [S(u)]α1 ... β1 ... = P (u)α1 σ1 · · · P (u)ρ1 β1 · · · S σ1 ... ρ1 ... . [£(u)X ]: uspatially projected Lie derivative. For any tensor T it is ρ σ [£(u)X T ]α... β... = P (u)α σ · · · P (u) β · · · [£X T ] . . . ρ... .
∇(u)lie = £(u)u : uspatial Lie temporal derivative. ∇(u) = P (u)∇: uspatially projected covariant derivative. D
: uspatially projected FermiWalker derivative along a curve with P (u) (fw,X) ds unit tangent vector X. d(u) = P (u)d: uspatially projected exterior derivative. “·u ”: uspatial inner product, i.e. X ·u Y = P (u)αβ X α Y β . “×u ” : uspatial cross product, i.e. [X ×u Y ]α = η(u)α ρσ X ρ Y σ . η(u)α ρσ = uβ η βα ρσ : uspatial 4volume. gradu = ∇(u): uspatial gradient. curlu = ∇(u)×u : uspatial curl. divu = ∇(u)·u : uspatial divergence. Scurlu : Symmetrized curlu , i.e. [Scurlu A]αβ = η(u)γδ(α ∇(u)γ Aβ) δ . C(fw)ab : FermiWalker rotation coeﬃcients, i.e. C(fw)ab = eb · ∇u ea . C(lie) b a : Lie rotation coeﬃcients, i.e. C(lie) b a = ω b (£(u)u ea ). ∇(u)(fw) = P (u)∇u : uspatial FermiWalker temporal derivative. ∇(u)(tem) . ≡ ∇(u)(fw) or ∇(u)(lie) ν(U, u): Relative spatial velocity of U with respect to u. ν(u, U ): Relative spatial velocity of u with respect to U . γ(U, u) = γ(u, U ) = γ: Lorentz factor of the two observers u and U . νˆ(u, U ): Unitary relative velocity vector of u with respect to U . ζ: Angular velocity. ω(k, u): Frequency of the light ray k with respect to the observer u. ν(U, u) = ν(u, U ) = ν: Magnitude of the relative velocity of the two observers u and U .
Notation
xv
B(U, u): Relative boost from u to U . P (U, u) = P (U )P (u): Mixed projector operator from LRSu into LRSU . B(lrs) (U, u) = P (U )B(U, u)P (u): Boost from LRSu into LRSU . B(lrs) (U, u)−1 = B(lrs) (u, U ): Inverse boost from LRSU to LRSu . B(lrs) u (U, u) = P (U, u)−1 D(lie,U ) X dτU
B(lrs) (U, u).
= [U, X]: Lie derivative of X along CU .
τ(U, u) : Relative standard time parameter, i.e. dτ(U, u) = γ(U, u)dτU . (U, u) : Relative standard length parameter, i.e. d (U, u) = γ(U, u)ν(U, u)dτU . D(fw,U,u) dτ(U, u)
D = P (u) dτ(U, : Projected absolute covariant derivative along U . u)
a(fw,U,u) = P (u) Dν(U,u) dτ(U,u) : Relative acceleration of U with respect to u. (∇X)αβ ≡ ∇β Xα .
Physical dimensions We are using geometrized units with G = 1 = c, G and c being Newton’s gravitational constant and the speed of light in vacuum, respectively. Symbols are as they appear in the text; the reader is advised that more than one symbol may be used for the same item and conversely the same symbol may refer to diﬀerent items. Reference is made to the observers (U, u). Time t Space r, x, y, z Mass M, m, μ0 Angular velocity ζ Energy E Speciﬁc energy E, γ (in units of mc2 ) Speciﬁc angular momentum (in units of mc) L, λ, Λ, of a rotating source (Kerr) a Spin S Speciﬁc spin S/m 4velocity U, u Relative velocity ν(U, u) Force F Acceleration a(U ) Expansion Θ(U )
→ → → → →
[L]1 [L]1 [L]1 [L]−1 [L]1
→
[L]0
→ → → → → → → → →
[L]1 [L]1 [L]2 [L]1 [L]0 [L]0 [L]0 [L]−1 [L]−1
xvi
Notation → → → → → → → →
Vorticity (ω(U )αβ ω(U )αβ )1/2 Electric charge Q, e Spacetime curvature (Rαβγδ Rαβγδ )1/2 Electric ﬁeld E(U ) Magnetic ﬁeld B(U ) Frequency ω(k, u) Gravitational wave amplitude h+,× Strain S(U )
[L]−1 [L]1 [L]−2 [L]−1 [L]−1 [L]−1 [L]0 [L]−2 .
Conversion factors We list here conversion factors from conventional CGS to geometrized units. For convenience we denote the quantities in CGS units with a tilde (∼). Name
CGS units
Mass
˜ M
M=
Electric charge
˜ Q
˜ Q=Q
Velocity
v˜
Acceleration
a ˜
Force
F˜
ν = vc˜ a = ca˜2 ˜ F = Gc4F
Electric ﬁeld
˜ E
˜ E=E
Magnetic ﬁeld
˜ H
˜ H=H
Energy
E˜
E=
E˜ ˜ c2 M
E M
Speciﬁc energy Angular momentum ˜c Angular momentum in units of M
Geometrized units ˜ GM c2
G 4π0 c4
4π0 G c4 4π0 G c2
GE˜ c4
≡γ=
˜ L
L=
˜ L ˜c M
λ=
E˜ ˜ c2 M
˜ GL c3 ˜ L = ˜L M Mc
1 Introduction
A physical measurement requires a collection of devices such as a clock, a theodolite, a counter, a light gun, and so on. The operational control of this instrumentation is exercised by the observer, who decides what to measure, how to perform a measurement, and how to interpret the results. The observer’s laboratory covers a ﬁnite spatial volume and the measurements last for a ﬁnite interval of time so we can deﬁne as the measurement’s domain the spacetime region in which a process of measurement takes place. If the background curvature can be neglected, then the measurements will not suﬀer from curvature eﬀects and will then be termed local. On the contrary, if the curvature is strong enough that it cannot be neglected over the measurement’s domain, the response of the instruments will depend on the position therein and therefore they require a careful calibration to correct for curvature perturbations. In this case the measurements carrying a signature of the curvature will be termed nonlocal.
1.1 Observers and physical measurements A laboratory is mathematically modeled by a family of nonintersecting timelike curves having u as tangent vector ﬁeld and denoted by Cu ; this family is also termed the congruence. Each curve of the congruence represents the history of a point in the laboratory. We choose the parameter τ on the curves of Cu so as to make the tangent vector ﬁeld u unitary; this choice is always possible for nonnull curves. Let Σ be a spacelike threedimensional section of Cu spanned by the curves which cross a selected curve γ∗ of the congruence orthogonally. The concepts of unitarity and orthogonality are relative to the assumed background metric. The curve γ∗ will be termed the ﬁducial curve of the congruence and referred to as the world line of the observer. Let the point of intersection of Σ with γ∗ be γ∗ (τ ); as τ varies continuously over γ∗ , the section Σ spans a fourdimensional volume which represents the spacetime history of the observer’s laboratory. Whenever we limit the extension of Σ to a range much smaller than
2
Introduction
the average radius of its induced curvature, we can identify Cu with the single curve γ∗ and Σ with the point γ∗ (τ ). Any timelike curve γ with tangent vector u can then be identiﬁed as the world line of an observer, which will be referred to as “the observer u.” If the parameter τ on γ is such as to make the tangent vector unitary, then its physical meaning is that of the proper time of the observer u, i.e. the time read on his clock in units of the speed of light in vacuum. This concept of observer, however, needs to be specialized further, deﬁning a reference frame adapted to him. A reference frame is deﬁned by a clock which marks the time as a parameter on γ, as already noted, and by a spatial frame made of three spacelike directions identiﬁed at each point on γ by spacelike curves stemming orthogonally from it. While the time direction is uniquely ﬁxed by the vector ﬁeld u, the spatial directions are deﬁned up to spatial rotations, i.e. transformations which do not change u; obviously there are inﬁnitely many such spatial perspectives. The result of a physical measurement is mathematically described by a scalar, a quantity which is invariant under general coordinate transformations. A scalar quantity, however, is not necessarily a physical measurement. The latter, in fact, needs to be deﬁned with respect to an observer and in particular to one of the inﬁnitely many spatial frames adapted to him. The aim of the relativistic theory of measurement is to enable one to devise, out of the tensorial representation of a physical system and with respect to a given frame, those scalars which describe speciﬁc properties of the system. The measurements are in general observerdependent so, as stated, a criterion should also be given for comparing measurements made by diﬀerent observers. A basic role in this procedure of comparison is played by the Lorentz group of transformations. A measurement which is observerindependent is termed Lorentz invariant. Lorentz invariant measurements are of key importance in physics.
1.2 Interpretation of physical measurements The description of a physical system depends both on the observer and on the chosen frame of reference. In most cases the result of a measurement is aﬀected by contributions from the background curvature and from the peculiarity of the reference frame. As long as it is not possible to discriminate among them, a measurement remains plagued by an intrinsic ambiguity. We shall present a few examples where this situation arises and discuss possible ways out. The most important among the observerdependent measurements is that of time intervals. Basic to Einstein’s theory of relativity is the relativity of time. Hence we shall illustrate this concept ﬁrst, dealing with inertial frames for the sake of clarity.
1.3 Clock synchronization and relativity of time The theory of special relativity, formally issued in 1905 (Einstein, 1905), presupposes that inertial observers are fully equivalent in describing physical laws. This
1.3 Clock synchronization and relativity of time
3
requirement, known as the principle of relativity, implies that one has to abandon the concepts of absolute space and absolute time. This step is essential in order to envisage a model of reality which is consistent with observations and in particular with the behavior of light. As is well known, the speed of light c, whose value in vacuum is 2.997 924 58 × 105 km s−1 , is independent of the observer who measures it, and therefore is an absolute quantity. Since time plays the role of a coordinate with the same prerogatives as the spatial ones, one needs a criterion for assigning a value of that coordinate, let us say t, to each spacetime point. The criterion of time labeling, also termed clock synchronization, should be the same in all frames if we want the principle of relativity to make sense, and this is assured by the universality of the velocity of light. In fact, one uses a light ray stemming from a ﬁducial point with spatial coordinates x0 , for example, and time coordinate equal to zero, then assigns to each point of spatial coordinates x0 + Δx crossed by the light ray the time t = Δx/c. In this way, assuming the connectivity of spacetime, we can label each of its points with a value of t. Clearly one must be able to ﬁx for each of them the spatial separation Δx from the given ﬁducial point, but that is a nontrivial procedure which will be discussed later in the book. The relativity of time is usually stated by saying that if an observer u compares the time t read on his own clock with that read on the clock of an observer u moving uniformly with respect to u and instantaneously coincident with it, then u ﬁnds that t diﬀers from t by some factor K, as t = Kt.1 On the other hand, if the comparison is made by the observer u , because of the equivalence of the inertial observers he will ﬁnd that the time t marked by the clock of u diﬀers from the time t read on his own clock when they instantaneously coincide, by the same factor, as t = Kt . The factor K, which we denote as the relativity factor, is at this stage unknown except for the obvious facts that it should be positive, it should depend only on the magnitude of the relative velocity for consistency with the principle of relativity, and ﬁnally that it should reduce to one when the relative velocity is equal to zero. Our aim is to ﬁnd the factor K and explain why it diﬀers in general from one. A similar analysis can be found in Bondi’s Kcalculus (Bondi, 1980; see also de Felice, 2006). In what follows we shall not require knowledge of the Lorentz transformations nor of any concept of relativity. Let us consider an inertial frame S with coordinates (x, y, z) and time t. The time axes in S form a congruence of curves each representing the history of a static observer at the corresponding spatial point. Denote by u the ﬁducial observer of this family, located at the spatial origin of S. At each point of S there exists a clock which marks the time t of that particular event and which would be read by the static observer spatially ﬁxed at that point. All static observers 1
The choice of a linear relation is justiﬁed a posteriori since it leads to the correct theory of relativity.
4
Introduction
in S are equivalent to each other since we require that their time runs with the same rate. A clock which is attached to each point of S will be termed an Sclock. Let S be another inertial frame with spatial coordinates (x , y , z ) and time t . We require that S moves uniformly along the xaxis of S with velocity ν. The xaxis of S will be considered spatially coincident with the x axis of S , with the further requirement that the origins of x and x coincide at the time t = t = 0. In this case the relative motion is that of a recession. In the frame S the totality of time axes forms a congruence of curves each representing the history of a static observer. At each point of S there is a clock, termed an S clock, which marks the time of that particular event and is read by the static observer ﬁxed at the corresponding spatial position. The S clocks mark the time t with the same rate; hence the static observers in S are equivalent to each other. Finally we denote by u the ﬁducial observer of the above congruence of time axes, ﬁxed at the spatial origin of S . Let the systems S and S be represented by the 2planes (ct, x) and (ct , x ) respectively;2 we then assume that from the spatial origin of S and at time tu , a light signal is emitted along the xaxis and towards the observer u . The light signal reaches the observer u at the time marked by the local Sclock, given by tu . (1.1) tu = 1 − ν/c At this event, the observer u can read two clocks which are momentarily coincident, namely the Sclock which marks the time tu as in (1.1) and his own clock which marks a time tu . In general the time beating on a given clock is driven by a sequence of events; in our case the time read on the clock of the observer u, at the spatial origin of S, follows the emission of the light signals. If these are emitted with continuity3 then the time marked by the clock of the observer u will be a continuous function on S which we still denote by tu . The time read on the Sclocks which are crossed by the observer u along his path marks the instants of recording by u of the light signals emitted by u. The events of reception by u , however, do not belong to the history of one observer only, but each of them, having a diﬀerent spatial position in S, belongs to the history of the static observer located at the corresponding spatial point. Let us now consider the same process as seen in the frame S . The spacetime of S is carpeted by S clocks each marking the time t read by the static observers ﬁxed at each spatial point of S . The observer u , at the spatial origin of S , receives at time tu the light signal emitted by the observer u who is seen receding along the negative direction of the x axis. The emission of the light signals by 2 3
This is possible without loss of generality because of the homogeneity and isotropy of space. By continuity here we mean that the time interval between any two events of emission (or of reception) goes to zero.
1.3 Clock synchronization and relativity of time
5
the observer u occurs at times tu read on the S clocks crossed by u along his path and given by tu =
tu . 1 + ν/c
(1.2)
The time on the clock of the observer u , denoted by tu , runs continuously with the recording of the light signals emitted by u. Meanwhile the observer u can read two clocks, momentarily coincident, namely his own clock which marks a time tu and the S clock which is crossed by u during his motion which marks a time tu . Also in this case we have to remember that tu is not the time read on the clock of one single observer but is the time read at each instant on an S clock belonging to the static observer ﬁxed at the corresponding spatial position in S . To summarize, the time read on the Sclocks set along the path of u in S is tu while the time marked by the clock carried by u is tu . Analogously the time read on the S clocks set along the path of u in S is tu while the time read by u on his own clock is given by tu . Our aim is to ﬁnd the relation between t u and tu in the frame S and that between tu and tu in the frame S . In both cases we are comparing times read on clocks which are in relative motion but instantaneously coincident. The observers u and u located at the spatial origins of S and S respectively cannot read each other’s clocks because they will be far apart after the initial time t = t = 0 when they are assumed to coincide. In order to ﬁnd the relation between tu and tu one has to go through the intermediate steps where (i) the observer u at the spatial origin of S correlates the time tu , read on his own clock at the emission of the light signals, to the time tu , marked by the Sclocks when they are reached by the light signals and simultaneously crossed by the observer u along his path in S; (ii) the observer u at the spatial origin of S correlates the time tu , read on his own clock, to the time tu marked by the S clocks when a light signal was emitted and simultaneously crossed by u along his path in S . The two points of view are not symmetric; in fact, the light signals are emitted by u and received by u in both cases. These intermediate steps allow us to establish the relativity of time. The principle of relativity ensures the complete equivalence of the inertial observers in the sense that they will always draw the same conclusions from an equal set of observations. In our case, comparing the points of view of the two observers, we deduce that the ratio between the time tu that u reads on each Sclock when he crosses it, and the time tu that he reads on his own clock at the same instant, is the same as the ratio between the time tu that u reads on
6
Introduction
each S clock which he crosses during his motion in S , and the time tu that he reads on his own clock at the same instant, namely: tu tu = . tu tu
(1.3)
Taking into account (1.1), relation (1.3) becomes tu = tu
ν t2u tu . 1 − = tu tu c
Then, from (1.2), tu =
t2u tu
1−
ν2 c2
(1.4)
.
(1.5)
Along the path of u in S, we have tu
=
1−
ν2 tu . c2
(1.6)
Similarly, from (1.3) and (1.2) we have tu = tu
ν t2 tu . = u 1+ tu tu c
(1.7)
ν2 1− 2 . c
(1.8)
Hence, from (1.1), tu =
t2 u tu
Along the path of u in S we ﬁnally have
ν2 t . (1.9) c2 u
Thus the factor K turns out to be equal to 1 − (ν/c)2 . The above considerations have been made under the assumption that the observers u and u are receding from each other. However the above result should still hold if the observers u and u are approaching instead. We shall prove that this is actually the case. Indeed the time rates of their clocks depend on the sense of the relative motion. In fact, if the two observers move away from each other the light signals emitted by one of them will be seen by the other with a delay, hence at a slower rate, because each signal has to cover a longer path than the previous one. If the observers instead approach each other then the signal emitted by one will be seen by the other with an anticipation due to the relative approaching motion, and so at a faster rate. This is what actually occurs to the time rates of the tu =
1−
1.3 Clock synchronization and relativity of time
7
clocks carried by the observers u and u , namely tu and t u . In fact, from (1.6) and (1.1) we deduce, from the point of view of the observer u, that 1 + ν/c tu . t u = (1.10) 1 − ν/c Hence, if ν > 0 (u recedes from u) then tu > tu , i.e. u judges the clock of u to be ticking at a slower rate with respect to his own; if ν < 0 (u approaching u) then tu < tu , that is u now judges the clock of u to be ticking at a faster rate with respect to his own. Despite this, the diﬀerence marked by the clocks of the two frames when they are instantaneously coincident must be independent of the sense of the relative motion. This will be shown in what follows. Let us consider two frames S and S approaching each other with velocity ν along the respective coordinate axes x and x . Let u be the observer at rest at the spatial origin of S and u the one at rest at the spatial origin of S . From the point of view of S, the observer u approaches u along a straight line of equation: x = −νt + x0
(1.11)
where x0 is the spatial position of u at the initial time t = 0. The observer u emits light signals along the xaxis at times tu . These signals move towards the observer u and meet him at the events of observation at times tu read by u on the Sclocks that he crosses along his path. The equation of motion of the light signals will be in general x = c(t − tu )
(1.12)
and so the instant of observation by u is given by the intersection of the line (1.12), which describes the motion of the light ray, and the line (1.11) which describes that of the observer u , namely c (tu − tu ) = −νtu + x0
(1.13)
which leads to tu =
tu + x0 /c . 1 + ν/c
(1.14)
Let us stress what was said before: while times tu are read on the clock of u at rest in the spatial origin of S, the instants tu are marked by the Sclocks spatially coincident with the position of the observer u when he detects the light signals. Let us now illustrate how the same process is seen in the frame S . In this case the observer u approaches u along the axis x with relative velocity ν and therefore along a straight line of equation x = νt − x0 .
(1.15)
8
Introduction
The position of u at the initial time t = 0 is given by some value of the coordinate x which we set equal to −x0 , with x0 positive. This value is related to x0 , which appears in (1.11), by an explicit relation that we here ignore.4 The observer u sends light signals at times tu . These reach the observer u set in the spatial origin of S at the instants tu read on his own clock. The motion of these signals is described by a straight line whose equation is given in general by x = c (t − tu ).
(1.16)
The time of emission by the observer u is ﬁxed by the intersection of the line (1.16) which describes the motion of the light signal with the line (1.15) which describes the motion of u, namely tu =
tu − x0 /c . 1 − ν/c
(1.17)
Let us recall again here that while tu is the time read by u on his own clock set stably at the spatial origin of S , the time tu is marked by the S clocks which are instantaneously coincident with the moving observer u. Here we exploit the equivalence between inertial frames regarding the reading of the clocks which lead to (1.3). After some elementary mathematical steps we obtain ν2 ν x0 x 2 tu + 0 tu . (1.18) (tu ) = 1 − 2 (tu )2 − 1 − c c c c This relation, deduced in the case of approaching observers, does not coincide with the analogous relation (1.6) deduced in the case of observers receding from each other. Although we do not know what the relation between x0 and x0 is, we can prove the symmetry between this case and the previously discussed one. Let us consider the extension of the light trajectories stemming from u to u in the frames S and S , until they intersect the world line of static observers located at x0 and x0 respectively. Let us denote these observers as u0 and u0 . In the frame S, the light signals intercept the observer u0 at times, read on the clock of u0 , given by tu0 = tu + x0 /c.
(1.19)
Then Eq. (1.14) can also be written as tu =
tu0 . 1 + ν/c
(1.20)
The time marked by the clock of u0 at the arrival of the light signals runs with the same rate as that of the time marked by the clock of u at the emission of the 4
The quantities x0 and x0 are related by a Lorentz transformation but here we cannot introduce it because the latter presupposes that one knows the relation between the time rates of the spatially coincident clocks of the frames S and S , which instead we want to deduce.
1.3 Clock synchronization and relativity of time
9
same signals, since u and u0 have zero relative velocity and therefore are to be considered as the same observer located at diﬀerent spatial positions. Then we can still denote tu0 as tu and write relation (1.20) as tu =
tu . 1 + ν/c
(1.21)
A similar argument can be repeated in the frame S . The time read on the clock of u0 , at the intersections of the light rays with the history of the observer u0 , is equal to tu0 = tu − x0 /c.
(1.22)
Relation (1.17) can be written as tu =
tu
0
1 − ν/c
.
(1.23)
The time read on the clock of u0 runs at the same rate as that of u since u0 and u are to be considered as the same observer but located at diﬀerent spatial positions. From this it follows that (1.23) can also be written as tu =
tu . 1 − ν/c
(1.24)
We clearly see that the relative motion of approach of u to u is equivalent to a relative motion of recession between the observers u and u0 in S and between u and u0 in S . The result of this comparison is the same as that shown in the relations (1.6) and (1.9), which are then independent of the sense of the relative motion, as expected. Moreover, this conclusion implies for consistency that, setting in (1.18) ν2 tu = 1 − 2 tu , (1.25) c it follows that
ν x0 x tu + 0 tu = 0. − 1− c c c
From this and (1.25) we further deduce that x0 x0 =
(1 − ν/c). 1 − ν 2 /c2
(1.26)
(1.27)
One should notice here that (1.26) must be solved with respect to x0 and not with respect x0 because the corresponding relation (1.25) between times, which implies (1.26), is relative to the situation where the observer u is the one who observes the moving frame S ; hence we have to express all quantities of S in terms of the coordinates of S. Equations (1.6) and (1.9) are the starting point for the arguments which lead to the Lorentz transformations. From the latter, one deduces a posteriori that Eq. (1.27) is just the Lorentz transform of the spatial
10
Introduction
coordinate of the point of S with coordinates
(x0 , tu = x0 /c); the corresponding point of S will have coordinates (x0 , tu = 1 − ν 2 /c2 tu ). The above analysis shows the important fact that the relativity of time is the result of the conspiracy of three basic facts, namely the ﬁnite velocity of light, the equivalence of the inertial observers, and the uniqueness of the clock synchronization procedure.
2 The theory of relativity: a mathematical overview
In the theory of relativity space and time loose their individuality and become indistinguishable in a continuous network termed spacetime. The latter provides the unique environment where all phenomena occur and all observers and observables live undisclosed until they are forced to be distinguished according to their role. Unlike other interactions, gravity is not generated by a ﬁeld of force but is just the manifestation of a varied background geometry. A variation of the background geometry may be induced by a choice of coordinates or by the presence of matter and energy distributions. In the former case the geometry variations give rise to inertial forces which act in a way similar but not fully equivalent to gravity; in the latter case they generate gravity, whose eﬀects however are never completely disentangled from those generated by inertial forces.
2.1 The spacetime A spacetime is described by a fourdimensional diﬀerentiable manifold M endowed with a pseudoRiemannian metric g. Any open set U ∈ M is homeomorphic to 4 meaning that it can be described in terms of local coordinates xα , for example, with α = 0, 1, 2, 3. These coordinates induce a coordinate basis {∂/∂xα ≡ ∂α } for the tangent space T M over U with dual {dxα }. It is often convenient to work with noncoordinate bases {eα } with dual {ω α } satisfying the duality condition ω α (eβ ) = δ α β .
(2.1)
If one expresses these vector ﬁelds in terms of coordinate components they are given by eα = eβ α ∂β ,
ω α = ω α β dxβ ,
(2.2)
12
The theory of relativity: a mathematical overview
where identical indices set diagonal to one another indicate that sums are to be taken only over the range of identical values. In (2.2) the matrices {eβ α } and {ω α β } are inverse to each other, i.e. eα β ω β μ = δ α μ ,
eα β ω μ α = δ μ β .
(2.3)
dxα = eα β ω β .
(2.4)
From the above relations it follows ∂α = ω β α eβ ,
A ﬁeld of bases {eα } is said to form a frame. The structure functions of {eα } are the Lie brackets of the basis vectors [eα , eβ ] = C γ αβ eγ ,
(2.5)
C γ αβ = −2e[α ω γ σ eσ β]
(2.6)
and are deﬁned from (2.2) as
where eγ ( · ) denotes a frame derivative. Here square brackets mean antisymmetrization (to be deﬁned shortly) of the enclosed indices, and indices between vertical bars are ignored by this operation. A general tensor ﬁeld can be expressed in terms of frame components as β S = S α... β... eα ⊗ · · · ⊗ ω ⊗ · · · ,
α S α... β... = S(ω , . . . , eβ , . . .).
(2.7)
It is convenient to adopt the notation ω α1 ...αp = p!ω [α1 ⊗ · · · ⊗ ω αp ] .
(2.8)
Hence a pform K – a totally antisymmetric ptimes covariant tensor ﬁeld, also referred to as a p0 tensor – can be written as K=
1 Kα ...α ω α1 ...αp . p! 1 p
(2.9)
pvector ﬁelds, namely the A similar notation eα1 ...αp can be used to express totally antisymmetric ptimes contravariant p0 tensor ﬁelds. A tensor ﬁeld whose components are antisymmetric in a subset of p covariant indices is termed a tensorvalued pform β γ1 ⊗ · · · ⊗ ω γp S = S α... β...[γ1 ...γp ] eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ ω
=
1 α... S eα ⊗ · · · ⊗ ω β ⊗ · · · ⊗ ω γ1 ...γp . p! β...γ1 ...γp
(2.10)
Right and left contractions Tensor products are deﬁned by a suitable contraction of the tensorial indices. Contraction of one index of each tensor is represented by the symbol (right
2.1 The spacetime
13
contraction) or (left contraction). If S and T are two arbitrary tensor ﬁelds, the left contraction S T denotes a contraction between the rightmost contravariant ... index of S and the leftmost covariant index of T (i.e. S ...α ... T α... ), and the right contraction S T denotes a contraction between the rightmost covariant index ... α... of S and the leftmost contravariant index 1 of T (i.e. S ...α T ... ), assuming in each case that such indices exist. If B is a 1 tensor ﬁeld, namely B = B α β eα ⊗ ω β ,
(2.11)
and it acts on a vector ﬁeld X by right contraction, we have X = [B X]α eα = B α β X β eα . (2.12) Similarly, if A and B are two 11 tensor ﬁelds, their right contraction is given by B
A B = [A B]α β eα ⊗ ω β = [Aα γ B γ β ] eα ⊗ ω β .
(2.13)
The identity transformation is represented by the unit tensor or Kronecker delta tensor δ: δ = δ α β eα ⊗ ω β . The trace of a
1 1
tensor is deﬁned as Tr A = Aα α .
Any
(2.14)
(2.15)
1 1
tensor can be decomposed into a puretrace and a tracefree (TF) part 1 A = A(TF) + [Tr A]δ. 4
(2.16)
Contraction over p indices of two general tensors is symbolically described as (right pcontraction) S
p
T ≡ S α... β1 ...βp T β1 ...βp ...
(2.17)
S
p
T ≡ S α... β1 ...βp Tβ1 ...βp ... .
(2.18)
or (left pcontraction)
1
Change of frame
Given a nondegenerate 1 tensor ﬁeld A, i.e. such that the determinant det(Aα β ) is everywhere nonvanishing, one can prove that the elements of the inverse matrix are the components of a tensor A−1 which is termed the inverse tensor of A: −1 α γ α A−1 A = δ, A (2.19) γ A β = δ β.
14
The theory of relativity: a mathematical overview
Nondegenerate
1
e¯α = A−1
1
tensor ﬁelds induce frame transformations as β eα = A−1 α eβ ,
ω ¯ α = ωα
A = Aα β ω β .
(2.20)
The spacetime metric Let (gαβ ) be a symmetric nondegenerate matrix of signature +2 and denote by (g αβ ) its inverse. Then g = gαβ ω α ⊗ ω β ,
g−1 = g αβ eα ⊗ eβ
(2.21)
locally deﬁne a pseudoRiemannian metric tensor (g) and its inverse (g−1 ) satisfying g−1
g = δ,
g αγ gγβ = δ α β .
(2.22)
Using the notation g = det(gαβ ) one ﬁnds det(gαβ ) = −g and d ln g = g αβ dgαβ = Tr [g−1
dg].
(2.23)
˜ Let us choose an orientation on M , namely an everywhere nonzero 4form O; ˜ a frame {eα } is termed oriented if O(e0 , e1 , e2 , e3 ) > 0. Since the metric is nondegenerate, it determines an isomorphism between the tangent and cotangent spaces at each point of the manifold which in indexnotation corresponds to “raising” and “lowering” indices. For a vector ﬁeld X and a 1form θ one has X = g X, Xα = gαβ X β , θ = g−1 θ, θα = g αβ θβ ,
(2.24)
where, as is customary, we use the sharp () and ﬂat () notation for an arbitrary tensor to mean the tensor obtained by raising or lowering, respectively, all of the indices which are not already of the appropriate type.
Unit volume 4form The LeviCivita permutation symbols α1 ...α4 and α1 ...α4 are totally antisymmetric with 0123 = 1 = 0123 ,
(2.25)
so that they vanish unless α1 . . . α4 is a permutation of 0 . . . 3, in which case their value is the sign of the permutation. The components of the unit volume 4form are related to the LeviCivita symbols by ηα1 α2 α3 α4 = g 1/2 α1 α2 α3 α4 ,
η α1 α2 α3 α4 = −g −1/2 α1 α2 α3 α4 .
(2.26)
2.1 The spacetime
15
Generalized Kronecker deltas The following identities deﬁne the generalized Kronecker deltas and relate them to the LeviCivita alternating symbols and to the unit volume 4form η: 4 = α1 ...α4 β1 ...β4 = −η α1 ...α4 ηβ1 ...β4 , δ α1β...α 1 ...β4 1 α ...α α ...α γ ...γ δ 1 p p+1 4 δ 1β1 ...βp p = (4 − p)! β1 ...βp γp+1 ...γ4 1 η α1 ...αp γp+1 ...γ4 ηβ1 ...βp γp+1 ...γ4 . =− (4 − p)!
(2.27)
A familiar alternative deﬁnition is given by ⎛
δ
α1 ...αp β1 ...βp
···
δ α1 β1 ⎜ = det ⎝ ... δ
αp
···
β1
⎞ δ α1 βp .. ⎟= p!δ α1 · · · δ αp . [β1 βp ] . ⎠ δ
αp
(2.28)
βp
Note that the second of the relations (2.27) holds in a fourdimensional Riemannian manifold with signature +2; in a threedimensional Euclidean manifold one has instead δ
α1 ...αp β1 ...βp
=
1 η α1 ...αp γp+1 ...γ3 ηβ1 ...βp γp+1 ...γ3 . (3 − p)!
(2.29)
Symmetrization and antisymmetrization Given an object with only covariant or contravariant indices, or a subset of only covariant or contravariant indices from those of a mixed object, one can always project out the purely symmetric and purely antisymmetric parts. For example for a p0 tensor ﬁeld one has 1 β1 ...βp δ Sβ ...β , p! α1 ...αp 1 p 1 = Sσ(α1 ...αp ) , p! σ
[ALT S]α1 ...αp = S[α1 ...αp ] = [SYM S]α1 ...αp = S(α1 ...αp )
(2.30)
where the sum is over all the permutations σ of {α1 . . . αp }. If T is a 02 tensor, its symmetric and antisymmetric parts are given by 1 (Tμν + Tνμ ) 2 1 = (Tμν − Tνμ ) . 2
T(μν) =
(2.31)
T[μν]
(2.32)
16
The theory of relativity: a mathematical overview Exterior product
The exterior or wedge product of p 1forms ω αi (i = 1, . . . p) is deﬁned as ω α1 ∧ · · · ∧ ω αp ≡ p! ω [α1 ⊗ · · · ⊗ ω αp ] =δ
α1 ...αp β1 β1 ...βp ω α1 ...αp
=ω
⊗ · · · ⊗ ω βp
,
(2.33)
where notation (2.8) has been used. From Eq. (2.33), the wedge product of a pform S with a qform T has the following expression [S ∧ T ]α1 ...αp+q =
(p + q)! S[α1 ...αp Tαp+1 ...αp+q ] , p!q!
(2.34)
so that the relation S ∧ T = (−1)pq T ∧ S
(2.35)
holds identically.
Hodge duality operation The Hodge duality operation associates with a pform S the (4 − p)form ∗S with components 1 (2.36) [∗S]αp+1 ...α4 = Sα1 ...αp η α1 ...αp αp+1 ...α4 , p! i.e. with η at the righthand side of the pform S and with a proper contraction of the ﬁrst indices of η (right duality convention). From the above deﬁnition one ﬁnds that a pform S satisﬁes the identity ∗∗
S ≡ (−1)p−1 S.
(2.37)
The duality operation applied to the wedge product of a pform S and a qform T (with q ≥ p) implies the following identity: ∗
(S ∧ ∗T ) = (−1)1+(4−q)(q−p)
1 S p!
p
T.
(2.38)
With p = 0 and q = 1 this reduces to (2.37), while when p = q one has ∗
(S ∧ ∗T ) = (−1)
1 S p!
p
T = ∗η
1 S p!
p
T,
(2.39)
since ∗η = (−1). Removing the duality operation from each side leads to S ∧ ∗T =
1 (S p!
p
T ) η.
(2.40)
2.2 Derivatives on a manifold
17
2.2 Derivatives on a manifold Diﬀerentiation on a manifold is essential to derive physical laws and deﬁne physical measurements. It stems from appropriate criteria for comparison between the algebraic structures at any two points of the manifold. Clearly diﬀerentiation allows one to deﬁne new quantities, as we shall see next.
Covariant derivative and connection The covariant derivative ∇eγ S of an arbitrary pq tensor S along the eγ frame p tensor with components direction is a q+1 α1 ...αp β1 ...βq
[∇eγ S]
α ...α
≡ ∇γ S 1β1 ...βp q (2.41) α ...α ... − · · ·. = eγ S 1β1 ...βp q + Γα1 δγ S δ...β1 ... + · · · − Γδ β1 γ S α1δ...
Here the coeﬃcients Γα γδ are the components of a linear connection on M deﬁned as ∇eα eβ = Γγ βα eγ
↔
∇eα ω β = −Γβ γα ω γ .
(2.42)
The covariant derivative satisﬁes the following rules. For any pair of vector ﬁelds X and Y and of real functions f and h, we have ∇f X+hY = f ∇X + h∇Y .
(2.43)
∇X S = X α ∇eα S ≡ X α ∇α S.
(2.44)
From this it follows that
A connection is said to be compatible with the metric if the latter is constant under covariant diﬀerentiation, that is 0 = ∇γ gαβ = eγ (gαβ ) − gδβ Γδ αγ − gαδ Γδ βγ = eγ (gαβ ) − Γβαγ − Γαβγ ,
(2.45)
where we set Γαβγ ≡ gασ Γσ βγ .
(2.46)
Given two vector ﬁelds X and Y , consider the covariant derivative of Y in the direction of X as (2.47) ∇X Y = X α eα (Y δ ) + Γδ βα X α Y β eδ . From (2.47) and (2.5) we deduce the following relation ∇X Y − ∇Y X = [X, Y ] − [2Γδ [βα] − C δ αβ ]X α Y β eδ .
(2.48)
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ]
(2.49)
The quantity
18
The theory of relativity: a mathematical overview
is a 13 tensor termed torsion. In the theory of relativity the torsion is assumed to be identically zero. Hence, X and Y being arbitrary, the following relation holds 1 Γδ [βα] = C δ αβ . (2.50) 2 From this, Eq. (2.45) can be inverted, yielding Γα βγ =
1 αδ g [eβ (gδγ ) + eγ (gβδ ) − eδ (gγβ ) + Cδγβ + Cβδγ − Cγβδ ] , (2.51) 2
where we set Cαβγ = gασ C σ βγ . In the event that the structure functions Cαβγ are all zero – in which case the bases eα are termed holonomous – the components of the linear connection are known as Christoﬀel symbols.
Curvature The curvature or Riemann tensor is deﬁned as R(eγ , eδ )eβ = [∇eγ , ∇eδ ] − ∇[eγ ,eδ ] eβ = Rα βγδ eα .
(2.52)
Hence the components in a given frame are Rα βγδ = eγ (Γα βδ ) − eδ (Γα βγ ) − C γδ Γα β + Γα γ Γ βδ − Γα δ Γ βγ .
(2.53)
This 1 tensor is explicitly antisymmetric in its last pair of indices and so deﬁnes a 1 tensorvalued 2form Ω: 1 Ω = eα ⊗ ω β ⊗ Ωα β = eα ⊗ ω β ⊗ Rα βγδ ω γδ . 2
(2.54)
Covariant derivative and curvature A remarkable property of the covariant derivative is that it does not commute with itself. In fact, given a vector X and using coordinate components (C γ αβ = 0) we have 1 ∇[α ∇β] X γ = Rγ σαβ X σ , (2.55) 2 while given a 1form ω we have ∇[α ∇β] ωγ =
1 ωσ Rσ γβα . 2
(2.56)
Relations (2.55) and (2.56) can be generalized to arbitrary tensors; for a tensor Tρ σ we have ∇[α ∇β] Tρ σ =
1 σ μ 1 Tμ R ρβα + Rσ μαβ Tρ μ . 2 2
1 1

(2.57)
2.2 Derivatives on a manifold
19
Applying the above relation to the metric tensor we have ∇[α ∇β] gρσ =
1 1 gμσ Rμ ρβα + gρμ Rμ σβα ≡ 0. 2 2
(2.58)
Hence Rσρβα + Rρσβα = 0,
(2.59)
implying that the totally covariant (or contravariant) Riemann tensor is antisymmetric with respect to the ﬁrst pair of indices as well.
Ricci and Bianchi identities The Ricci identities are given by 3Rα [βγδ] = Rα βγδ + Rα γδβ + Rα δβγ = 0,
(2.60)
whereas the Bianchi identities are given by 3∇[ Rγδ]β α = ∇ Rγδβ α + ∇δ R γβ α + ∇γ Rδ β α = 0.
(2.61)
Finally, if we make the algebraic sum of the Ricci identities (2.60) for each permutation of the indices of a totally covariant Riemann tensor, Rα[βγδ] + Rδ[αβγ] − Rγ[δαβ] − Rβ[γδα] = 0 ,
(2.62)
Rαβγδ = Rγδαβ ,
(2.63)
it follows that
i.e. the Riemann tensor is symmetric under exchange of the two pairs of indices.
Ricci tensor and curvature scalar From the symmetries of the curvature tensor its ﬁrst contraction generates the Ricci curvature tensor Rαβ = Rγ αγβ ,
(2.64)
which is symmetric due to the Ricci identities. Its trace deﬁnes the curvature scalar R ≡ Rα α = Rγα γα .
(2.65)
20
The theory of relativity: a mathematical overview Weyl and Einstein tensors
The Weyl tensor is a suitable combination of the curvature tensor, the Ricci tensor, and the curvature scalar, as follows: C αβ γδ = Rαβ γδ − 2R[α [γ δ β] δ] +
R αβ δ 6 γδ
= Rαβ γδ − 2δ [α [γ S β] δ] ,
(2.66)
1 S β α = Rβ α − Rδ β α . 6
(2.67)
where
The Weyl tensor is tracefree, i.e. C α βαδ = 0, and is invariant under conformal transformations of the metric. The onefold contraction of the Bianchi identities (using the tracefree property of the Weyl tensor (2.66)) reduces to 0 = ∇[ Rγδ] αβ δ γ α = ∇α Rδ αβ + 2∇[ Rδ] β R = ∇α Cδ αβ + ∇[ Rβ δ] − δ β δ] . 6
(2.68)
The twofold contraction of the Bianchi identities leads to 0 = ∇[ Rαβ γδ] δ γ α δ δ β .
(2.69)
Deﬁning the Einstein tensor as 1 Gδ ≡ Rδ − δ δ R, 2
(2.70)
relation (2.69) implies identically that ∇δ Gδ = 0.
(2.71)
Cotton tensor The Cotton tensor (Cotton, 1899; Eisenhart, 1997) is deﬁned as R β β β R δ ≡ 2∇[ R δ] − δ δ] = 2∇[ S β δ] . 6
(2.72)
Hence the divergence of the Weyl tensor can be written in a more compact form as 1 ∇α Cδ αβ = − Rβ δ . (2.73) 2
2.2 Derivatives on a manifold
21
The property of the Einstein tensor of being divergencefree leads to the tracefree property of the Cotton tensor: Rβ αβ = ∇β Gβ α = 0.
(2.74)
The totally antisymmetric part of the fully covariant (or contravariant) Cotton tensor is zero due to the symmetry of the Ricci and metric tensors, that is 3R[αβγ] = Rαβγ + Rβγα + Rγαβ = 0.
(2.75)
Finally from (2.71) it follows that the divergence of the Cotton tensor is also zero: ∇β Rβ δ = 0.
(2.76)
Absolute derivative along a world line Given a curve γ parameterized by λ and denoting by γ˙ the corresponding tangent vector ﬁeld, the absolute derivative of any tensor ﬁeld S(λ) deﬁned along the curve is DS = ∇γ˙ S. dλ
(2.77)
Given a local coordinate system {xα }, the curve is described by the functions γ (λ) = xα (γ(λ)) ≡ xα (λ), so that α
α [γ(λ)] ˙ =
dxα (λ). dλ
For a vector ﬁeld X deﬁned on γ, its absolute derivative takes the form dX α (λ) DX = + Γα γβ (λ)γ˙ β (λ)X γ (λ) eα . dλ dλ
(2.78)
Parallel transport and geodesics A tensor ﬁeld S deﬁned on a curve γ with parameter λ is said to be parallel transported along the curve if its absolute derivative along γ is zero: DS = 0. dλ
(2.79)
A curve γ whose tangent vector γ˙ satisﬁes the equation Dγ˙ (λ) = f (λ)γ(λ), ˙ dλ
(2.80)
22
The theory of relativity: a mathematical overview
where f (λ) is a function deﬁned on γ, is termed geodesic. It is always possible to reparameterize the curve with σ(λ) such that Eq. (2.80) becomes Dγ˙ (σ) = 0, dσ
(2.81)
where dxα α . γ˙ = dσ In this case the parameter σ is said to be aﬃne and it is deﬁned up to linear transformations. From Eq. (2.81) we see that a characteristic feature of an aﬃne geodesic is that the tangent vector is parallel transported along it.
FermiWalker derivative and transport Let us introduce a new parameter s along a nonnull curve γ such that the tangent vector of the curve uα = dxα /ds is unitary, that is, u · u = ±1. If the curve is timelike then u · u = −1 and the parameter s is termed proper time, as already noted. The curvature vector (or acceleration) of the curve is deﬁned as a(u) =
Du . ds
(2.82)
Consider a vector ﬁeld X(s) on γ. We deﬁne the FermiWalker derivative of X along γ as the vector ﬁeld D(fw,u) X/ds having components
D(fw,u) X ds
α
α DX ≡ ± [a(u)α (u · X) − uα (a(u) · X)] ds α DX ± [a(u) ∧ u]α γ X γ , = ds
(2.83)
where the signs ± should be chosen according to the causal character of the curve: + for a timelike curve and − for a spacelike 1 one. The generalization to a tensor ﬁeld T ∈ 1 , for example, is the following:
D(fw,u) T ds
α
≡
β
DT ds
α ± ([a(u) ∧ u]α γ T γ β β
− [a(u) ∧ u]γ β T α γ ) .
(2.84)
The tensor ﬁeld T is said to be FermiWalker transported along the curve if its FermiWalker derivative is identically zero.
2.2 Derivatives on a manifold
23
Lie derivative The Lie derivative along a congruence CX of curves with tangent vector ﬁeld X is denoted £X and deﬁned as follows: (i) For a scalar ﬁeld f , £X f = X(f ).
(2.85)
£X Y = [X, Y ].
(2.86)
[£X ω]β = X γ eγ (ωβ ) + ωα eβ (X α ).
(2.87)
(ii) For a vector ﬁeld Y , (iii) For a 1form ω, (iv) For a general tensor S α... β... , [£X S]α... β... = X γ eγ (S α... β... ) − S μ... β... eμ (X α ) + . . . + S α... μ... eβ (X μ ) + . . . .
(2.88)
From (2.86), the Lie operator applied to the vectors of a frame leads to (2.5), that is £eα eβ = [eα , eβ ] = C γ αβ eγ .
(2.89)
A general tensor ﬁeld S is said to be Lie transported along a congruence CX if its Lie derivative with respect to X is identically zero.
Exterior derivative The exterior derivative is a diﬀerential operator d which associates a pform to a (p + 1)form according to the following properties: (i) d is additive, i.e. d(S + T ) = dS + dT for arbitrary forms S, T . (ii) If f is a 0form (i.e. a scalar function), df is the ordinary diﬀerential of the function f . If {xα } denotes a local coordinate system then df = ∂α f dxα . In a noncoordinate frame {eα } with dual {ω α } one has instead df = eα (f )ω α . (iii) If S is a pform and T a qform then the following relation holds: d(S ∧ T ) = dS ∧ T + (−1)p S ∧ dT.
(2.90)
(iv) d2 S = d(dS) = 0 for all forms S. As a ﬁrst application let us consider the exterior derivatives of the dual frame 1forms ω α . We have, recalling (2.6), dω α = d(ω α β dxβ ) = ∂μ (ω α β )dxμ ∧ dxβ = ∂μ (ω α β )eμ σ eβ ρ ω σ ∧ ω ρ = e[σ (ω α β )eβ ρ] ω σ ∧ ω ρ 1 = − C α σρ ω σ ∧ ω ρ . 2
(2.91)
24
The theory of relativity: a mathematical overview
One can use this result to evaluate the exterior derivative of ω αβ = ω α ∧ ω β ; from (2.90) and (2.33) we have dω αβ = dω α ∧ ω β − ω α ∧ dω β 1 = − C α μν ω μ ∧ ω ν ∧ ω β + (−1)C β μν ω α ∧ ω μ ∧ ω ν 2 1 = − C α μν ω μνβ + (−1)C β μν ω αμν . 2
(2.92)
From this it follows that the exterior derivative of a 2form S, S=
1 1 Sαβ ω α ∧ ω β ≡ Sαβ ω αβ , 2 2
(2.93)
is given by 1 [dS]βμν ω βμν 3! 1 Sαβ ω α ∧ ω β =d 2 1 1 1 = eγ (Sαβ ) ω γαβ + Sαβ − C α μν ω μν ∧ ω β 2 2 2 1 1 − Sαβ ω α ∧ − C β μν ω μν 2 2 1 1 = eγ (Sαβ )ω γαβ − Sαβ C α μν ω μνβ 2 4 1 β αμν + Sαβ C μν ω 4 1 1 = e[β (Sμν] )ω βμν − Sαβ C α μν ω βμν 2 2 1 1 e[β (Sμν] ) − Sαβ C α μν ω βμν ; = 2 2
dS =
hence [dS]βμν
1 α = 3 e[β (Sμν] ) − Sαβ C μν . 2
(2.94)
(2.95)
From the above analysis a general formula for the exterior derivative of the basis pforms ω α1 ...αp follows as 1 1 dω α1 ...αp = − C α1 ρσ ω ρσα2 ...αp + · · · − (−1)p−1 C αp ρσ ω α1 ...αp−1 ρσ 2 2 p 1 =− (−1)i−1 C αi βγ ω α1 ...αi−1 βγαi+1 ...αp . (2.96) 2 i=1 As a consequence, the exterior derivative of a pform S, S=
1 Sα ...α ω α1 ...αp , p! 1 p
(2.97)
2.2 Derivatives on a manifold
25
is a (p + 1)form deﬁned as dS =
1 [dS]α1 ...αp+1 ω α1 ...αp+1 , (p + 1)!
(2.98)
where the components [dS]α1 ...αp+1 are given by [dS]α1 ...αp+1 = (p + 1)(e[α1 (Sα2 ...αp+1 ] ) 1 − p C β [α1 α2 Sβα3 ...αi−1 αi+1 ...αp+1 ] ). 2
(2.99)
In a coordinate frame the second term of (2.99) vanishes. Furthermore, this result may be reexpressed in terms of a covariant derivative as [dS]α1 ...αp+1 = (p + 1)∇[α1 Sα2 ...αp+1 ] .
(2.100)
In fact, applying condition (2.90) to (2.98) we obtain dS =
1 1 dSα1 ...αp ∧ ω α1 ...αp + Sα1 ...αp dω α1 ...αp . p! p!
(2.101)
By combining this relation with (2.96) we ﬁnally obtain (2.100).
The divergence operator The divergence operator δ of a pform is deﬁned as δT = ∗ [d∗ T ].
(2.102)
[∗ T ]αβγ = η δ αβγ Tδ
(2.103)
[d∗ T ]μαβγ = 4∇[μ (η δ αβγ] Tδ ),
(2.104)
If T = Tα ω α is a 1form we have
and
so that 1 μαβγ ∗ η [d T ]μαβγ 4! 1 = η μαβγ η δ αβγ ∇μ Tδ 3! 1 μαβγ = − δδαβγ ∇μ T δ 3! = −δδμ ∇μ T δ = −∇δ T δ .
δT = ∗ [d∗ T ] =
If T =
(2.105)
1 Tαβ ω α ∧ ω β is a 2form we have 2 [∗ T ]μν =
1 ημν αβ Tαβ 2
(2.106)
26
The theory of relativity: a mathematical overview
and [d∗ T ]λμν = so that
3 ∇[λ (ημν] αβ Tαβ ), 2
3 1 λμνσ αβ ημν Tαβ [δT ] = [ [d T ]] = η ∇λ 3 2 1 σλ = 2!δαβ ∇λ T αβ = −∇β T βσ . 4 σ
∗
∗
(2.107)
σ
(2.108)
Finally, if T is a pform we have [δT ]α2 ...αp = −∇α Tαα2 ...αp .
(2.109)
We introduce also the notation div T = −δT
(2.110)
and extend this operation to any tensor ﬁeld T = T α1 ...αn eα1 ⊗ · · · ⊗ eαn as div T = [∇α T αα2 ...αn ] eα2 ⊗ · · · ⊗ eαn .
(2.111)
De Rham Laplacian The de Rham Laplacian operator for pforms is deﬁned by Δ(dR) = δd + dδ.
(2.112)
It diﬀers from the Laplacian Δ, namely [ΔS]α1 ...αp = −∇α ∇α Sα1 ...αp ,
(2.113)
by curvature terms [Δ(dR) S]α1 ...αp = [ΔS]α1 ...αp +
p
Rβ αi Sα1 ...αi−1 β αi+1 ...αp
i=1
−
p
Rβ αi γ αj Sα1 ...αi−1 β αi+1 ...αj−1 γ αj+1 ...αp .
i=j=1
As an example let us consider the electromagnetic 1form A with F = dA, and dF = 0, δF = −4πJ. Then one ﬁnds Δ(dR) A − d(δA) = −4πJ,
Δ(dR) F = −4π(dJ),
(2.114)
showing that [Δ(dR) , d] = 0. Let us write explicitly the de Rham Laplacian of the Faraday 2form F . We have [Δ(dR) F ]αβ = [ΔF ]αβ − Rμ α Fμβ + Rμ β Fμα − Rμ α γ β Fμγ + Rμ β γ α Fμγ .
(2.115)
2.3 Killing symmetries
27
One can rederive the de Rham Laplacian of F in the form (2.114) starting from the homogeous Maxwell’s equations dF = 0 written in coordinate components: ∇γ Fαβ + ∇α Fβγ + ∇β Fγα = 0.
(2.116)
Covariant diﬀerentiation ∇γ of both sides gives − ΔFαβ + 2∇γ ∇[α Fβ]γ = 0.
(2.117)
Recalling the noncommutativity of the covariant derivatives acting in this case on Fμν , namely [∇γ , ∇α ]Fβ γ = Rβ μγ α Fμγ + Rγ μγ α Fβμ ,
(2.118)
replace the second term of Eq. (2.117) and use the nonhomogeous Maxwell’s equations ∇γ Fβγ = 4πJβ . Finally we obtain the relation Δ(dR) F = −4π(dJ). For a 3form S we have instead [Δ(dR) S]α1 α2 α3 = [ΔS]α1 α2 αp + Rβ α1 Sβα2 α3 + Rβ α2 Sα1 βα3 + Rβ α3 Sα1 α2 β − 2Rβ [α2 γ α3 ] Sβγα1 − 2Rβ [α1 γ α2 ] Sβγα3 + 2Rβ [α1 γ α3 ] Sβγα2 , which can also be written as [Δ(dR) S]α1 α2 α3 = [ΔS]α1 α2 αp + 3Rβ [α1 Sβα2 α3 ] − 6Rβ [α1 γ α2 Sβγα3 ] .
2.3 Killing symmetries The Lie derivative of a general tensor ﬁeld along a congruence of curves describes the “intrinsic” behavior of this ﬁeld along the congruence. If the ﬁeld is the metric tensor then the vanishing of its Lie derivative implies that the vector ﬁeld ξ, tangent to the congruence, is an isometry for the metric, and this ﬁeld is called a Killing vector ﬁeld. From the deﬁnitions of the covariant and Lie derivatives, a Killing vector ﬁeld satisﬁes the equations ∇(α ξβ) = 0.
(2.119)
In a remarkable paper, Papapetrou (1966) pointed out that a Killing vector ﬁeld ξ can be considered as the vector potential generating an electromagnetic ﬁeld Fαβ = ∇α ξβ ,
(2.120)
with current J α = Rα β ξ β (which vanishes in vacuum spacetimes) satisfying the Lorentz gauge ∇α ξ α = 0. Fayos and Sopuerta (1999) introduced the term Papapetrou ﬁeld for such an electromagnetic ﬁeld. Their result is that Papapetrou ﬁelds constitute a link between the Killing symmetries and the algebraic structure of a spacetime established by the alignment of the principal null directions (see the following subsection) of the Papapetrou ﬁeld with those of the Riemann tensor.
28
The theory of relativity: a mathematical overview A symmetric 2tensor Kμν is termed a Killing tensor if it satisﬁes the relation ∇λ Kμν + ∇ν Kλμ + ∇μ Kνλ = 3∇(λ Kμν) = 0.
(2.121)
2.4 Petrov classification The algebraic properties of the spacetime curvature and in particular of the Weyl tensor play a central role in Einstein’s theory. In the most general spacetime there exist four distinct null eigenvectors l of the Weyl tensor, known as principal null directions (PNDs), which satisfy the PenroseDebever equation, l[α Cβ]μν[γ lδ] lμ lν = 0.
(2.122)
When some of them coincide we have algebraically special cases. The multiplicity of the principal null directions leads to the canonical Petrov classiﬁcation: • • • • • •
Type Type Type Type Type Type
I (four distinct PNDs) II (one pair of PNDs coincides) D (two pairs of PNDs coincide) III (three PNDs coincide) N (all four PNDs coincide) 0 (no PNDs).
2.5 Einstein’s equations A physical measurement crucially depends on the spacetime which supports it. A spacetime is a solution of Einstein’s equations, which in the presence of a cosmological constant and for any source term are given by Gαβ + Λgαβ = 8πTαβ .
(2.123)
Here Λ is the cosmological constant; Tαβ is the energymomentum tensor of the source of the gravitational ﬁeld, and satisﬁes the local causality conditions and the energy conditions (Hawking and Ellis, 1973). The most relevant Tαβ are: (i) Tαβ = 0, empty spacetime. 1 [∇α φ∇β φ − 12 gαβ [∇μ φ∇μ φ + m2 φ2 ]], massive (m) scalar ﬁeld. (ii) Tαβ = 4π (iii) Tαβ = (μ + p)uα uβ + pgαβ , perfect ﬂuid, where μ is the energy density of the ﬂuid, p is its hydrostatic (isotropic) pressure, and uα is the 4velocity of the ﬂuid element. 1 [Fαμ Fβ μ − 14 gαβ Fμν F μν ], electromagnetic ﬁeld. In this case Ein(iv) Tαβ = 4π stein’s equations are coupled to Maxwell’s equations through the Faraday 2form ﬁeld F , which can be written as dF = 0,
−δF = 4πJ.
(2.124)
2.6 Exact solutions
29
The ﬁrst of these equations implies the existence of a 4potential A such that F = dA, while the second one leads to the conservation of the electromagnetic current δJ = 0 (or ∇α J α = 0). Equivalently, Eqs. (2.124) can be written in terms of covariant derivatives as ∇β ∗F αβ = 0,
∇β F αβ = 4πJ α .
(2.125)
In what follows we shall consider exact solutions of Einstein’s equations which are of physical interest.
2.6 Exact solutions We shall list the main exact solutions of Einstein’s equations which provide natural support for most of the physical measurements considered here.
Constant spacetime curvature solutions Let us consider a maximally symmetric spacetime which satisﬁes the relation Rαβ γδ =
1 Rδ αβ , 12 γδ
→
Rαβ =
1 Rgαβ , 4
(2.126)
where the spacetime Ricci scalar R is constant. Relation (2.126) is equivalent to the vanishing of the tracefree components of the Ricci tensor, that is, 1 [RTF ]αβ ≡ Rαβ − Rgαβ = 0. 4
(2.127)
In this case it follows that: (i) if Tαβ = 0 and Λ = 0 (empty spacetime), Einstein’s equations (2.123) are satisﬁed with Λ=
1 R; 4
(2.128)
(ii) if Tαβ = 0 describes a perfect ﬂuid and either Λ = 0 or Λ = 0, Einstein’s equations are satisﬁed with μ=
1 R = −p, 32π
→
Tαβ = pgαβ ,
a condition which physically describes the vacuum state.
(2.129)
30
The theory of relativity: a mathematical overview
Among this family of constant spacetime curvature solutions we distinguish three relevant cases (Stephani et al., 2003): (i) R > 0, de Sitter (dS): ds2 = −dt2 + α2 cosh2
t [dχ2 + sin2 χ(dθ2 + sin2 θdφ2 )], α
where t ∈ (−∞, +∞), χ ∈ [0, π], θ ∈ [0, π], φ ∈ [0, 2π]. (ii) R = 0, Minkowski (M). (iii) R < 0, Antide Sitter (AdS): t [dχ2 + sinh2 χ(dθ2 + sin2 θdφ2 )], ds2 = −dt2 + α2 cos2 α
(2.130)
(2.131)
where t ∈ (−∞, +∞), χ ∈ [0, π], θ ∈ [0, π], φ ∈ [0, 2π].
Constant spatial curvature solutions Spacetime solutions of Einstein’s equations having spatial sections t = constant, with constant curvature are the FriedmannRobertsonWalker solutions, with metric ds2 = −dt2 + a(t)2 [dχ2 + f (χ)2 (dθ2 + sin2 θdφ2 )].
(2.132)
Of course dS, AdS, and also M are special cases.
G¨ odel solution The G¨odel solution (G¨ odel, 1949) describes a rotating universe with metric 1 ds2 = −dt2 + dx2 − U 2 dy 2 − 2U dtdy + dz 2 , 2
(2.133)
√
where U = e 2ωx and ω is a constant. Metric (2.133) describes an empty nonexpanding but rotating universe and ω has the meaning of the angular velocity of a test particle with respect to a nearby ﬁducial particle. This is a type D solution with a pressureless perfect ﬂuid as a source. The 4velocity u of the ﬂuid is a Killing vector, which is not hypersurfaceforming.
Kasner solution This is an empty spacetime solution (Kasner, 1925) whose metric is given by ds2 = −dt2 + t2p1 dx2 + t2p2 dy 2 + t2p3 dz 2 ,
t > 0,
(2.134)
2.6 Exact solutions
31
where p1 , p2 , p3 are constant parameters satisfying the conditions p21 + p22 + p23 = 1 and p1 + p2 + p3 = 1. It describes a spacetime while approaching a curvature singularity at t = 0, the latter being either of cosmological nature or emerging from gravitational collapse. It is of Petrov type I, except for the case p2 = p3 = 2/3, p1 = −1/3, which corresponds to type D, and the case p1 = p2 = p3 = 0, which is Minkowski.
Schwarzschild solution This solution was obtained in 1915 by Schwarzschild (1916a; 1916b) and independently by Droste, a student of Lorentz, in 1916. In asymptotically spherical coordinates the spacetime metric is given by −1 2M 2M 2 dt + 1 − dr2 + r2 (dθ2 + sin2 θdφ2 ), ds = − 1 − r r 2
(2.135)
with t ∈ (−∞, +∞), r ∈ (2M, ∞), θ ∈ [0, π], φ ∈ [0, 2π], and M being the total mass of the source. It describes the spacetime outside a spherically symmetric, electrically neutral ﬂuid source. r = 0 is a spacelike curvature singularity which is hidden by an event horizon at r = 2M.
ReissnerNordstr¨ om solution This is a generalization of the Schwarzschild solution to an electrically charged sphere (Reissner, 1916; Nordstr¨ om, 1918). In asymptotically spherical coordinates the spacetime metric is given by
2M Q2 + 2 ds = − 1 − r r 2
−1 2M Q2 + 2 dt + 1 − dr2 r r 2
+ r2 (dθ2 + sin2 θdφ2 ),
(2.136)
where Q and M are the electric charge and the total mass of the source, respectively. The electromagnetic ﬁeld F and the 4vector potential A are given by F =−
Q dt ∧ dr, r2
A=−
Q dt. r
(2.137)
Here r = 0 is a timelike curvature singularity which may be hidden by an
event horizon at r = r+ = M + M2 − Q2 if M > Q.1 When M < Q we have no horizons and the singularity at r = 0 is termed naked . 1
The solution admits also an inner horizon at r = r− = M −
M2 − Q2 .
32
The theory of relativity: a mathematical overview Kerr solution
Written in BoyerLindquist coordinates, Kerr spacetime (Kerr, 1963; Kerr and Shild, 1967) is given by
ds2 = −dt2 + +
2Mr (a sin2 θdφ − dt)2 + (r2 + a2 ) sin2 θdφ2 Σ
Σ 2 dr + Σdθ2 , Δ
(2.138)
where Σ = r2 + a2 cos2 θ and Δ = r2 + a2 − 2Mr. Here M and a are respectively the total mass and the speciﬁc angular momentum of the source. This reduces to the Schwarzschild solution when a = 0. It admits an event horizon2 √ at r = r+ = M + M2 − a2 , if a ≤ M and a curvature ringsingularity in the above coordinates is found at r = 0 and θ = π/2. This singularity is timelike and allows the curvature invariant Rαβγδ Rαβγδ to be directional depending on how it is approached. The Kerr solution is most popular in the astrophysical community because, under suitable conditions, it describes the spacetime generated by a rotating black hole (Bardeen, 1970; Ruﬃni, 1973; Rees, Ruﬃni and Wheeler, 1974; Ruﬃni, 1978).
KerrNewman solution This solution is a generalization of Kerr’s to electrically charged sources (Newman et al., 1965). In Boyer and Lindquist coordinates the spacetime metric is given by 2Mr − Q2 2a sin θ(2Mr − Q2 ) dt2 − dtdφ ds2 = − 1 − Σ Σ a2 sin2 θ(2Mr − Q2 ) 2 2 2 + sin θ r + a + dφ2 Σ Σ + dr2 + Σdθ2 , Δ
(2.139)
where Σ = r2 + a2 cos2 θ and Δ = r2 + a2 − 2Mr + Q2 . Here M, a, and Q are respectively the total mass, the speciﬁc angular momentum, and the charge of the source. The horizon and the curvature singularity have the same properties as in the Kerr metric. 2
As in the ReissnerNordstr¨ om solution, the Kerr solution also admits an inner event horizon √ at r = r− = M − M2 − a2 .
2.6 Exact solutions
33
Single gravitational planewave solution A plane gravitational wave (Bondi, Pirani, and Robinson, 1959) is a nontrivial solution of the vacuum Einstein equations with a ﬁveparameter group of motion. The spacetime metric is given by ds2 = −dt2 + dz 2 + L2 (e2β dx2 + e−2β dy 2 ),
(2.140) √ with L, β depending only on the parameter u = (−t + z)/ 2 and such that d2 L/du2 + β 2 L = 0. This solution is of Petrov type N. It has the same symmetry properties as an electromagnetic plane wave in Minkowski spacetime; hence we deduce the associated Killing vectors considering the electromagnetic case. In this case the Faraday 2form is given by Fμν = A(u) ([k ∧ r]μν cos θ(u) + [k ∧ s]μν sin θ(u)) ,
(2.141)
where kμ is the null wave vector, rμ and sμ are unitary spacelike vectors which are mutually orthogonal and also orthogonal to k: k · r = k · s = s · r = 0,
s · s = 1 = r · r.
(2.142)
The wave amplitude A(u) and its polarization θ(u) are both functions of u = kμ xμ . The following properties hold: ∂ν F μν = 0 and ∂[ρ Fμν] = 0. Without loss of generality one can choose the coordinate system so that 1 k = √ (∂t + ∂z ), 2
r = ∂x ,
and u = kμ xμ =
s = ∂y ,
−t + z √ , 2
(2.143)
(2.144)
as already deﬁned. It is easy to verify that F μν is invariant under the action of a ﬁveparameter subgroup of the Lorentz group, with generators (i) (ii) (iii) (iv) (v)
ξ1 ξ2 ξ3 ξ4 ξ5
= k, generating translation on the plane −t + z = constant = r, generating translation along x = s, generating translation along y = xρ [sρ k μ + kρ sμ ], generating null rotations about s = xρ [rρ k μ + kρ rμ ], generating null rotations about r.
The only nonvanishing structure functions are C 1 24 = C 1 35 = 1.
(2.145)
3 Spacetime splitting
The concept of spacetime brings into a uniﬁed scenario quantities which, in the prerelativistic era, carried distinct notions like time and space, energy and momentum, mechanical power and force, electric and magnetic ﬁelds, and so on. In everyday experience, however, our intuition is still compatible with the perception of a threedimensional space and a onedimensional time; hence a physical measurement requires a local recovery of the prerelativistic type of separation between space and time, yet consistent with the principle of relativity. To this end we need a speciﬁc algorithm which allows us to perform the required splitting, identifying a “space” and a “time” relative to any given observer. This is accomplished locally by means of a congruence of timelike world lines with a futurepointing unit tangent vector ﬁeld u, which may be interpreted as the 4velocity of a family of observers. These world lines are naturally parameterized by the proper time τu deﬁned on each of them from some initial value. The splitting of the tangent space at each point of the congruence into a local time direction spanned by vectors parallel to u, and a local rest space spanned by vectors orthogonal to u (hereafter LRSu ), allows one to decompose all spacetime tensors and tensor equations into spatial and temporal components. (ChoquetBruhat, DillardBleick and DeWittMorette 1977).
3.1 Orthogonal decompositions Let g be the fourdimensional spacetime metric with signature +2 and components gαβ (α, β = 0, 1, 2, 3), ∇ its associated covariant derivative operator, and η the unit volume 4form which assures spacetime orientation (see (2.26)). Assume that the spacetime is timeoriented and let u be a futurepointing unit timelike (uα uα = −1) vector ﬁeld which identiﬁes an observer. The local splitting of the tangent space into orthogonal subspaces uniquely related to the given observer u is accomplished by a temporal projection operator T (u) which generates vectors parallel to u and a spatial projection operator P (u) which generates LRSu . These
3.1 Orthogonal decompositions
35
operators, in mixed form, are deﬁned as follows: T (u) = −u ⊗ u P (u) = I + u ⊗ u ,
(3.1)
where I ≡ δ α β is the identity on the tangent spaces of the manifold M . The deﬁnitions (3.1) imply that P (u) u = 0,
T (u) u = u ,
(3.2)
and P (u) P (u) = P (u),
T (u) T (u) = T (u),
T (u) P (u) = 0 .
(3.3)
In terms of components the above relations can be written T (u)α β = −uα uβ , P (u)α β = δ α β + uα uβ , P (u)α μ uμ = 0, T (u)α μ uμ = uα , P (u)α μ P (u)μ β = P (u)α β , T (u)α μ T (u)μ β = T (u)α β , T (u)α μ P (u)μ β ≡ 0.
(3.4)
r
Given a s tensor S, let us denote by [P (u)S] and [T (u)S] its fully spatial and temporal projections, obtained by acting with the corresponding operators on all of its indices. In terms of components these are given by γ... α δ [P (u)S]α... β... = P (u) γ · · · P (u) β · · · S δ...
(3.5)
[T (u)S]α... β... = T (u)α γ · · · T (u)δ β · · · S γ... δ... .
(3.6)
The splitting of S relative to a given observer is the set of tensors which arise from the spatial and temporal projection of each of its indices, as we shall now illustrate.
Splitting of a vector If S is a vector ﬁeld, its splitting gives rise to a scalar ﬁeld and a spatial vector ﬁeld: S
↔
{u · S, [P (u)S]} .
(3.7)
In terms of components these are given by Sα
↔
{uγ S γ , P (u)α γ S γ }.
(3.8)
36
Spacetime splitting
With respect to the observer u, the vector S then admits the following representation: S α = [T (u)S]α + [P (u)S]α = −(uγ S γ )uα + P (u)α γ S γ ,
(3.9)
also termed 1 + 3splitting.
1
Splitting of a
1 1
tensor
If S is a mixed 1 tensor ﬁeld, then its splitting consists of a scalar ﬁeld, a spatial vector ﬁeld, a spatial 1form, and a spatial 11 tensor ﬁeld, namely S α β ↔ {uδ uγ S γ δ , P (u)α γ uδ S γ δ , P (u)δ α uγ S γ δ , P (u)α γ P (u)δ β S γ δ }. In terms of these ﬁelds, the tensor S admits the following representation: S α β = [T (u)α γ + P (u)α γ ][T (u)δ β + P (u)δ β ]S γ δ = (uδ uγ S γ δ )uα uβ − uα uγ P (u)δ β S γ δ − uδ uβ P (u)α γ S γ δ + [P (u)S]α β ,
(3.10)
for any chosen observer u. The local spatial and temporal projections of a pq tensor are easily generalized. Let us consider the metric tensor gαβ . From (3.2) and (3.3) one ﬁnds that gαβ = P (u)αβ + T (u)αβ ; hence, the spatial metric [P (u)g]αβ = P (u)αβ is the only nontrivial spatial ﬁeld which arises from the splitting of the spacetime metric.
Splitting of pforms Given a pform S = S[α1 ...αp ] ω α1 ⊗ · · · ⊗ ω αp ≡
1 Sα ...α ω α1 ∧ . . . ∧ ω αp , p! 1 p
(3.11)
we deﬁne the electric part of S relative to the observer u by the quantity S (E) (u) = −uσ Sσα1 ...αp−1 (3.12) α1 ...αp−1
or, in a more compact form, S (E) (u) = −u S . Similarly we deﬁne the magnetic part of S by the quantity = P (u)β1 α1 . . . P (u)βp αp Sβ1 ...βp (3.13) S (M) (u) α1 ...αp
3.1 Orthogonal decompositions
37
or, in a more compact form, S (M) (u) = P (u)S. From the above deﬁnitions we deduce the representation of S in terms of its spatial and temporal decompositions: S = u ∧ S (E) (u) + S (M) (u),
(3.14)
Sα1 ...αp = p!u[α1 [S (E) (u)]α2 ...αp ] + [S (M) (u)]α1 ...αp .
(3.15)
or, in components,
Notice that the contraction u S is automatically spatial due to the antisymmetry of S (since u (u S) = 0 or equivalently uα uβ Sβαγ3 ...γp = 0). Consider the splitting of the unit volume 4form η. From its properties and having in mind that [P (u)η] ≡ 0, one deduces the following representation: η = −u ∧ η(u).
(3.16)
To express this in terms of components, let us recall (2.34) and the antisymmetrization rule for four indices. From (2.30), this is given by 1 Rα[βγδ] − Rβ[αγδ] + Rγ[αβδ] − Rδ[αβγ] 4 for any tensor R. Hence we have R[αβγδ] =
[u ∧ η(u)]αβγδ = 4u[α η(u)βγδ] = 2u[α η(u)β]γδ + 2u[γ η(u)δ]αβ ,
(3.17)
(3.18)
where the spatial unit volume 3form η(u)αβγ = uδ ηδαβγ
(3.19)
is the only nontrivial spatial ﬁeld which arises from the splitting of the volume 4form. From the above it then follows that ηαβγδ = −2u[α η(u)β]γδ − 2u[γ η(u)δ]αβ .
(3.20)
In (2.36) and (2.37) we saw that, using the spacetime (Hodge) duality operation (∗ ), one can associate with any pform S (with 0 ≤ p ≤ 4) a (4 − p)form. Similarly a spatial duality operation (∗(u) ) is deﬁned for a spatial pform S (u S = 0) replacing η with η(u), namely ∗(u)
Sα1 ...α3−p =
1 Sβ ...β η(u)β1 ...βp α1 ...α3−p . p! 1 p
(3.21)
As an example, given a spatial 2form S, its spatial dual is a vector given by [∗(u) S]α =
1 η(u)αβγ Sβγ . 2
(3.22)
This operation satisﬁes the property ∗(u) ∗(u)
S = S.
(3.23)
38
Spacetime splitting
Let us now consider the splitting of ∗ S where S is given by (3.14). We have ∗
S = u ∧ [∗ S](E) (u) + [∗ S](M) (u) = ∗ [u ∧ S (E) (u) + S (M) (u)] = ∗ [u ∧ S (E) (u)] + ∗ [S (M) (u)] = ∗(u) S (E) (u) + ∗ [∗(u) [∗(u) [S (M) (u)]]] = ∗(u) S (E) (u) + (−1)p−1 u ∧ ∗(u) [S (M) (u)],
(3.24)
where, in the third line, we have used (3.23). Comparing the ﬁrst and the last line we have [∗ S](E) (u) = (−1)p−1∗(u) [S (M) (u)],
[∗ S](M) (u) = ∗(u) S (E) (u).
(3.25)
Splitting of diﬀerential operators We saw in Chapter 2 that in general relativity one encounters several spacetime tensorial diﬀerential operators which act on tensor ﬁelds. Let us recall them: if T is a tensor ﬁeld of any rank, we have: (i) the Lie derivative of T along the direction of a given vector ﬁeld X: [£X T ] (ii) the covariant derivative of T : ∇T (iii) the absolute derivative of T along a curve with unit tangent vector X and parameterized by s: ∇X T ≡ DT /ds (iv) the FermiWalker derivative of T along a nonnull curve with unit tangent vector X and parameterized by s: D(fw,X) T /ds, deﬁned in (2.83). Finally if S is a pform, one has (v) the exterior derivative of S: dS. Application of the spatial projection into the LRSu of a family of observers u to the spacetime derivatives (i) to (v) yields new operators which can be more easily compared with those deﬁned in a threedimensional Euclidean space. Given a tensor ﬁeld T of components T α... β... we have in fact (i) the spatially projected Lie derivative along a vector ﬁeld X α...
[£(u)X T ]
σ...
β...
≡ P (u)α σ . . . P (u)ρ β . . . [£X T ]
ρ... ;
(3.26)
when X = u we also use the notation ∇(u)(lie) T ≡ £(u)u T, and this operation will be termed the spatialLie temporal derivative
(3.27)
3.2 Threedimensional notation
39
(ii) the spatially projected covariant derivative along any eγ frame direction ∇(u)γ T ≡ P (u)∇γ T,
(3.28)
namely [∇(u)γ T ]α... β... = P (u)α α1 . . . P (u)β1 β . . . P (u)σ γ ∇σ T α1 ... β1 ...
(3.29)
(iii) the spatially projected absolute derivative along a curve with unit tangent vector X [P (u)∇X T ]α... β... = P (u)α α1 . . . P (u)β1 β . . . [∇X T ]α1 ... β1 ...
(3.30)
(iv) the spatially projected FermiWalker derivative along a curve with unit tangent vector X and parameterized by s D(fw,X) T σ... D(fw,X) T α... = P (u)α σ . . . P (u)ρ β... (3.31) P (u) ds ds β... ρ... (v) the spatially projected exterior derivative of a pform S d(u)S ≡ P (u)dS,
(3.32)
[d(u)S]α1 ...αp β = P (u)β1 α1 . . . P (u)σ β [dS]β1 ...σ .
(3.33)
namely
Note that all these spatial diﬀerential operators are well deﬁned since they arise from the spatial projection of spacetime diﬀerential operators. Moreover, there are relations among them which we shall analyze below. From their deﬁnitions it is clear that both the FermiWalker and the Lie derivatives of the vector ﬁeld u along itself vanish identically (and so do the projections orthogonal to u of these derivatives). The only derivative of u along itself which is meaningful, i.e. diﬀerent than zero, is the covariant derivative P (u)∇u u = ∇u u = a(u).
(3.34)
3.2 Threedimensional notation Given a family of observers u, a vector X is termed spatial with respect to u when it lives in the local rest space of u, i.e. it satisﬁes the condition Xα uα = 0. It is then convenient to introduce 3dimensional vector notation for the spatial inner product and the spatial cross product of two spatial vector ﬁelds X and Y . The spatial inner product is deﬁned as X ·u Y = P (u)αβ X α Y β
(3.35)
while the spatial cross product is [X ×u Y ]α = η(u)α βγ X β Y γ , where η(u)α βγ = uσ η σα βγ as stated.
(3.36)
40
Spacetime splitting
In terms of the above deﬁnitions we can deﬁne spatial gradient, curl, and divergence operators of functions f and spatial vector ﬁelds X as gradu f =∇(u)f, curlu X=∇(u) ×u X, divu X =∇(u) ·u X.
(3.37)
In terms of components these relations are given by [gradu f ]α = ∇(u)α f = P (u)αβ eβ (f ), [curlu X]α = η(u)αβγ ∇(u)β Xγ = uσ ησ αβγ ∇β Xγ , [divu X]
= ∇(u)α X α = P (u)αβ ∇α Xβ .
(3.38)
It is useful to extend the above deﬁnitions to (i) (ii) (iii) (iv)
the the the the
spatial cross product of a vector X by a symmetric tensor A spatial cross product of two symmetric spatial tensors A and B spatial inner product of two symmetric spatial tensors A and B trace of the above tensor product
as shown below: [X ×u A]αβ = η(u) γδ(α Xγ Aβ) δ , [A ×u B]α = η(u)αβγ Aβ δ B δγ , [A ·u B]α β = Aαγ B γβ , Tr[A ·u B] = Aαβ B αβ .
(3.39)
One may also introduce a symmetric curl operation or Scurl . For a spatial and symmetric 20 tensor A this is deﬁned by [Scurlu A]αβ = η(u)γδ(α ∇(u)γ Aβ) δ . The Scurl operation can be extended to a spatial antisymmetric follows:
2 0
(3.40) tensor Ω, as
[Scurlu Ω]αβ = η(u)γδ(α ∇(u)γ Ωβ) δ = ∇(u)γ η(u)γδ(α η(u)β) δσ [∗(u) Ω]σ = −divu [∗(u) Ω]P (u)αβ + ∇(u)(α [∗(u) Ω]β) .
(3.41)
Similarly, a spatial divergence of spatial tensors can be deﬁned as in (3.38)3 , that is [divu X]α...β = ∇(u)σ X σα...β ,
X = P (u)X.
(3.42)
Finally, the above operations performed on spatial ﬁelds may be extended to nonspatial ones. For example, if X = X u + X ⊥ is a nonspatial ﬁeld we deﬁne
3.4 Adapted frames
41
[curl X]α ≡ η(u)αβγ ∇(u)β Xγ = 2X ω(u)α + [curlu X]α . Analogous deﬁnitions hold for the other spatial operations considered above.
3.3 Kinematics of the observer’s congruence Let us now consider the splitting of the covariant derivative ∇β uα . This operation generates two spatial ﬁelds, namely the acceleration vector ﬁeld a(u) and the kinematical tensor ﬁeld k(u), deﬁned as a(u) = P (u)∇u u, k(u) = −∇(u)u = ω(u) − θ(u),
(3.43)
where ω(u)αβ = P (u)μα P (u)νβ ∇[μ uν] , θ(u)αβ = P (u)μα P (u)νβ ∇(μ uν) 1 = [£(u)u P (u)]αβ , 2
(3.44)
are the components of tensor ﬁelds ω(u) and θ(u) having the meaning respectively of vorticity and expansion. From the above deﬁnitions, the tensor ﬁeld ∇β uα can be written as ∇β uα = −a(u)α uβ − k(u)α β .
(3.45)
The expansion tensor ﬁeld θ(u) may itself be decomposed into its tracefree and puretrace parts: 1 (3.46) θ(u) = σ(u) + Θ(u)P (u), 3 where the tracefree tensor ﬁeld σ(u) (σ(u)α α = 0) is termed shear and the scalar Θ(u) = ∇α uα
(3.47)
is termed the volumetric (or isotropic) scalar expansion. We can also deﬁne the vorticity vector ﬁeld ω(u) = 1/2 curlu u as the spatial dual of the spatial rotation tensor: ω(u)α =
1 1 η(u)αβγ ω(u)βγ = η σαβγ uσ ∇β uγ . 2 2
(3.48)
Although we use the same symbol for the vorticity tensor and the associated vector they can be easily distinguished by the context.
3.4 Adapted frames Given a ﬁeld of observers u, a frame {eα } with α = 0, 1, 2, 3 (with dual ω α ) is termed adapted to u if e0 = u and ea with a = 1, 2, 3 are orthogonal to u, that
42
Spacetime splitting
is u · ea = 0. From this it follows that ω 0 = −u . Obviously the components of u relative to the frame are simply uα = δ0α and the metric tensor is given by g = −u ⊗ u + P (u)ab ω a ⊗ ω b .
(3.49)
In this section all indices denote components relative to the frame {eα }. The evolution of the frame vectors along the world lines of u is governed by the relations ∇u eα = eσ Γσ α0 . Let us consider separately the cases α = 0 and α = a: (i) When α = 0 it follows that ∇u e0 = ∇u u = a(u) = eσ Γσ 00 .
(3.50)
Recalling that u · a(u) = 0, one ﬁnds that Γ0 00 = 0 and therefore that Γb 00 = a(u)b .
(3.51)
∇u ea = Γσ a0 eσ = Γ0 a0 e0 + Γc a0 ec .
(3.52)
(ii) When α = a we have
The dot product of these terms with −u gives − u · ∇u ea = Γ0 a0 = a(u)a .
(3.53)
We then have two expressions for the acceleration, namely a(u)a = Γ0 a0 ,
and a(u)a = Γa 00 .
(3.54)
Before proceeding, let us deﬁne Γαβγ ≡ eα · eσ Γσ βγ ;
(3.55)
expanding the sum we have Γαβγ = (eα · e0 )Γ0 βγ + (eα · eb )Γb βγ = δα0 g00 Γ0 βγ + δαc P (u)cb Γb βγ .
(3.56)
Hence it follows that Γ0βγ = −Γ0 βγ ,
Γcβγ = P (u)cb Γb βγ .
(3.57)
From the latter, lowering the upindices in (3.54) we obtain Γ0a0 + Γa00 = 2Γ(0a)0 = 0 ;
(3.58)
moreover the transport law for the inner product (ea · u) along u itself yields ∇u (ea · u) = −Γ0 a0 + Γa00 = 0
(3.59)
from (3.58). Therefore the antisymmetry of the ﬁrst pair of indices (0a) ensures that the spatial vectors of the frame remain orthogonal to u while moving along the world line of u itself. From (3.52) and (3.55) we have also Γba0 = eb · ∇u ea = eb · [P (u)∇u ea ] ≡ C(fw)ba ,
(3.60)
3.4 Adapted frames
43
where C(fw)ba are termed FermiWalker structure functions.1 From (3.60) relation (3.52) can be written as ∇u ea = a(u)a e0 + C(fw) b a eb ,
(3.61)
P (u)∇u ea = C(fw) b a eb .
(3.62)
implying that
Let us now prove that C(fw) b a can also we written as C(fw) b a = C(lie) b a − k(u)b a ,
(3.63)
C(lie) b a ≡ ω b (£(u)u ea ),
(3.64)
where
recalling that ω b is the dual basis; Eq. (3.64) implies that P (u)£u ea = £(u)u ea = C(lie) b a eb .
(3.65)
In terms of the structure functions of the frame, we have by deﬁnition £(u)u ea ≡ P (u)£u ea = P (u)[u, ea ] = P (u) C α 0a eα = C b 0a eb ,
(3.66)
where we have used the relations P (u)e0 = 0 and P (u)eb = eb ; therefore C(lie) b a = C b 0a .
(3.67)
£u ea = ∇u ea − ∇ea u = ∇u ea + k(u)c a ec ,
(3.68)
Using (3.45) we obtain
so, projecting both sides orthogonally to u yields £(u)u ea = P (u)∇u ea + k(u)c a ec ,
(3.69)
which leads to (3.63). Using (3.45), one obtains ∇ea e0 = Γγ 0a eγ = −k(u)b a eb , ∇ea eb = Γγ ba eγ = −k(u)ba e0 + Γc ba ec .
(3.70)
Summarizing, we have Γa 00 = a(u)a , Γb 0a = −k(u)b a , 1
Γ0 a0 = a(u)a , Γ0 ba = −k(u)ba .
Γb a0 = C(fw) b a ,
(3.71)
A more correct notation for the FermiWalker structure functions should be C(fw,u,ec )ba , i.e. a symbol which includes explicit dependence on the frame. Clearly when the frame is ﬁxed the simpliﬁed notation proves useful.
44
Spacetime splitting
We can also express the structure functions in terms of kinematical quantities. In fact, from the deﬁnition eα C α βγ = [eβ , eγ ] = ∇eβ eγ − ∇eγ eβ
(3.72)
we have eα C α 0b = ∇u eb − ∇eb u = a(u)b u + [C(fw) c b + k(u)c b ]ec = a(u)b u + C(lie) c b ec ,
(3.73)
so that C 0 0b = a(u)b ,
C c 0b = C(lie) c b .
(3.74)
Similarly eα C α bc = ∇eb ec − ∇ec eb = 2ω(u)bc u + 2Γd [cb] ed ,
(3.75)
so that C 0 bc = 2ω(u)bc ,
C d bc = 2Γd [cb] .
(3.76)
We recall that the structure functions satisfy the Jacobi identities [[eα , eβ ], eρ ] + [[eβ , eρ ], eα ] + [[eρ , eα ], eβ ] = 0,
(3.77)
which implies the following diﬀerential relations: e[ρ (C σ αβ] ) − C σ ν[ρ C ν αβ] = 0.
(3.78)
Splitting such relations using an observeradapted frame gives 0 = −∇(u)(lie) ω(u)ab + ∇(u)[a a(u)b] , 0 = −∇(u)(lie) C d ab + 2e[a C(lie) f b] + 2a(u)[a C(lie) f b] , 0 = −∇(u)[r ω(u)ba] + a(u)[r ω(u)ba] , 0 = −e [r C c ab] + C c s[r C s ab] + 2C(lie) c [r ω(u)ab] .
(3.79)
The ﬁrst of these relations can also be written as 1 [∇(u)(lie) + Θ(u)]ω(u)s − [curlu a(u)]s = 0, 2
(3.80)
where we have used the property ∇(u)(lie) η(u)abc = Θ(u)η(u)abc .
(3.81)
3.4 Adapted frames
45
The third equation becomes − divu ω(u) + a(u) · ω(u) = 0
(3.82)
after contraction of both sides with η(u)rba .
SpatialFermiWalker and spatialLie temporal derivatives In the previous subsection we introduced the FermiWalker structure functions C(fw) b a , as follows: P (u)∇u ea = C(fw) b a eb .
(3.83)
A similar relation was deﬁned for the Lie derivative projected along u, which we also termed a spatialLie temporal derivative (see (3.27)): £(u)u ea = P (u)£u ea = C(lie) b a eb = C b 0a eb .
(3.84)
It is useful to handle both these operations with a uniﬁed notation2 ∇(u)(tem) ea = C(tem) b a eb ,
(3.85)
where “tem” refers either to “fw” or to “lie”, with the following deﬁnitions: ∇(u)(fw) ea ≡ P (u)∇u ea ,
∇(u)(lie) ea ≡ P (u)£u ea = £(u)u ea .
(3.86)
Therefore if X is a vector ﬁeld orthogonal to u, i.e. X · u = 0, we have dX a ∇(u)(tem) X = ∇(u)(tem) (X a ea ) = ea + X a C(tem) b a eb dτu dX b b a = + X C(tem) a eb dτu = (∇(u)(tem) X b )eb .
(3.87)
The operation ∇(u)(fw) = P (u)∇u is termed a spatialFermiWalker temporal derivative. It can be extended to nonspatial ﬁelds. If we apply this operation to the vector ﬁeld u itself we have ∇(u)(fw) u = P (u)∇u u = a(u),
(3.88)
and if X = f u we have ∇(u)(fw) X = ∇(u)(fw) (f u) = f a(u), ∇(u)(lie) X = ∇(u)(lie) (f u) = 0. 2
(3.89)
This notation has been introduced by R. T. Jantzen, who also considered two more spatial temporal derivatives, corotating FermiWalker and Lie . Details can be found in Jantzen, Carini, and Bini (1992).
46
Spacetime splitting
Hence the temporal derivatives so deﬁned through their action on purely spatial and purely temporal ﬁelds can now act on any spacetime ﬁeld.
Frame components of the Riemann tensor From the deﬁnition eα Rα βγδ = [∇eγ , ∇eδ ]eβ − C σ γδ ∇eσ eβ ,
(3.90)
we have eα Rα 0b0 = [∇eb , ∇u ]u − C σ b0 ∇eσ u
(3.91)
= ∇eb (a(u) ec ) − ∇u (∇eb u) − C b0 a(u) ec − C b0 ∇ec u = [∇(u)b + a(u)b ]a(u)c + ∇(u)(fw) k(u)c b − [k(u)2 ]c b ec , c
0
c
c
where [k(u)2 ]c b ≡ k(u)c s k(u)s b , so that Rc 0b0 = [∇(u)b + a(u)b ]a(u)c + ∇(u)(fw) k(u)c b − [k(u)2 ]c b ,
(3.92)
∇(u)(fw) k(u)c b = ∇u k(u)c b + C(fw) c f k(u)f b − C(fw) f b k(u)c f .
(3.93)
where
Similarly eα Rα bcd = [∇ec , ∇ed ]eb − C 0 cd ∇u eb − C f cd ∇ef eb
(3.94)
= ∇ec (∇ed eb ) − ∇ed (∇ec eb ) + 2ω(u)cd ∇u eb − C f cd ∇ef eb = ∇ec (−k(u)bd u + Γf bd ef ) − ∇ed (−k(u)bc u + Γf bc ef ) + 2ω(u)cd (a(u)b u + C(fw) f b ef ) − C f cd (−k(u)bf u + Γr bf er ),
(3.95)
that is, eα Rα bcd = [−ec (k(u)bd ) − Γs bd k(u)sc + ed (k(u)bc ) + Γs bc k(u)sd − 2ω(u)cd a(u)b + C s cd k(u)bs ]u + [2e[c (Γf bd] ) + 2Γf s[c Γs bd] − 2k(u)b[c k(u)f d] − 2ωcd C(fw) f b − C s cd Γf bs ]ef .
(3.96)
Analyzing the time and space components of the Riemann tensor relative to the basis, we have R0 bcd = −2[∇(u)[c k(u)bd] + ω(u)cd a(u)b ]
(3.97)
3.4 Adapted frames
47
and Rf bcd = R(fw) f bcd − 2k(u)b[c k(u)f d] = R(fw) f bcd + 2k(u)f [c k(u)bd] ,
(3.98)
where R(fw) f bcd = 2e[c Γf bd] + 2Γs b[c Γf sd] − C s cd Γf bs − 2ω(u)cd C(fw) f b .
(3.99)
This tensor is termed a FermiWalker spatial Riemann tensor;3 it can be written in invariant form as follows R(fw) (u)(X, Y )Z = {[∇(u)X , ∇(u)Y ]Z − ∇(u)[X,Y ] }Z − 2ω(u)(X, Y )∇(u)(fw) Z,
(3.100)
where X, Y , and Z are spatial ﬁelds with respect to u and we note that [X, Y ] = P (u)[X, Y ] − 2ω (X, Y ) u.
(3.101)
The FermiWalker spatial Riemann tensor does not have all the symmetries of a threedimensional Riemann tensor. For instance it does not satisfy the Ricci identities. In fact we have (3.102) 0 = Rf [bcd] = R(fw) f [bcd] − 2k(u)[bc k(u)f d] . 0 Recalling that the antisymmetric part of a 3 tensor Xbcd can be written as 1 X[bc]d + X[cd]b + X[db]c 3 1 Xb[cd] + Xc[db] + Xd[bc] , = 3
X[bcd] =
(3.103)
it follows that (3.102) can be written more conveniently as R(fw) f [bcd] = 2k(u)f [b ω(u)cd] .
(3.104)
From the latter one can construct a new Riemann tensor with all the necessary symmetries. Ferrarese (1965) has shown that the symmetryobeying Riemann tensor, denoted by R(sym) ab cd , is related to the FermiWalker Riemann tensor (3.100) by R(sym) ab cd = R(fw) ab cd − 2ω(u)ab ω(u)cd − 4θ(u)[a [c ω(u)b] d] = Rab cd + 2k(u)b [c k(u)a d] − 2ω(u)ab ω(u)cd − 4θ(u)[a [c ω(u)b] d] , 3
(3.105)
A Lie spatial Riemann tensor can be deﬁned similarly, replacing the FermiWalker structure functions C(fw) fb with the corresponding Lie structure functions C(lie) fb according to (3.63).
48
Spacetime splitting
or Rab cd = R(sym) ab cd + 2θ(u)a [c θ(u)b d] + 2ω ab ωcd + 2ω(u)a [c ω(u)b d] ,
(3.106)
where (3.98) has been used. In the special case of a Bornrigid congruence of observers u, i.e. θ(u) = 0, we ﬁnd that (3.105) can be written as R(sym) ab cd = R(fw) ab cd − 2ω(u)ab ω(u)cd = Rab cd + 2ω(u)b [c ω(u)a d] − 2ω(u)ab ω(u)cd ,
(3.107)
whereas in the special case of a vorticityfree congruence, i.e. ω(u) = 0, we have R(sym) ab cd = R(fw) ab cd = Rab cd + 2θ(u)b [c θ(u)a d] .
(3.108)
Together with the spatial symmetric Riemann tensor R(sym) ab cd we also introduce the spatial symmetric Ricci tensor, R(sym) a b = R(sym) ca cb , as well as the associated scalar R(sym) = R(sym) a a .
3.5 Comparing families of observers Let u and U be two unitary timelike vector ﬁelds. Deﬁne the relative spatial velocity of U with respect to u as ν(U, u)α = −(uσ Uσ )−1 P (u)α β U β = −(uσ Uσ )−1 [U α + (uρ Uρ )uα ] .
(3.109)
The spacelike vector ﬁeld ν(U, u)α , orthogonal to u, can also be written as ν (U, u)α , ν(U, u)α = ν(U, u)ˆ
(3.110)
where νˆ(U, u) is the unitary vector giving the direction of ν(U, u) in the rest frame of u. Similar formulas hold for the relative velocity of u with respect to U ; hence the 4velocities of the two observers admit the following decomposition: U = γ(U, u)[u + ν(U, u)] = γ(U, u)[u + ν(U, u)ˆ ν (U, u)],
(3.111)
and u = γ(u, U )[U + ν(u, U )] = γ(u, U )[U + ν(u, U )ˆ ν (u, U )]. Both the spatial relative velocity vectors have the same magnitude, ν(U, u) = [ν(U, u)α ν(U, u)α ]1/2 = [ν(u, U )α ν(u, U )α ]1/2 .
(3.112)
3.5 Comparing families of observers
49
The common gamma factor is related to that magnitude by γ(U, u) = γ(u, U ) = [1 − ν(U, u)2 ]−1/2 = −Uα uα ;
(3.113)
hence we recognize it as the relative Lorentz factor. It is convenient to abbreviate γ(U, u) by γ and ν(U, u) by ν when their meaning is clear from the context and there are no more than two observers involved. Let us notice here that by substituting (3.111) into (3.112) we obtain the following relation: − νˆ(u, U ) = γ[ˆ ν (U, u) + νu],
(3.114)
U = γ[u + ν νˆ(U, u)]
(3.115)
which together with
yields a unique relative boost B(U, u) from u to U , namely B(U, u)u = U = γ[u + ν νˆ(U, u)] B(U, u)ˆ ν (U, u) = −ˆ ν (u, U ) = γ[ˆ ν (U, u) + νu].
(3.116)
The inverse relations hold by interchanging U with u. The boost acts as the identity on the intersection of their local rest spaces LRSu ∩ LRSU .
Maps between LRSs The spatial measurements of two observers in relative motion can be compared by relating their respective LRSs. Let U and u be two such observers and LRSU and LRSu their LRSs. There exist several maps between these LRSs, as we now demonstrate. Combining the projection operators P (U ) and P (u) one can form the following “mixed projection” maps: (i) P (U, u) from LRSu into LRSU , deﬁned as P (U, u) = P (U )P (u) : LRSu → LRSU ,
(3.117)
P (U, u)−1 : LRSU → LRSu
(3.118)
with its inverse:
(ii) P (u, U ) from LRSU into LRSu , deﬁned as P (u, U ) = P (u)P (U ) : LRSU → LRSu ,
(3.119)
P (u, U )−1 : LRSu → LRSU .
(3.120)
with its inverse:
50
Spacetime splitting
Note that P (U, u) = P (u, U )−1 shown as, from their representations, P (U, u) −1
= P (u) + γνU ⊗ νˆ(U, u),
P (U, u)
= P (U ) + νU ⊗ νˆ(u, U ),
P (u, U )
= P (U ) + γνu ⊗ νˆ(u, U ),
−1
P (u, U )
= P (u) + νu ⊗ νˆ(U, u).
(3.121)
Let us deduce each of the above relations. (a) Relation (3.121)1 is easily veriﬁed. Consider a vector X ∈ LRSu ; then, by deﬁnition, P (U, u)X = P (U )P (u)X = P (U )X = X + U (U · X) ;
(3.122)
but U · X = γ(u · X + ν νˆ(U, u) · X) = γν νˆ(U, u) · X. Hence P (U, u)X = X + γνU νˆ(U, u) · X = [P (u) + γνU ⊗ νˆ(U, u)] X,
(3.123)
which completes the proof. (b) Let us now verify (3.121)2 . Consider a vector X = P (U, u)Y ∈ LRSU , with Y ∈ LRSu ; then, by using the relation u = γ(U + ν νˆ(u, U )) and the property Y · u = 0, we have Y · U = −νY · νˆ(u, U ). Hence Y = P (U )Y − U (U · Y ) = P (U, u)Y + νU [Y · νˆ(u, U )] = P (U, u)Y + νU [P (U, u)Y · νˆ(u, U )].
(3.124)
Substituting Y = P (U, u)−1 X gives P (U, u)−1 X = P (U )X + νU [X · νˆ(u, U )] = [P (U ) + νU ⊗ νˆ(u, U )] X,
(3.125)
which completes the proof. (c) Relations (3.121)3 and (3.121)4 are straightforwardly veriﬁed by exchanging U with u in (3.121)1 and (3.121)2 respectively. One can then show that P (U, u)ˆ ν (U, u) = −γ νˆ(u, U ), 1 P (u, U )−1 νˆ(U, u) = − νˆ(u, U ). γ
(3.126)
Note that, from the property (3.3) of the projection operator P (U ), we have P (U, u) = P (U ) P (u) = P (U ) P (U, u) = P (U, u) P (u).
(3.127)
3.5 Comparing families of observers
51
This is a mixed tensor which is spatial with respect to u in its covariant index and with respect to U in its contravariant index.4 Moreover the following relations hold: P (U ) = P (U, u) P (U, u)−1 , P (u) = P (U, u)−1
P (U, u).
(3.128)
Boost maps Similarly to what we have done in combining projection maps, the boost B(U, u) induces an invertible map between the local rest spaces of the given observers deﬁned as B(lrs) (U, u) ≡ P (U )B(U, u)P (u) : LRSu → LRSU .
(3.129)
It also acts as the identity on the intersection of their subspaces LRSU ∩ LRSu . Because the boost is an isometry, exchanging the role of U and u in (3.129) leads to the inverse boost: B(lrs) (U, u)−1 ≡ B(lrs) (u, U ) : LRSU → LRSu .
(3.130)
Before giving explicit representations of B(lrs) (U, u) and its inverse B(lrs) (U, u)−1 we note that any map between the local rest spaces of two observers may be expressed only in terms of quantities which are spatial with respect to those observers. The representations of the boost and its inverse can be given in terms of the associated tensors B(lrs)u (U, u),
B(lrs)U (U, u),
B(lrs)u (u, U ),
B(lrs)U (u, U ),
(3.131)
deﬁned by B(lrs)u (U, u) = P (U, u)−1
B(lrs) (U, u),
B(lrs)U (U, u) = B(lrs) (U, u) P (U, u)−1 ,
(3.132)
with the corresponding expressions for the inverse boost obtained simply by exchanging the roles of U and u, and with B(lrs) (U, u) = B(lrs)U (U, u) P (U, u) = P (U, u) B(lrs)u (U, u).
(3.133)
The explicit expression for B(lrs)u (U, u), for example, is given by B(lrs)u (U, u) = P (u) + 4
1−γ νˆ(U, u) ⊗ νˆ(U, u). γ
(3.134)
This is a kind of twopoint tensor that Schouten (1954) termed a connecting tensor.
52
Spacetime splitting
This can be shown as follows. Let X ∈ LRSu ; then B(lrs)u (U, u)X = P (U, u)−1 [B(lrs) (U, u)X] = P (U, u)−1 [B(lrs) (U, u)X  νˆ(U, u) + X ⊥ ],
(3.135)
where X  = X · νˆ(U, u) and X ⊥ = X − X  νˆ(U, u), that is X = X  νˆ(U, u) + X ⊥ .
(3.136)
We have also used the fact that the boost reduces to the identity for vectors not belonging to the boost plane, as X ⊥ . In this case the boost plane is spanned by the vectors u and νˆ(U, u). Taking into account (3.116), that is, B(lrs) (U, u)ˆ ν (U, u) = −ˆ ν (u, U ),
(3.137)
as well as the linearity of the boost map, we have B(lrs)u (U, u)X = P (U, u)−1 [X − X  νˆ(u, U ) − X  νˆ(U, u)] ν (u, U ) + νˆ(U, u)] = P (U, u)−1 X − X  P (U, u)−1 [ˆ 1 − γ  X νˆ(U, u), =X+ (3.138) γ where P (U, u)−1 X = X because X ∈ LRSu : P (U, u)−1 X = P (U, u)−1 P (u)X = P (U, u)−1 P (U )P (u)X = P (U, u)−1 P (U, u)X = P (u)X = X.
(3.139)
Hence P (U, u)−1 νˆ(U, u) = νˆ(U, u), because ν(U, u) belongs to LRSu . Moreover, from (3.126), by exchanging the roles of U and u, we ﬁnd 1 1 ν (u, U ) P (U, u)−1 νˆ(u, U ) = − νˆ(U, u) = B(u, U )ˆ γ γ 1 = B(U, u)−1 νˆ(u, U ). γ
(3.140)
Therefore 1 B(lrs)u (U, u)X = X − X − + 1 νˆ(U, u) γ 1 = X − (X · νˆ(U, u)) − + 1 νˆ(U, u) γ γ−1 νˆ(U, u) ⊗ νˆ(U, u) X, = P (u) − γ 
which is equivalent to (3.134).
(3.141)
3.6 Splitting of derivatives along a timelike curve
53
Similarly, for the inverse boost B(lrs) (u, U ), one has γ ν(U, u) ⊗ ν(U, u) γ+1 γ−1 νˆ(U, u) ⊗ νˆ(U, u) , = P (u) − γ γ ν(u, U ) ⊗ ν(u, U ) B(lrs)U (u, U ) = P (U ) − γ+1 γ−1 νˆ(u, U ) ⊗ νˆ(u, U ) . = P (U ) − γ B(lrs)u (u, U ) = P (u) −
(3.142)
Thus, if S is a vector ﬁeld such that S ∈ LRSU , then its inverse boost is the vector belonging to LRSu : B(lrs) (u, U )S = [P (u) − γ(γ + 1)−1 ν(U, u) ⊗ ν(U, u) ] P (u, U )S. (3.143) In components, the latter relation becomes B(lrs) (u, U )α β S β = [P (u)α μ − γ(γ + 1)−1 ν(U, u)α ν(U, u)μ ]P (u, U )μ β S β . (3.144)
3.6 Splitting of derivatives along a timelike curve Consider a congruence of curves CU with tangent vector ﬁeld U and proper time τU as parameter. We know, at this stage, that the evolution along CU of any tensor ﬁeld can be speciﬁed by one of the following spacetime derivatives: (i) the absolute derivative along CU : D/dτU = ∇U (ii) the FermiWalker derivative along CU : D(fw,U ) /dτU (iii) the spacetime Lie derivative along CU : £U , for which we use also the notation D(lie,U ) /dτU = £U . The actions of the FermiWalker and Lie derivatives on a vector ﬁeld X are related to the absolute derivative as follows: D(fw,U ) X = ∇U X + a(U )(U · X) − U (a(U ) · X) dτU = P (U )∇U X − U ∇U (X · U ) + a(U )(X · U ), D(lie,U ) X = [U, X] = ∇U X + a(U )(U · X) + k(U ) X, dτU
(3.145)
where k(U ) = ω(U )−θ(U ) is the kinematical tensor of the congruence CU , deﬁned in (3.43). For X = U we have D(fw,U ) U = 0, dτU
D(lie,U ) U = 0, dτU
(3.146)
54
Spacetime splitting
whereas DU = a(U ). dτU
(3.147)
If X is spatial with respect to U , i.e. X · U = 0, we have instead D(fw,U ) X = P (U )∇U X, dτU D(lie,U ) X = ∇U X − k(U ) X dτU = P (U )∇U X + U (a(U ) · X) + k(U ) X.
(3.148)
The projection of D(lie,U ) X/dτU orthogonal to U gives, as in (3.148), P (U )
D(lie,U ) X = P (U )∇U X + k(U ) X dτU D(fw,U ) X + k(U ) X. = dτU
(3.149)
Let u be another family of observers whose world lines have as parameter the proper time τu . One can introduce, on the congruence CU whose unit tangent vector ﬁeld can be written as U = γ(U, u)[u + ν(U, u)],
(3.150)
two new parameterizations τ(U,u) and (U,u) as follows: dτ(U,u) = γ(U, u), dτU
d (U,u) = γ(U, u)ν(U, u), dτU
(3.151)
where τ(U,u) corresponds to the proper times of the observers u when their curves are crossed by a given curve of CU , and (U,u) corresponds to the proper length on CU . The projection orthogonal to u of the absolute derivative along U is expressed as P (u)
D = P (u)∇U = γ[P (u)∇u + P (u)∇ν(U,u) ] dτU = γ[P (u)∇u + ∇(u)ν(U,u) ].
(3.152)
We note that in the above equation the derivative operation P (u)∇u is just what we have termed the spatialFermiWalker temporal derivative, i.e. ∇(u)(fw) , in (3.86). For a vector ﬁeld X we can then write
3.6 Splitting of derivatives along a timelike curve P (u)
D(fw,U ) X D(fw,U,u) X ≡ dτU dτU DX = P (u) + [P (u, U )a(U )](U · X) dτU − γν(U, u)(a(U ) · X),
P (u)
55
(3.153)
D(lie,U ) X D(lie,U,u) X ≡ dτU dτU DX = P (u) + P (u, U )a(U )(U · X) dτU + P (u)[k(U ) X].
(3.154)
We shall now examine these three projected absolute derivatives in detail. Projected absolute derivative Consider the absolute derivative of u along U , namely ∇U u. Since u is unitary, then Du u· =0 dτU and we can write, from (3.43), Du Du = P (u) = γ[P (u)∇u u + P (u)∇ν(U,u) u] dτU dτU = γ[∇(u)(fw) u + P (u)∇ν(U,u) u] = γ[a(u) − k(u) ν(U, u)] = γ[a(u) + ω(u) ×u ν(U, u) + θ(u) ν(U, u)].
(3.155)
Let us denote the above quantity as a (minus) FermiWalker gravitational force, that is Du (G) F(fw,U,u) = − dτU = −γ[a(u) + ω(u) ×u ν(U, u) + θ(u) ν(U, u)]. (3.156) (G)
It should be stressed here that, although F(fw,U,u) is referred to as a gravitational force, it contains contributions by true gravity and by inertial forces. Consider now the case of X orthogonal to u, i.e. X · u = 0. The projection onto LRSu of the absolute derivative of X along U gives P (u)
DX = γ[P (u)∇u X + ∇(u)ν(U,u) X] dτU D(fw,U,u) X . ≡ dτU
(3.157)
This diﬀerential operator plays an important role since both FermiWalker and Lie derivatives along U can be expressed in terms of it. It is therefore worthwhile
56
Spacetime splitting
to write down the above expression in terms of (adapted) frame components: P (u)
dX a DX = ea + γ X a P (u)∇u ea + ν(U, u)b X a ∇(u)b ea dτU dτU dX b eb , + γ X a C(fw) b a + ν(U, u)c Γb ac = dτU
(3.158)
where we set X = X a ea . Introducing the relative standard time parameterization τ (U, u) deﬁned in (3.151), we have D(fw,U,u) X D(fw,U,u) X a DX P (u) = = ea (3.159) dτ(U,u) dτ(U,u) dτ(U,u) or, in components, D(fw,U,u) X b dX b = + X a C(fw) b a + ν(U, u)c Γb ac . dτ(U,u) dτ(U,u)
(3.160)
A particular vector ﬁeld which is orthogonal to u and is deﬁned all along CU is the ﬁeld of relative velocities, ν(U, u). We introduce the acceleration of U relative to u as D(fw,U,u) D ν(U, u) = γ −1 P (u) ν(U, u). (3.161) a(fw,U,u) = dτ(U,u) dτU Considering instead the unit vector νˆ(U, u), this quantity can be written as a(fw,U,u) = P (u)
D dτ(U,u)
[ν νˆ(U, u)],
(3.162)
where ν = ν(U, u). Finally we have a(fw,U,u) = νˆ(U, u)
dν D + νP (u) νˆ(U, u). dτ(U,u) dτ(U,u)
(3.163)
It is therefore quite natural to denote the ﬁrst term as a tangential FermiWalker (T ) acceleration a(fw,U,u) of U relative to u, and the second as a centripetal Fermi(C)
Walker acceleration a(fw,U,u) of U relative to u: (T )
(C)
a(fw,U,u) = a(fw,U,u) + a(fw,U,u) ,
(3.164)
where (T )
dν , dτ(U,u) D = ν P (u) νˆ(U, u) dτ(U,u) D(fw,U,u) =ν νˆ(U, u). dτ(U,u)
a(fw,U,u) = νˆ(U, u) (C)
a(fw,U,u)
(3.165)
3.6 Splitting of derivatives along a timelike curve
57
To generalize the classical mechanics notion of centripetal acceleration we need to convert the relative standard time parameterization into an analogous relative standard length parameterization:5 d (U,u) = νdτ(U,u) .
(3.166)
With this parameterization we have (C)
a(fw,U,u) = ν 2 P (u) =ν =
D νˆ(U, u) d (U,u)
D(fw,U,u) νˆ(U, u) dτ(U,u) ν2
R(fw,U,u)
ηˆ(fw,U,u)
= ν 2 k(fw,U,u) ηˆ(fw,U,u) ,
(3.167)
where ηˆ(fw,U,u) is a unit spacelike vector orthogonal to νˆ(U, u), k(fw,U,u) is the FermiWalker relative curvature, and R(fw,U,u) is the curvature radius of the curve such that ηˆ(fw,U,u) D = P (u) νˆ(U, u). (3.168) k(fw,U,u) ηˆ(fw,U,u) = R(fw,U,u) d (U,u) Clearly, whether geometrically or physically motivated, one can replace the spatialFermiWalker temporal derivative with the spatialLie temporal derivative deﬁning the corresponding quantities. Doing this, one really understands the power of the notation used. For example (and for later use) one can deﬁne the Lie relative curvature of a curve and the associated curvature radius: k(lie,U,u) ηˆ(lie,U,u) =
ηˆ(lie,U,u) D(lie,U,u) = νˆ(U, u). R(lie,U,u) d (U,u)
(3.169)
Projected FermiWalker derivative The projected FermiWalker derivative for a general vector ﬁeld X is given by (3.153). In the case X · u = 0, and using the standard decomposition of U = γ(u + ν(U, u)), we have P (u)
5
D(fw,U,u) X D(fw,U ) X ≡ dτU dτU DX = P (u) + γP (u, U )a(U )(ν(U, u) · X) dτU − γν(U, u)(P (u, U )a(U ) · X).
(3.170)
In fact the Euclidean space deﬁnition involves spatial orbits parameterized by the (spatial) curvilinear abscissa.
58
Spacetime splitting
Setting P (u, U )a(U ) = γF (U, u),
(3.171)
we ﬁnd that P (u)
D(fw,U ) X DX = P (u) + γ 2 F (U, u)(ν(U, u) · X) dτU dτU − γ 2 ν(U, u)(F (U, u) · X) DX = P (u) dτU 2 + γ X ×u (F (U, u) ×u ν(U, u)).
(3.172)
Projected Lie derivative Finally, for the projected Lie derivative we have, from (3.154), P (u)
D(lie,U,u) X D(lie,U ) X ≡ dτU dτU DX = P (u) + γ 2 F (U, u)(ν(U, u) · X) dτU + P (u)[k(U ) X].
(3.173)
4 Special frames
The deﬁnition and interpretation of a physical measurement are better achieved if reference is made to a system of Cartesian axes which matches an instantaneous inertial frame. With respect to such a frame, a measurement is expressed in terms of the projection on its axes of the tensors or tensor equations characterizing the phenomenon under investigation. Two diﬀerent Cartesian frames are connected by a general Lorentz transformation; hence the properties of the Lorentz group play a key role in the theory of measurements. In Newtonian mechanics the local rest spaces of all observers coincide as a result of the absoluteness of time but in relativistic mechanics the local rest spaces do not coincide, because of relativity of time; hence a comparison between any two quantities which belong to the local rest spaces of diﬀerent observers requires a nontrivial mapping among them. This is what we are going to discuss now.
4.1 Orthonormal frames Let u be the 4velocity of an observer; at any point of his world line let us choose ˆ = 1, 2, 3. in LRSu three mutually orthogonal unit spacelike vectors e(u)aˆ with a In what follows these vectors will be simply denoted by eaˆ whenever it is not necessary to specify the timelike vector u which is orthogonal to them. They satisfy the relation eaˆ · eˆb = δaˆˆb .
(4.1)
It is clear that the triad {eaˆ }aˆ=1, 2, 3 is an orthonormal basis for LRSu . ˆ = 0, 1, 2, 3 itself forms an orthonorIf we write u ≡ eˆ0 , then the set {eαˆ } with α mal basis for the local tangent space and is termed a tetrad (Pirani, 1956b). The vectors eαˆ satisfy the condition eαˆ · eβˆ = ηαˆ βˆ ≡ diag [−1, 1, 1, 1],
(4.2)
60
Special frames
which ensures that {eαˆ } is an instantaneous inertial frame, as stated. Hereafter the orthonormal frame vectors as well as all tetrad components will be denoted by hatted indices. From (4.2), hatted indices are raised and lowered by the Minkowski metric ηαˆ βˆ . Let eαˆ , e¯αˆ be two orthonormal frames deﬁned at a certain point of the spacetime manifold. Then eαˆ · eβˆ = e¯αˆ · e¯βˆ = ηαˆ βˆ ,
(4.3)
where dot deﬁnes the inner product induced by the background metric, that is eαˆ · eβˆ ≡ gμν eαˆ μ eβˆ ν . If L is a matrix which generates a change of frame, we have ˆ
e¯αˆ = eβˆ Lβ αˆ .
(4.4)
The orthonormality of both frames places restrictions on the components of the matrix L, i.e. ˆ
ηαˆ βˆ = Lγˆ αˆ Lδ βˆ ηγˆ δˆ.
(4.5)
The set of matrices L which satisfy (4.5) forms a group known as the Lorentz group of transformations. We shall impose the conditions ˆ
L0 ˆ0 > 0,
det(L) = 1,
(4.6)
which imply that futurepointing timelike vectors remain futurepointing. Lorentz transformations which satisfy conditions (4.6) are termed orthochronous and proper. If u is a vector ﬁeld whose integral curves form a congruence Cu , and {eαˆ } is a ﬁeld of orthonormal bases over the above congruence, then the transport law for any vector of the tetrad along any direction of the frame itself is given by ∇eαˆ eβˆ = Γγˆ βˆαˆ eγˆ .
(4.7)
Here Γγˆ βˆαˆ are the frame components of the connection coeﬃcients, also termed Ricci rotation coeﬃcients. The requirement that the frame remains orthonormal over the congruence is fulﬁlled by the condition Γ(ˆγ β) ˆ α ˆ = 0,
(4.8)
Γγˆ βˆαˆ ≡ ηγˆ σˆ Γσˆ βˆαˆ .
(4.9)
where we set
Since the frame components of the metric are constant, from (2.51) it follows that the connection coeﬃcients have the form 1 αˆ δˆ Γαˆ βˆ Cδˆ (4.10) ˆγ = η ˆγ βˆ + Cβˆδˆ ˆγ − Cγ ˆ ˆ ˆβδ , 2
4.1 Orthonormal frames
61
where C γˆ βˆδˆ are the structure functions of the given frame. Using (3.45) and (3.60), and following the reasoning of Section 3.2, we have ∇eˆ0 eˆ0 = Γaˆ ˆ0ˆ0 eaˆ = a(u)aˆ eaˆ , ˆ
∇eˆ0 eaˆ = Γγˆ aˆˆ0 eγˆ = a(u)aˆ eˆ0 + C(fw) b aˆ eˆb , ˆ
∇eaˆ eˆ0 = Γγˆ ˆ0ˆa eγˆ = −k(u)b aˆ eˆb , ∇eaˆ eˆb = Γγˆ ˆbˆa eγˆ = −k(u)ˆbˆa eˆ0 + Γcˆˆbˆa ecˆ, ˆ
(4.11)
ˆ
where C(fw) b aˆ are related to C(lie) b aˆ by (3.63) with hatted indices. Summarizing, we have Γaˆ ˆ0ˆ0 = a(u)aˆ , ˆ ˆ Γb ˆ0ˆa = −k(u)b aˆ ,
ˆ
Γ0 aˆˆ0 = a(u)aˆ , ˆ Γ0ˆbˆa = −k(u)ˆbˆa ,
ˆ
ˆ
Γb aˆˆ0 = C(fw) b aˆ ,
(4.12)
where each component has a precise physical meaning. Let U be the unit timelike vector tangent to the world line γ of a particle and let it be analyzed by a family of observers u represented by a congruence of curves Cu . Let {eαˆ } be a ﬁeld of tetrads which are adapted to u throughout the congruence Cu . Clearly γ intersects at each of its points a curve of the congruence Cu . We denote by uγ and {eaˆ }γ the restrictions on γ of the vector ﬁelds u = eˆ0 and {eaˆ }. With respect to the observer u, the vector ﬁeld U admits the following representation: U = γ(U, u)[u + ν(U, u)aˆ eaˆ ] ≡ γ[u + ν νˆ(U, u)aˆ eaˆ ].
(4.13)
It is physically relevant to calculate the transport law for the vectors of {eαˆ }γ along γ as judged by the observer u himself. From Eqs. (4.13) and (4.11) we have ∇U eaˆ = γ∇u eaˆ + γν cˆ∇ecˆ eaˆ ˆ ˆ = γ a(u)aˆ − k(u)aˆcˆν cˆ u + γ C(fw) d aˆ + ν cˆΓd aˆcˆ edˆ, ∇U u = γ∇u u + γν cˆ∇ecˆ u ˆ ˆ = γ a(u)b − k(u)b cˆν cˆ eˆb ,
(4.14)
where we have set ν(U, u)aˆ = ν aˆ to simplify notation. Projecting onto LRSu and recalling (3.156), we obtain D(fw,U,u) ˆ ˆ P (u)∇U eaˆ ≡ eaˆ = γ C(fw) d aˆ + ν cˆΓd aˆcˆ edˆ, dτU (G)
ˆ
P (u)∇U u = ∇U u = −F(fw,U,u) b eˆb .
(4.15)
By using as a parameter the relative standard time τ(U,u) , the ﬁrst line of (4.15) becomes D(fw,U,u) ˆ ˆ eaˆ = C(fw) d aˆ + ν cˆΓd aˆcˆ edˆ. (4.16) dτ(U,u)
62
Special frames
This equation represents the transport law along γ of the vectors of the triad {eaˆ }γ as judged by u. Clearly the vectors {eaˆ }γ undergo a rotation described by the tensor ˆ
ˆ
ˆ
Rd aˆ = C(fw) d aˆ + ν cˆΓd aˆcˆ.
(4.17)
The antisymmetry of the FermiWalker structure functions C(fw) dˆ ˆa allows us to deﬁne a FermiWalker angular velocity vector ζ(fw) by ˆ
f C(fw) dˆ ˆa = −dˆ ˆafˆζ(fw) ,
(4.18)
which depends entirely on the properties of the tetrad {eαˆ }. Here we set dˆ ˆafˆ = η(u)dˆ ˆafˆ from (3.19). It is easy to verify that ˆ
C(fw) d aˆ edˆ = ζ(fw) ×u eaˆ .
(4.19)
In addition, from the antisymmetry of the ﬁrst two indices of the Ricci rotation coeﬃcients, the second term on the righthand side of (4.16) can be written as ˆ
ˆ
ˆ
f ν cˆΓd aˆcˆ = −d aˆfˆζ(sc) ,
(4.20)
where ζ(sc) is an instantaneous angular velocity vector, termed the spatial curvature angular velocity, which illustrates the behavior of the vector ﬁeld eaˆ γ along ν(U, u) in LRSu . It is easy to verify that ˆ
ν cˆΓd aˆcˆedˆ = ζ(sc) ×u eaˆ .
(4.21)
In terms of (4.18) and (4.20), Eq. (4.16) can be written as D(fw,U,u) eaˆ = [ζ(fw) + ζ(sc) ] ×u eaˆ . dτ(U,u)
(4.22)
Clearly, (4.22) allows us to describe the behavior along γ of any tensor deﬁned in LRSu . Consider as an example the unitary relative velocity direction νˆ(U, u) = νˆ(U, u)aˆ eaˆ ≡ νˆ. From (4.16) it follows that D(fw,U,u) νˆ = dτ(U,u)
D(fw,U,u) νˆ dτ(U,u)
dˆ edˆ
ˆ d νˆd ˆ ˆ = + νˆaˆ C(fw) d aˆ + ν cˆΓd aˆcˆ edˆ. dτ(U,u)
(4.23)
Recalling the deﬁnition of the centripetal part of the FermiWalker acceleration of the particle U relative to u given in (3.165), namely ν
D(fw,U,u) νˆ (C) = a(fw,U,u) , dτ(U,u)
(4.24)
4.1 Orthonormal frames we have
(C) a(fw,U,u)
=ν
63
d νˆd e ˆ + ζ(fw) + ζ(sc) ×u ν. dτ(U,u) d ˆ
(4.25)
Tetrads {u = eˆ0 , eaˆ } may be further speciﬁed by particular transport laws for the triad {eaˆ } along the world line of u itself, as we will show next.
FermiWalker frames A vector ﬁeld X is said to be FermiWalker transported along the congruence Cu when its FermiWalker derivative along Cu is equal to zero, i.e., from (2.83) D(fw,u) X = ∇u X + [a(u) ∧ u] X = 0. dτu
(4.26)
If X = u, then the above relation holds identically. If X = eaˆ we have instead D(fw,u) eaˆ = ∇u eaˆ − a(u)aˆ u dτu ˆ
= a(u)aˆ u + C(fw) b aˆ eˆb − a(u)aˆ u ˆ
= C(fw) b aˆ eˆb = 0.
(4.27)
ˆ
Since {eˆb } is a basis in LRSu , then C(fw) b aˆ = 0. In this case the tetrad frame is termed FermiWalker.
Absolute FrenetSerret frames A FrenetSerret frame {Eαˆ } (α = 0, 1, 2, 3) along a timelike curve with unit tangent vector U is deﬁned as a solution of the following equations: DEˆ0 = κEˆ1 , dτU
DEˆ1 = κEˆ0 + τ1 Eˆ2 , dτU
DEˆ2 = −τ1 Eˆ1 + τ2 Eˆ3 , dτU
DEˆ3 = −τ2 Eˆ2 , dτU
(4.28)
where Eˆ0 = U and ˆ
κ = Γ1 ˆ0ˆ0 ,
ˆ
τ1 = C(fw) 2 ˆ1 ,
ˆ
τ2 = C(fw) 3 ˆ2
(4.29)
are respectively the magnitude of the acceleration and the ﬁrst and second torsions of the world line. τ1 and τ2 are the components of the FrenetSerret angular velocity ω(FS) = τ1 Eˆ3 + τ2 Eˆ1 of the spatial triad {Eaˆ } with respect to a FermiWalker frame deﬁned along U . The frame {Eαˆ } (with dual W αˆ ) is termed an absolute FrenetSerret frame.
64
Special frames In compact form we have DEαˆ ˆ = Eβˆ C β αˆ , dτU
(4.30)
C = CK + CT .
(4.31)
with
Here ˆ
ˆ
CK = κ(Eˆ1 ⊗ W 0 + Eˆ0 ⊗ W 1 ),
(4.32)
and ˆ
ˆ
ˆ
ˆ
CT = τˆ1 (Eˆ1 ⊗ W 2 + Eˆ2 ⊗ W 1 ) + τ2 (Eˆ2 ⊗ W 3 − Eˆ3 ⊗ W 2 ),
(4.33)
W αˆ being the dual frame of Eαˆ . Therefore ˆ
ˆ
CK = −κ[W 0 ∧ W 1 ] = −[a(U ) ∧ U ] ,
(4.34)
and we have the following relation for the FermiWalker derivative along U of a generic vector X: D(fw,U )X DX = − CK dτU dτU
X.
(4.35)
This form of the FermiWalker derivative along a timelike world line will be used to generalize to the case of a null world line (see, for instance, Bini et al., 2006).
Relative FrenetSerret frames Let us consider again a curve γ with tangent vector ﬁeld U and a family of observers u represented by the congruence Cu of its integral curves. These cross the world line of U and at each of its points one deﬁnes a vector representing the (unique) 3velocity of the particle U , as measured locally by the observer u, namely ν(U, u) = ν νˆ(U, u),
(4.36)
recalling that U = γ[u + ν νˆ(U, u)]; clearly ν(U, u) ∈ LRSu . We shall now construct, along the world line of U , a frame in LRSu following a FrenetSerret procedure. The ﬁrst vector of this frame is chosen as the unit spacelike direction νˆ(U, u). The remaining two vectors are the relative normal ηˆ(fw,U,u) which, apart from the sign, is deﬁned as the normalized P (u)projected covariant derivative of νˆ(U, u) along U and its cross product with νˆ(U, u), namely the relative binormal ν (U, u), ηˆ(fw,U,u) , βˆ(fw,U,u) } βˆ(fw,U,u) = νˆ(U, u) ×u ηˆ(fw,U,u) . Let {E(fw,U,u)ˆa } = {ˆ be this frame.
4.1 Orthonormal frames
65
Projecting the covariant derivatives along U of these vectors onto LRSu and reparameterizing the derivatives with respect to the spatial arc length d (U,u) = γ(U, u)ν(U, u)dτU ,
(4.37)
D(fw,U,u) D(fw,U,u) E(fw,U,u)ˆa = γν E(fw,U,u)ˆa , dτU d (U,u)
(4.38)
namely
leads to the relative FrenetSerret equations: D(fw,U,u) νˆ(U, u) = k(fw,U,u) ηˆ(fw,U,u) , d (U,u) D(fw,U,u) ηˆ(fw,U,u) = −k(fw,U,u) νˆ(U, u) + τ(fw,U,u) βˆ(fw,U,u) , d (U,u) D(fw,U,u) ˆ β(fw,U,u) = −τ(fw,U,u) ηˆ(fw,U,u) . d (U,u)
(4.39) (4.40) (4.41)
Here the coeﬃcients k(fw,U,u) and τ(fw,U,u) are respectively the relative FermiWalker curvature and torsion of the world line of U as measured by u. The frame E(fw,U,u)ˆa is termed a relative FrenetSerret frame. Equation (4.39) does not allow one to ﬁx the signs of the curvature k(fw,U,u) and of the frame vector ηˆ(fw,U,u) individually. Once these signs are ﬁxed, although arbitrarily, the third vector βˆ(fw,U,u) = νˆ(U,u) ×u ηˆ(fw,U,u) is chosen so as to make the frame righthanded, and this ﬁxes the sign of the torsion. The relative FrenetSerret equations can be written in a more compact form as D(fw,U,u) E(fw,U,u)ˆa = ω(fw,U,u) ×u E(fw,U,u)ˆa , d (U,u)
(4.42)
ω(fw,U,u) = τ(fw,U,u) νˆ(U, u) + k(fw,U,u) βˆ(fw,U,u)
(4.43)
where
deﬁnes the relative FrenetSerret angular velocity of the spatial frame. The world line with 4velocity U is said to be urelatively straight if the FermiWalker relative curvature vanishes, k(fw,U,u) = 0; and urelatively ﬂat if the FermiWalker relative torsion vanishes, τ(fw,U,u) = 0. When the relative curvature vanishes identically the relative FrenetSerret frame can still be deﬁned by adopting an appropriate limiting procedure. On the other hand, when it vanishes only at an isolated point, one must allow it to have either sign when it is nonzero in order to extend the relative normal continuously through this point.
Comoving relative FrenetSerret frame Consider again a test particle with 4velocity U being the target of an observer u. The centripetal acceleration of the particle is measured by the observer u in
66
Special frames
his own LRSu , while the centrifugal acceleration is an “inertial acceleration” measured by the particle U in its own LRSU . In Newtonian mechanics these rest frames coincide, the time being absolute; therefore one can simply consider the centripetal acceleration as opposite to the centrifugal one. In general relativity the observer and the test particle in relative motion have diﬀerent rest frames, so a comparison between the two accelerations is only possible with a suitable geometrical representation of the centrifugal acceleration of the test particle in the rest frame of the observer. If u is a vector ﬁeld and Cu is the congruence of its integrable curves, at each point where the particle U crosses a curve of Cu the vectors u and U deﬁne a boost of (u, ν(U, u)) into (U, −ν(u, U )) and leaves the orthogonal 2space LRSu ∩LRSU invariant, as discussed in Section 3.3. Let ˆ U ) = −ˆ V(u, ν (u, U ) = γ(νu + νˆ(U, u))
(4.44)
be the negative of the relative velocity of u with respect to U as in (3.114). The projection into the rest space of U of the covariant derivative of any vector ﬁeld X deﬁned on the world line of U and lying in LRSU leads to the usual FermiWalker derivative along U : D(fw,U ) X D(fw,U ) X DX = P (U ) = γν , dτU dτU d (U,u)
X ∈ LRSU .
(4.45)
ˆ U ) by adding One can construct a FrenetSerretlike frame starting from V(u, ˆ(fw,u,U ) being the normal and Bˆ(fw,u,U ) two new vectors, both lying in LRSU , N the binormal of the world line of U . They are solutions of the FrenetSerret relations D(fw,U ) ˆ ˆ(fw,u,U ) , V(u, U ) = K(fw,u,U ) N d (U,u) D(fw,U ) ˆ ˆ U ) + T(fw,u,U ) Bˆ(fw,u,U ) , N(fw,u,U ) = −K(fw,u,U ) V(u, d (U,u) D(fw,U ) ˆ ˆ(fw,u,U ) . B(fw,u,U ) = −T(fw,u,U ) N d (U,u)
(4.46)
Let us now prove that the following relation holds: ˆ(fw,u,U ) = γk(fw,U,u) ηˆ(fw,U,u) K(fw,u,U ) N (G)
+ νˆ(U, u) ×u [ˆ ν (U, u) ×u F(fw,U,u) ].
(4.47)
First recall that the FermiWalker derivative along U of a vector orthogonal to U reduces to the projection on LRSU of its covariant derivative along U itself. Hence, diﬀerentiating (4.44) with respect to the parameter (U, u), we obtain
4.1 Orthonormal frames
67
ˆ U) D(fw,U ) V(u, D 1 P (U ) = [γ(νu + νˆ(U, u))] d (U,u) γν dτU d(γν) Du dγ Dˆ ν (U, u) 1 P (U ) u + γν + νˆ(U, u) +γ = γν dτU dτU dτU dτU dγ dγ Dˆ ν (U, u) 1 (G) P (U ) u − γνF(fw,U,u) + νˆ(U, u) +γ = γν νdτU dτU dτU dγ u Dˆ ν (U, u) 1 (G) P (U ) + νˆ(U, u) − γνF(fw,U,u) + γ , = γν dτU ν dτU (4.48) where we have used (3.156) and the relation dγ dν = γ3ν . dτU dτU
(4.49)
The ﬁrst term in square brackets in the last line of (4.48) is along U and therefore vanishes after contraction with P (U ); hence we have ˆ U) D(fw,U ) V(u, 1 Dˆ ν (U, u) (G) P (U ) −γνF(fw,U,u) + γ . = d (U,u) γν dτU
(4.50)
The last term on the righthand side of (4.48) can be written as Dˆ ν (U, u) Dˆ ν (U, u) Dˆ ν (U, u) = P (u) − uu · dτU dτU dτU Dˆ ν (U, u) Du = P (u) + uˆ ν (U, u) · dτU dτU Dˆ ν (U, u) (G) = P (u) − uˆ ν (U, u) · F(fw,U,u) dτU (G)
= γνk(fw,U,u) ηˆ(fw,U,u) − uˆ ν (U, u) · F(fw,U,u) .
(4.51)
The projection orthogonal to U of this quantity is
P (U )
Dˆ ν (U, u) (G) = γνk(fw,U,u) ηˆ(fw,U,u) − νˆ(U, u) · F(fw,U,u) [P (U )u], dτU
(4.52)
noting that P (U )ˆ η(fw,U,u) = ηˆ(fw,U,u) + U U · ηˆ(fw,U,u) = ηˆ(fw,U,u) ,
(4.53)
68
Special frames
since U = γ[u + ν νˆ(U, u)] and νˆ(U, u) · ηˆ(fw,U,u) = 0. From the above it then follows that ˆ U) D(fw,U ) V(u, (G) = −P (U )F(fw,U,u) + γk(fw,U,u) ηˆ(fw,U,u) d (U,u) (G)
νˆ(U, u) · F(fw,U,u)
(u − γU ) ν (G) (G) = γk(fw,U,u) ηˆ(fw,U,u) − F(fw,U,u) − U γν νˆ(U, u) · F(fw,U,u) −
(G)
νˆ(U, u) · F(fw,U,u)
(u − γU ) ν (G) = γk(fw,U,u) ηˆ(fw,U,u) − F(fw,U,u) u − γU (G) − νˆ(U, u) · F(fw,U,u) γνU + ν −
(G)
(G)
= γk(fw,U,u) ηˆ(fw,U,u) − F(fw,U,u) + νˆ(U, u) · F(fw,U,u) νˆ(U, u). Finally, ˆ U) D(fw,U ) V(u, = γk(fw,U,u) ηˆ(fw,U,u) d (U,u) (G) + νˆ(U, u) ×u νˆ(U, u) ×u F(fw,U,u) ,
(4.54)
which completes the proof. ˆ U ) yields The cross product of (4.47) with V(u, ˆ U ) ×U N ˆ(fw,u,U ) = γk(fw,U,u) V(u, ˆ U ) ×U ηˆ(fw,U,u) K(fw,u,U ) V(u, (G)
ˆ U ) ×U νˆ(U, u) + (ˆ ν (U, u) · F(fw,U,u) )V(u, (G)
ˆ U ) ×U F − V(u, (fw,U,u) ,
(4.55)
that is ˆ U ) ×U ηˆ(fw,U,u) K(fw,u,U ) Bˆ(fw,u,U ) = γk(fw,U,u) V(u, (G) ˆ U ) ×U νˆ(U, u) + (ˆ ν (U, u) · F(fw,U,u) )V(u, (G)
ˆ U ) ×U F − V(u, (fw,U,u) .
(4.56)
To proceed further it is necessary to evaluate the cross product in LRSU of a vector X(U ), for example, which belongs to LRSU , with a vector Y (u), for example, which belongs to LRSu . The result is the following: X(U ) ×U Y (u) = γ {[P (u, U )X(U ) ×u Y (u)] − (ν · P (u, U )X(U ))(ν ×u Y ) − u[P (u, U )X(U ) · (ν ×u Y (u))]} .
(4.57)
4.2 Null frames
69
ˆ U ) and In our case X(U ) = V(u, ˆ U ) = γ νˆ(U, u) ; P (u, U )V(u,
(4.58)
therefore we have ˆ U ) ×U Y (u) = γ 2 [ˆ V(u, ν (U, u) ×u Y (u)] − γ 2 ν 2 [ˆ ν (U, u) ×u Y (u)] = νˆ(U, u) ×u Y (u).
(4.59)
From (4.56) it then follows that K(fw,u,U ) Bˆ(fw,u,U ) = γk(fw,U,u) νˆ(U, u) ×u ηˆ(fw,U,u) (G)
− νˆ(U, u) ×u F(fw,U,u) (G) = γk(fw,U,u) βˆ(fw,U,u) − νˆ(U, u) ×u F(fw,U,u) .
(4.60)
The spatial triad ˆ(fw,u,U ) } ˆ U ), N ˆ(fw,u,U ) , Bˆ(fw,u,U ) = V(u, ˆ U ) ×U N {E(fw,u,U )ˆa } = {V(u,
(4.61)
is termed a comoving relative FrenetSerret frame. In compact form the comoving relative FrenetSerret relations become D(fw,U ) E(fw,u,U )ˆa = Ω(fw,u,U ) ×U E(fw,u,U )ˆa , d (U,u)
(4.62)
ˆ U ) + K(fw,u,U ) Bˆ(fw,u,U ) deﬁnes the comoving relwhere Ω(fw,u,U ) = T(fw,u,U ) V(u, ative angular velocity of this new frame with respect to a FermiWalker spatial frame. Again no assumption is made about the choice of signs for the curvature and torsion in this general discussion.
4.2 Null frames Null frames include null vectors, and are particularly convenient for treating null ﬁelds such as electromagnetic and gravitational radiation. These frames can be complex or real.
Complex (NewmanPenrose) null frames A (complex, null) NewmanPenrose frame {l, n, m, m} ¯ (Newman and Penrose, 1962) can be associated with an orthonormal frame {eαˆ } in the following way: 1 l = √ [eˆ0 + eˆ1 ], 2 1 m = √ [eˆ2 + ieˆ3 ], 2
1 n = √ [eˆ0 − eˆ1 ], 2 1 m ¯ = √ [eˆ2 − ieˆ3 ], 2
(4.63)
with l ·n = −1 and m· m ¯ = 1. The four vectors l, n, m, m ¯ are null vectors (l ·l = 0, n · n = 0, m · m = 0, m ¯ ·m ¯ = 0); two of them are real (l, n) and the other two
70
Special frames
are complex conjugate (m, m). ¯ The frame components of the metric are then equal to ⎛ ⎞ 0 −1 0 0 ⎜−1 0 0 0⎟ ⎟. (gab ) = (g ab ) = ⎜ (4.64) ⎝0 0 0 1⎠ 0 0 1 0 Real null frames Instead of a complex null frame it is useful to introduce a real null frame made up of two null vectors and two spacelike vectors. Choosing the spatial vectors orthogonal to each other and normalized to unity, one constructs a quasiorthogonal real null frame. Denoting such a frame as {Eα } (α = 1, 2, 3, 4), the standard relation with an orthonormal frame {eαˆ } is the following: 1 E1 = √ (eˆ0 + eˆ2 ), 2 E2 = eˆ1 , 1 E3 = √ (eˆ0 − eˆ2 ), 2 E4 = eˆ3 . The frame components of the metric are given by ⎛ 0 0 −1 ⎜0 1 0 (gab ) = (Ea · Eb ) = ⎜ ⎝−1 0 0 0 0 0
(4.65) ⎞ 0 0⎟ ⎟ = (g ab ). 0⎠
(4.66)
1
Let us denote by Π and Π the vector spaces generated by (E1 , E3 ) and (E2 , E4 ), respectively. The null vectors E1 and E3 are deﬁned up to a multiplicative factor, while the spatial vectors E2 and E4 can be arbitrarily rotated in the 2plane Π . Therefore there arises a set of equivalent frames, each being a suitable FrenetSerret frame along a null curve. This frame satisﬁes the following set of FrenetSerret evolution equations: DE1 = −KE2 , dλ
DE2 = T1 E1 − KE3 , dλ
DE4 DE3 = T1 E2 + T2 E4 , = T2 E1 , (4.67) dλ dλ where λ is an arbitrary parameter along E1 and the quantities K, T1 , and T2 play roles analogous to the FrenetSerret curvature and torsions in the timelike case. The connection matrix DEa = Eb C b a (4.68) dλ
4.2 Null frames has components ⎛ 0 ⎜ −K (C a b ) = ⎜ ⎝ 0 0
T1 0 −K 0
0 T1 0 T2
71
⎛
⎞ T2 0⎟ ⎟, 0⎠ 0
0 ⎜ −T 1 (C ab ) = ⎜ ⎝ 0 −T2
T1 0 −K 0
0 K 0 0
⎞ T2 0⎟ ⎟. 0⎠ 0
(4.69)
The invariants of the matrix are given by I1 =
1 Cab C ab = 2KT1 , 2
where (∗ C ab ) =
1 abcd η Ccd 2
I2 =
1 Cab ∗ C ab = 2KT2 , 2
(4.70)
⎞ −T1 0 ⎟ ⎟. −K ⎠ 0
(4.71)
⎛
0 ⎜−T2 =⎜ ⎝ 0 T1
T2 0 0 0
0 0 0 K
Finally, one can evaluate the four complex eigenvalues of the matrix (C a b ), λ1,2 = ±ω ≡ [−KT1 + K(T12 + T22 )1/2 ]1/2 , λ3,4 = ±iχ ≡ i[KT1 + K(T12 + T22 )1/2 ]1/2 ,
(4.72)
which in turn deﬁne two nonnegative quantities ω and χ. Let us consider the decompositions C = CK + CT , C = CK + CT
(4.73)
with CK = −K(E2 ⊗ W 1 + E3 ⊗ W 2 ), = K(E2 ∧ E3 ), CK
(4.74)
and CT = [T1 (E1 ⊗ W 2 + E2 ⊗ W 3 ) + T2 (E1 ⊗ W 4 + E4 ⊗ W 3 )], T1 E2 + T2 E4 2 2 CT = T1 + T2 E1 ∧ 2 , T1 + T22
(4.75)
W a being the dual frame of Ea . The curvature part CK of the connection matrix generates a null rotation by an angle K in the timelike hyperplane of E1 , E2 , E3 , which leaves the null vector E3 ﬁxed (CK E3 = 0). The torsion part CT of the connection matrix generates a null rotation by an angle T12 + T22 in the timelike hyperplane of E1 , E3 , (T12 + T22 )−1/2 (T1 E2 + T2 E4 ), which leaves the null vector E1 ﬁxed (CT E1 = 0).
72
Special frames
Along a Killing trajectory the FrenetSerret scalars are all constant, leading to considerable simpliﬁcation of the above formulas. This is the situation for null circular orbits in stationary axisymmetric spacetimes, for example. There still remains the problem that the whole FrenetSerret frame machinery is still subject to reparameterization, under which only the relative ratios of the curvature and torsions are invariant. The curvature is just the norm of the second derivative, K=
D2 x D2 x · dλ2 dλ2
1/2 ≥ 0,
(4.76)
and it transforms as the square of the related rate of change of the parameters ˆ that is λ → λ, 2 ˆ dλ ˆ . (4.77) K=K dλ ˆ is invariant (when nonzero), like the arc length ˆ 1/2 dλ Thus dΛ = K1/2 dλ = K in the nonnull case, and can be used to introduce a preferred unit curvature parameterization when the curvature is everywhere nonvanishing, deﬁned up to an additive constant like the arc length in the null case. This was already introduced by Vessiot for the Riemannian case in 1905 (Vessiot, 1905), who called Λ the pseudoarc length. The full transformation of all the FrenetSerret quantities is easily calculated. Letting Λ = dΛ/dλ and using a hatted notation for the new quantities, one ﬁnds that the frame vectors undergo a boost and a null rotation, with the invariance of the last vector conﬁrming the invariant nature of the osculating hyperplane for which it is the unit normal: ˆ2 = E2 + ζE1 , ˆ1 = (Λ )−1 E1 , E E ζ ˆ ˆ 4 = E4 , E3 = Λ E3 + ζ E1 + E2 , E 2
(4.78)
where ζ = Λ /(KΛ ), and the new FrenetSerret quantities are given by ˆ = (Λ )−2 K, K
K Tˆ1 = T1 + ζ + ζ 2 , 2
Tˆ2 = T2 .
(4.79)
The pseudoarc length parameterization is obtained by setting Λ = K1/2 (so that ζ = K /(2K2 )), and the new form of the FrenetSerret equations, once hats are dropped, can be obtained from (4.67) simply by making the substitutions K → 1 and λ → Λ. Note that, in the case of constant curvature, Λ = 0 (and ζ = 0); this is just a simple aﬃne parameter transformation and both torsions remain unchanged, while the frame undergoes a simple boost constant rescaling.
4.2 Null frames
73
FermiWalker transport along the null world line Exactly as in the case of a timelike curve, one may retain only the torsion part of the connection matrix while absorbing the curvature part into the derivative to deﬁne a corresponding generalized FermiWalker transport along the null world line, as follows: D(fw) X α DX α = − CK α β X β = 0. (4.80) dλ dλ
5 The world function
Spatial and temporal intervals between any two points in spacetime are physically meaningful only if their measurement is made along a curve which joins them. A curve is a natural bridge which allows one to connect the algebraic structures at diﬀerent points; therefore it is an essential tool to any measurement procedure. The mathematical quantities which glue together the concepts of points and curves are the twopoint functions.1 The most important of these is the world function, ﬁrst introduced by Ruse (1931) and then used by Synge (1960). As Synge himself realized, the world function is well deﬁned only locally; however, a global generalization has recently been found by Cardin and Marigonda (2004), opening the way to a better understanding of its potential in the theory of relativity. In de Felice and Clarke (1990) the world function was exploited to deﬁne spatial and temporal separations between points, and to ﬁnd curvature eﬀects in the measurement of angles and of relative velocities. Following their work we shall recall the main properties and applications of a world function, starting from its very deﬁnition. Consider a smooth curve γ, parameterized by s ∈ . Let γ˙ be the ﬁeld of vectors tangent to γ. The quantity ! 1 γ˙ · γ ˙ 2 ds (5.1) L= γ
does not depend on the parameter s and therefore it provides the seed for a measurement of length on the curve. Given two points P0 and P1 in the manifold, there exist an inﬁnite number of smooth curves γ joining them. Hence, setting P0 = γ(s0 ) and P1 = γ(s1 ), the quantity ! s1 1 γ˙ · γ ˙ 2 ds (5.2) L(s0 , s1 ; γ) = s0 1
One should say manypoint functions in general but we shall only consider two and threepoint functions.
5.1 The connector
75
is a pathdependent function of the two points; clearly it vanishes if the curve is null. We use the term world function for the quantity Ω(s0 , s1 ; γ) =
1 2 L , 2
(5.3)
which is a real twopoint function. The world function is not a map between tensor ﬁelds at diﬀerent points; therefore meaningful applications require a transport law along any given curve.
5.1 The connector Transport laws underlie the diﬀerential operations on the manifold; examples of these are the Lie and absolute derivatives. For the purpose of our analysis we need to discuss in some detail the transport law which leads to the absolute derivative. Given a curve γ with parameter s and two points on it, P0 = γ(s0 ) and P1 = γ(s1 ), we use the term connector on γ for a map Γ(s0 , s1 ; γ) : TP0 (M ) → TP1 (M ) which carries vectors from
to
P0
P1 .
(5.4)
The eﬀect of this map is described as
ˇ(1) Γ(s0 , s1 ; γ)u(0) = u
(5.5)
for any u0 ∈ TP0 (M ) and with u ˇ1 ∈ TP1 (M ). Following de Felice and Clarke (1990), we summarize the main properties of this map. The connector is subject to the following conditions: (i) Linearity. We require Γ to be a linear map, i.e. for any set of real numbers cA and vectors u(A) at P0 , we have Γ(s0 , s1 ; γ)(cA u(A) ) = cA Γ(s0 , s1 ; γ)u(A) . (ii) Consistency. If γ joins P2 = γ (s2 ) then
P0
= γ(s0 ) to
P1
= γ(s1 ) and γ joins
Γ(s1 , s2 ; γ )Γ(s0 , s1 ; γ) = Γ(s0 , s2 ; γ ◦ γ ),
(5.6) P1
= γ (s1 ) to (5.7)
where the symbol ◦ means the concatenation of the curves. Hence the eﬀect of carrying a vector from P0 to P1 along γ and then from P1 to P2 along γ is equivalent to carrying that vector from P0 to P2 along the curve γ ◦ γ . (iii) Parameterization independence. The result of the action of Γ along a given curve does not depend on its parameterization. (iv) Diﬀerentiability. We require that the result of the application of Γ(s0 , s1 ; γ) varies smoothly if we vary the points P0 and P1 and deform the whole path between them. The consequences of the above conditions allow one to describe the connector as a tensorlike twopoint function. Let {eα } be a ﬁeld of bases on some open set of
76
The world function
the manifold containing the pair of points and the curve connecting them. From linearity we have Γ(s0 , s1 ; γ)u(0) = uα0 Γ(s0 , s1 ; γ)eα0 = uα0 Γ(s0 , s1 ; γ)α0β1 eβ1 =u ˇβ1 eβ1 ,
(5.8)
where the indices with subscripts 0 and 1 refer to quantities deﬁned at the points P0 and P1 respectively. From the property of a basis we have u ˇβ1 = uα0 Γ(s0 , s1 ; γ)α0β1 ,
(5.9)
where the coeﬃcients Γ(s0 , s1 ; γ)α0β1 are the components of the connector. We shall adopt the convention that the ﬁrst index refers to the tangent basis at the ﬁrst endpoint of Γ and the second index to the tangent basis at the second endpoint. From the above relation it follows that Γ(s0 , s0 ; γ)α0β0 = δαβ00 .
(5.10)
The requirement of consistency by concatenation implies that Γ(s0 , s1 ; γ)σ0 μ1 Γ(s1 , s2 ; γ)μ1 ν2 = Γ(s0 , s2 ; γ)σ0 ν2 .
(5.11)
Finally the requirement of diﬀerentiability, coupled with all the other properties of the connector, leads to the following law of diﬀerentiation (see de Felice and Clarke, 1990, for details): " " d ρ" Γ(s0 , s; γ)α0 " = −Γ(s0 , s; γ)α0 μ Γρ μν (s)γ˙ ν (s), (5.12) ds s with the initial condition given by (5.10). Here Γρ μν are the connection coeﬃcients on the manifold; they only depend on the point and not on the curves crossing it. It is well established that the connection coeﬃcients are not the components of a 12 tensor; nevertheless they behave as the components of such a tensor under linear coordinate transformations. This follows from the transformation properties of the components of the connector Γ(s0 , s1 ; γ)α0 β1 ; the latter behave as a covector at P0 and as a vector at P1 . From (5.9) and with respect to a general coordinate transformation x (x) we have Γ (s0 , s1 ; γ)α0 β1 =
β1 ∂xμ0 ν1 ∂x Γ(s , s ; γ) . 0 1 μ 0 ∂xα0 ∂xν1
(5.13)
Hence from (5.12) it follows that Γλ μν (x ) =
∂xλ ∂xβ ∂xγ α ∂xλ ∂ 2 xρ Γ (x) − . βγ ∂xα ∂xμ ∂xν ∂xρ ∂xμ ∂xν
(5.14)
A basic property of spacetime geometry is the compatibility of the connection with the metric, expressed by the identity ∇α gμν ≡ 0.
(5.15)
5.2 Mathematical properties of the world function
77
This relation stems from the requirement that the connector preserves the scalar product. Along any curve connecting points P0 and P1 we have u · vˇ)1 , (u · v)0 = (ˇ
(5.16)
ˇ, vˇ ∈ TP1 (M ). with u, v ∈ TP0 (M ) and u Let us conclude this brief summary by recalling the concept of geodesic on the manifold. A curve γ connecting P0 to P1 is geodesic if Γ(s0 , s1 ; γ)γ˙ (0) = f (s1 )γ˙ (1) ,
(5.17)
where f (s) is a diﬀerentiable function on γ. As stated, a geodesic can always be reparameterized with s (s) so that f (s (s)) = 1; in this case the parameter s is termed aﬃne. An aﬃne parameter is deﬁned up to linear transformations.
5.2 Mathematical properties of the world function The world function behaves as a scalar at each of its endpoints and can be diﬀerentiated at each of them separately; in this case it generates new functions which may behave as a scalar at one point and a vector or 1form at the other, or as a 1form at the ﬁrst point and as a vector at the other, and so on. To deﬁne the derivatives of a world function we shall consider only those whose endpoints are connected by a geodesic. If we further restrict our analysis to normal neighborhoods then the geodesic connecting any two points is unique. The world function can be written as 1 (5.18) Ω(s0 , s1 ; Υ) = (s1 − s0 )2 X · X, 2 ˙ denotes the tangent vector to the unique geodesic Υ joining P0 to P1 where X = Υ and parameterized by s. We shall now deduce the derivatives of Ω with respect to variations of its two endpoints. Let P0 and P1 belong to smooth curves γ˜0 and γ˜1 , parameterized by t, and let Υt ≡ Υγ˜0 (t)→˜γ1 (t) be the geodesic connecting points of γ˜0 to points of γ˜1 . We then require that the curve connecting the points P0 and P1 as they vary independently on the curves γ˜0 and γ˜1 is the unique geodesic ΥP0 →P1 . In this case the points we are considering belong also to the geodesic Υt so they can be referred to as P0 = Υt (s0 ) and P1 = Υt (s1 ). This situation is depicted in Fig. (5.1). We then have a oneparameter family of geodesics CX with connecting vector Y = γ˜˙ , namely £X Y = 0. By deﬁnition we have DX = 0, ds To pursue our task let us write
DX DY = . dt ds
d ∂Ω α0 ∂Ω α1 Ω(Υt (s0 ), Υt (s1 ); Υt ) = Y + Y α α1 0 dt ∂x ∂x DX ·X , = (s1 − s0 )2 dt
(5.19)
(5.20)
78
The world function
ϒt + δt
γ∼ (t + δt)
P0′= 0
γ∼ (t + δt)
P1 ′= 1
Y X
ϒt
γ∼ (t)
P0 = 0
γ∼0 (t)
γ∼1 (t)
γ∼ (t)
P1 = 1
Fig. 5.1. The points P0 = γ˜0 (t) and P1 = γ˜1 (t) vary on the curves γ˜0 and γ˜1 and remain connected by the unique geodesic Υγ˜0 (t)→˜γ1 (t) . The parameter on the geodesic Υt is s with tangent vector X while that on the curves γ˜0 and γ˜1 is t with tangent vector Y .
where {xα0 = xα (s0 )} and {xα1 = xα (s1 )} are the local coordinates for Υt (s0 ) and Υt (s1 ) respectively. From the equation for geodesic deviation, D2 Y = R(X, Y )X, ds2
(5.21)
D2 Y · X ≡ 0. ds2
(5.22)
we deduce identically that
From (5.19)1 we then have DY d d D X· = (X · Y ) = 0, ds ds ds ds
(5.23)
which implies that d (X · Y ) = κ, ds
(5.24)
where κ is a constant along Υt (s). Integration along Υt (s) from s0 to s1 yields s
κ(s1 − s0 ) = [X · Y ]s10 ;
(5.25)
hence we can write (5.24) more conveniently as d s (X · Y ) = (s1 − s0 )−1 [X · Y ]s10 . ds
(5.26)
Setting Ωα0 =
∂Ω , ∂xα0
Ωα1 =
∂Ω , ∂xα1
(5.27)
5.2 Mathematical properties of the world function and recalling (5.19)2 , Eq. (5.20) becomes dΩ = Ωα0 Y α0 + Ωα1 Y α1 = (s1 − s0 )2 dt = (s1 − s0 ) [Xα1 Y α1 − Xα0 Y α0 ] ;
DY ·X ds
79
(5.28)
thus Ωα0 = −(s1 − s0 )Xα0 ,
Ωα1 = (s1 − s0 )Xα1 .
(5.29)
From the latter we have Ωα0 Ωα0 = Ωα1 Ωα1 = (s1 − s0 )2 X · X = 2Ω,
(5.30)
g α0 β0 Ωα0 Ωβ0 = g α1 β1 Ωα1 Ωβ1 = 2Ω.
(5.31)
or equivalently
The quantities in (5.29) are the components of 1forms deﬁned respectively at Υt (s0 ) and Υt (s1 ). Hence diﬀerentiating one of them, say Ω (s0 ), with respect to t, we obtain DΩ (0) DX β0 β1 = Ωα0 β0 Y + Ωα0 β1 Y = −(s1 − s0 ) , (5.32) dt dt α0 α0 where Ωα0 β0 = ∇β Ωα (s0 ) etc. From (5.19)2 we have
DX dt
= s0
DY ds
,
(5.33)
s0
but now we need to know the derivative of the connecting vector ﬁeld Y . This can be obtained from the general solution of the equation for geodesic deviation (de Felice and Clarke, 1990); we shall omit here the detailed derivation, which can be found in the cited literature, and provide only the results. To ﬁrst order in the curvature, α0 DY = −αY α0 + αΓβ1 α0 Y β1 ds ! s1 − α2 Y ρ0 (s1 − s)2 K β τ Γρ0 τ Γβ α0 ds s ! 0s1 2 ρ1 (s1 − s)(s − s0 )K σ τ Γρ1 τ Γσ α0 ds −α Y s0
+ O(Riem2 ),
(5.34)
where O(Riem2 ) means terms of order 2 or larger in the curvature, α ≡ (s1 − s0 )−1 , and K α ρ = Rα μνρ X μ X ν .
(5.35)
80
The world function
To simplify notation, we also set Γα0 β ≡ Γ(s0 , s; Υt )α0 β , these being the components of the connector deﬁned on the curve Υt with parameter s. From (5.19)1 and (5.32) we ﬁnally have ! s1 Ωα0 β0 = gα0 β0 + α gα0 γ0 (s1 − s)2 K ρ τ Γρ γ0 Γβ0 τ ds s0
+ O(Riem2 ),
!
Ωα0 β1 = −gα0 γ0 Γβ1 γ0 + α gα0 γ0
(5.36) s1
(s1 − s)(s − s0 )K ρ τ Γβ1 τ Γρ γ0 ds
s0
+ O(Riem2 ).
(5.37)
The values of the world function and its derivatives in the limit of coincident endpoints are easily obtained as lim Ω = lim Ωα = 0
s1 →s0
s1 →s0
lim Ωαβ = gαβ (s0 ).
s1 →s0
(5.38)
It is worth pointing out here that these limiting values do not depend on the path along which the endpoints have been made to coincide. When the spacetime admits a set of n Killing vectors, say ξ(a) with a = 1 . . . n, a world function connecting points on a geodesic satisﬁes the following property: α0 α1 Ωα0 + ξ(a) Ωα1 = 0. ξ(a)
(5.39)
Using (5.39) and (5.31), one obtains the explicit form of the world function in special situations. The simplest example is found in Minkowski spacetime. Since in this case the geodesics are straight lines, the world function is just Ωﬂat (x0 , x1 ) =
1 β β α ηαβ (xα 0 − x1 )(x0 − x1 ), 2
(5.40)
whatever geodesic connects P0 to P1 . Recent applications of the world function to detect the time delay and frequency shift of light signals are due to Teyssandier, Le PoncinLaﬁtte, and Linet (2008). Because of the potential of the world function approach to measurements, we judge it useful to derive its analytical form in various types of spacetime metrics.
5.3 The world function in Fermi coordinates Consider a general spacetime metric and introduce a Fermi coordinate system (T, X, Y, Z) in some neighborhood of an accelerated world line γ with (constant) acceleration A; the spatial coordinates X, Y, Z span the axes of a triad which is FermiWalker transported along γ while T measures proper time at the origin
5.3 The world function in Fermi coordinates
81
of the spatial coordinates. Up to terms linear in the spatial coordinates, one has (see (6.18) of Misner, Thorne, and Wheeler, 1973) ds2 = (ηαβ + 2AXδα0 δβ0 )dX α dX β = −(1 − 2AX)dT 2 + dX 2 + dY 2 + dZ 2 + O(2),
(5.41)
a form which is valid within a world tube of radius 1/A so that AX 1 is the condition for this approximation to be correct. In what follows we shall give general expressions for both timelike and null geodesics of the Fermi metric (5.41), as well as the expression for the world function (Bini et al., 2008). To ﬁrst order in the acceleration parameter A, the timelike geodesics can be written explicitly in terms of an aﬃne parameter λ as T (λ) = T (0) + Cλ + ACλ(C X λ + X(0)), 1 X(λ) = X(0) + C X λ + AC 2 λ2 , 2 Y Y (λ) = Y (0) + C λ, Z(λ) = Z(0) + C Z λ,
(5.42)
where C, C X , C Y , C Z are integration constants. α α and XB be the Fermi coordinates of two general spacetime points A Let XA and B connected by a geodesic. By using the explicit expressions (5.42) of the geodesics, and from the deﬁnition of the world function, we have 1 [−C 2 + (C X )2 + (C Y )2 + (C Z )2 ], (5.43) 2 where the orbital parameters have to be replaced by the coordinates of the initial and ﬁnal points. Choosing the aﬃne parameter in such a way that λ = 0 corresponds to XA and λ = 1 to XB we get the conditions Ω(XA , XB ) =
0 XA = T (0),
1 XA = X(0),
2 XA = Y (0),
3 XA = Z(0),
(5.44)
and 0 0 1 XB = XA + C + AC(C X + XA ), 1 1 1 XB = XA + C X + AC 2 , 2 2 2 XB = XA + CY , 3 3 = XA + CZ . XB
(5.45)
Next, solving the latter equations for C, C X , C Y , C Z yields C CX CY
0 0 1 (XB − XA )(1 − AXB ), 1 1 1 0 0 2 (XB − XA ) − A(XB − XA ) , 2 2 2 = XB − XA ,
3 3 − XA C Z = XB
(5.46)
82
The world function
to ﬁrst order in A. Substituting then in Eq. (5.43) gives the following ﬁnal expression for the world function: α 1 β β 1 1 α ηαβ + A(XA + XB )δα0 δβ0 (XA − XB )(XA − XB ) Ω(XA , XB ) 2 1 1 1 0 0 2 + XB )(XA − XB ) , (5.47) = Ωﬂat (XA , XB ) + A(XA 2 to ﬁrst order in the acceleration parameter A.
5.4 The world function in de Sitter spacetime In isotropic and homogeneous coordinates, the de Sitter metric is given by (Stephani et al., 2003): ds2 = −dt2 + e2H0 t (dx2 + dy 2 + dz 2 ),
(5.48)
where H0 is the Hubble constant. Metric (5.48) satisﬁes Einstein’s vacuum ﬁeld equations with nonvanishing cosmological constant Λ = 3H02 and Weyl ﬂat spatial sections. It describes an empty expanding and nonrotating universe. Let us denote by {t, xi } ({xi } = {x, y, z}, i = 1, 2, 3) the coordinates of an arbitrary event P of this spacetime. A ﬁrst integration of the geodesic equations is easily obtained using the Killing symmetries of the spacetime. In terms of an aﬃne parameter s, it gives 2 dt dxi i −2H0 t =C e , = − + C 2 e−2H0 t , (5.49) ds ds where = 0, −1, +1 correspond to null, timelike, and spacelike geodesics respectively, and {C i } are constants with C 2 = δij C i C j . Introduce a family of ﬁducial observers n = −dt,
n = ∂t ,
(5.50)
with the adapted orthonormal frame etˆ = ∂t ,
eˆi = e−H0 t ∂i .
(5.51)
ˆ The frame components uα ( ) of the vector tangent to the geodesics are given by i −H0 t
C e α ˆ 2 −2H t 0 u( ) = u( ) eαˆ = − + C e n+ √ (5.52) eˆi . − + C 2 e−2H0 t
Equations (5.49) can be easily integrated for the diﬀerent values of (i.e. for any causal character) and the results can be summarized as follows: √ C eH0 t = √ sin[ (σH0 s + α0 )], √ Ci √ xi − xi0 = − 2 cot[ (σH0 s + α0 )], C σH0
(5.53)
5.4 The world function in de Sitter spacetime
83
where xi0 are integration constants and σ is a sign indicator, corresponding to futurepointing (σ = 1) or pastpointing (σ = −1) geodesics, and the null case = 0 is intended in the sense of the limit. Let us require now that s = 0 corresponds to the spacetime point A and s = 1 to B. This implies √ √ C C eH0 tA = √ sin[ α0 ], eH0 tB = √ sin[ (σH0 + α0 )], √ Ci √ xiA − xi0 = − 2 cot[ α0 ], C σH0 √ Ci √ xiB − xi0 = − 2 cot[ (σH0 + α0 )]. C σH0 Moreover, from (5.2)
!
1
L=
ds =

(5.54)
(5.55)
0
and, from (5.3), Ω(XA , XB ) =
1 . 2
(5.56)
Using relations (5.54) one can then obtain as a function of the coordinates of A and B, and hence Ω. To this end it is convenient to write down an equivalent set of equations in place of (5.54) namely √ √ H0 tA e , α0 = arcsin C √ √ √ H0 tB H0 tA e e − arcsin , σH0 = arcsin C C ξ 2 ≡ δij (xiB − xiA )(xjB − xjA ) √ √ = 2 2 [cot[ (σH0 + α0 )] − cot[ α0 ]]2 , C H0
(5.57)
in the three unknowns α0 , C 2 , and . Using the the ﬁrst two of these equations in the third one gives √ √ 2 H0 tB H0 tA 2 2 e e − cot arcsin , (5.58) H0 ξ = 2 cot arcsin C C C √ which can be formally inverted to obtain the quantity /C in terms of tA , tB , xiA , xiB . To see this in detail let us use the notation √ (5.59) w = /C, a = eH0 tA , b = eH0 tB ; we then have
H02 ξ 2 = w2
1 −1− b2 w 2
2
1 −1 a2 w2
,
(5.60)
84
The world function
from which [(a + b)2 − H02 ξ 2 a2 b2 ][H02 ξ 2 a2 b2 − (a − b)2 ] . (5.61) 4ξ 2 a4 b4 H02 √ Backsubstituting this expression for w = /C in the second of Eqs. (5.57) then gives , √ 1 = [arcsin(bw) − arcsin(aw)], (5.62) σH0 w2 =
and hence Ω, from (5.56).
5.5 The world function in G¨ odel spacetime In Cartesianlike coordinates xα = (t, x, y, z), G¨ odel’s metric takes the form (2.133) that we recall here (G¨ odel, 1949): 1 ds2 = −dt2 + dx2 − U 2 dy 2 − 2U dtdy + dz 2 , 2
(5.63)
√
where U = e 2ωx and ω is a constant. The symmetries of this metric are summarized by ﬁve Killing vector ﬁelds, √ ξtμ = ∂t , ξyμ = ∂y , ξzμ = ∂z , ξ4μ = ∂x − 2ωy∂y , √ √ √ (5.64) ξ5μ = −2e− 2ωx ∂t + 2ωy∂x + (e−2 2ωx − ω 2 y 2 )∂y . Let us consider a geodesic Υ parameterized by an aﬃne parameter λ with tan˙ x, ˙ y, ˙ z), ˙ where dot means diﬀerentiation with respect to λ. gent vector P μ = (t, The geodesic equations are √ √ t¨ + 2 2ω t˙x˙ + 2ωU x˙ y˙ = 0, √ 1 √ x ¨ + 2ωU t˙y˙ + U 2 2ω y˙ 2 = 0, 2 √ −1 ˙ y¨ − 2 2ωU tx˙ = 0, z¨ = 0.
(5.65)
Using the Killing symmetries, this system of equations can be fully integrated. First we recall that dxα ≡ −E = constant, dλ α dx py = gyα ≡ py = constant, dλ dz ≡ pz = constant pz = dλ pt = gtα
(5.66)
5.5 The world function in G¨ odel spacetime
85
are conserved quantities. Then, from the above and the metric form (5.63), it follows that 2py t˙ = −E − , U
y˙ =
2E 2py + 2, U U
(5.67)
and z(λ) = z0 + pz λ.
(5.68)
Using these relations in (5.65) we obtain √ √ 2 2ωp2y 2 2ωEpy − = 0. x ¨− U U2 Multiplying both sides of (5.69) by 2x˙ we can write it as √ √ d 2 x˙ + 4Epy e− 2ωx + 2p2y e−2 2ωx = 0, dλ
(5.69)
(5.70)
which implies that √
x˙ 2 + 4Epy e−
2ωx
√
+ 2p2y e−2
2ωx
= C1 ,
(5.71)
where C1 is a constant. The normalization condition p·p = −, with = (1, 0, −1) for timelike, null, and spacelike orbits respectively, allows one to ﬁx the constant C1 as C1 = −2 − E 2 − p2z ;
(5.72)
then Eq. (5.71) can be integrated by standard methods. Use of this solution in both Eqs. (5.67) leads us to the full integration of the geodesic equations. Clearly this allows us to write the exact world function along any nonnull geodesic. In a remarkable paper, Warner and Buchdahl (1980) were able to derive the exact form of the world function for G¨ odel’s spacetime, following an alternative approach. Their aim was to ﬁnd the world function as a solution of a set of diﬀerential equations. Following their arguments, let us ﬁrst recall from (5.39) that, given a set of Killing vectors ξ(a) and a pair of events with coordinates {xμ } and {xμ }, the world function satisﬁes the relation
μ μ ∂μ Ω + ξ(a) ∂μ Ω = 0. ξ(a)
(5.73)
We then have ∂t Ω + ∂t Ω = 0, ∂y Ω + ∂y Ω = 0, ∂z Ω + ∂z Ω = 0,
(5.74)
which imply for Ω a dependence of the type 1 Ω(t − t, x, x , y − y, z − z; Υ) = Ω1 (τ, x, x , ξ; Υ) + (z − z)2 , 2
(5.75)
86
The world function
where Ω1 is a new function to be determined and τ = t − t,
ξ = y − y.
(5.76)
Using the remaining Killing vectors ξ(4) and ξ(5) we have from (5.64) the additional equations √ √ (5.77) 0 = ∂x Ω − 2ωy∂y Ω + ∂x Ω − 2ωy ∂y Ω, √ √ √ 0 = 2ωy∂x Ω + (e−2 2ωx − ω 2 y 2 )∂y Ω − 2e− 2ωx ∂t Ω √ √ + 2ωy ∂x Ω + (e−2 2ωx − ω 2 y 2 )∂y Ω √
− 2e−
2ωx ∂t Ω.
(5.78)
Introducing the new variables √
v = e−
2ωx
√
v = e−
,
2ωx
,
(5.79)
v∂v Ω1 + v ∂v Ω1 + ξ∂ξ Ω1 = 0,
(5.80)
and recalling (5.75), Eq. (5.77) gives
whose solution is
Ω1 = Ω1
v v τ, , ξ ξ
.
(5.81)
v , ξ
(5.82)
Using this form of Ω1 in (5.78) and denoting α=
v , ξ
β=
we obtain α(ω 2 − β 2 + α2 )∂α Ω1 − β(ω 2 − α2 − β 2 )∂β Ω1 + 2 (α − β)∂τ Ω1 = 0.
(5.83)
This equation can be considerably simpliﬁed with the further change of variables s+r =
α , ω
s−r =
β , ω
τ1 = ωτ,
(5.84)
leading to s(4r2 + 1)∂r Ω1 + r(4s2 + 1)∂s Ω1 + 4r∂τ1 Ω1 = 0.
(5.85)
This equation can be solved with the method of separation of variables. Setting Ω1 = Ω1 (r) + Ω1 (s) + Ω1 (τ1 ), we obtain 2 4r + 1 1 1 −1 ¯ +K tan (2s) − τ1 , (5.86) Ω1 = K ln 8 4s2 + 1 2 ¯ are separation constants. Hence the full world function (5.75) where K and K follows.
5.6 The world function of a weak gravitational wave
87
5.6 The world function of a weak gravitational wave Consider the metric of a weak gravitational plane wave propagating along the x direction of a coordinate frame with “+, ×” polarization states, written in the form ds2 = −dt2 + dx2 + (1 − h+ )dy 2 + (1 + h+ )dz 2 − 2h× dxdz,
(5.87)
where the wave amplitudes h+/× = h+/× (t − x) are functions of (t − x). Let them be given by h+ = A+ sin ω(t − x),
h× = A× cos ω(t − x),
(5.88)
where linear polarization is characterized by A+ = 0 or A× = 0, whereas circular polarization is assured by the condition A+ = ±A× . It is also useful to introduce the polarization angle, ψ = tan−1 (A× /A+ ). The geodesics of this metric are given by (de Felice, 1979) U(geo) =
1 [(2 + f + E 2 )∂t + (2 + f − E 2 )∂x ] 2E + [α(1 + h+ ) + βh× ]∂y + [β(1 − h+ ) + αh× ]∂z ,
(5.89)
where α, β, and E are conserved Killing quantities, 2 = 1, 0, −1 correspond to timelike, null, and spacelike geodesics respectively, and to ﬁrst order in the wave amplitudes f α2 (1 + h+ ) + β 2 (1 − h+ ) + 2αβh× .
(5.90)
The parametric equations of the geodesics are then easily obtained: t(λ) = Eλ + t0 + x(λ) − x0 , 1 λ − [(α2 − β 2 )A+ cos ω(Eλ + t0 − x0 ) 2E 2ωE 2 − 2αβA× sin ω(Eλ + t0 − x0 )] + x0 ,
x(λ) = (2 + α2 + β 2 − E 2 )
y(λ) = αλ + y0 1 [αA+ cos ω(Eλ + t0 − x0 ) − βA× sin ω(Eλ + t0 − x0 )], − ωE z(λ) = βλ + z0 1 [βA+ cos ω(Eλ + t0 − x0 ) + αA× sin ω(Eλ + t0 − x0 )] , (5.91) + ωE where λ is an aﬃne parameter and x0 , y0 , z0 are integration constants. We can then evaluate the world function for two general points P0 and P1 connected by a geodesic Υ. A direct calculation gives Ω(s0 , s1 ; Υ) = Ω(s0 , s1 )ﬂat A+ cos ω(t1 − x1 ) − cos ω(t0 − x0 ) [(y1 − y0 )2 − (z1 − z0 )2 ] + 2ω t1 − x1 − (t0 − x0 ) sin ω(t1 − x1 ) − sin ω(t0 − x0 ) A× (y1 − y0 )(z1 − z0 ) , (5.92) − ω t1 − x1 − (t0 − x0 )
88
The world function
where Ω(s0 , s1 )ﬂat =
1 ηαβ (x0 − x1 )α (x0 − x1 )β 2
(5.93)
is the world function in Minkowski spacetime (see Bini et al., 2009).
5.7 Applications of the world function: GPS or emission coordinates Let us brieﬂy review the standard construction of GPS coordinates in a ﬂat spacetime (Rovelli, 2002). Consider Minkowski spacetime in standard Cartesian coordinates (t, x, y, z), ds2 = ηαβ dxα dxβ = −dt2 + dx2 + dy 2 + dz 2 ,
(5.94)
and four satellites, represented by test particles in geodesic motion. With the above choice of coordinates, timelike geodesics are straight lines: A α A α xα A (τ ) = UA τ + xA (0),
A
= 1, . . . , 4,
(5.95)
where UA = γA [∂t + νA niA ∂i ] = cosh αA ∂t + sinh αA niA ∂i
(5.96)
are their (constant) 4velocities and τ A is the proper time parameterization along each world line. In (5.96), γA is the Lorentz factor and the linear velocities νA are relative to any of the four particles chosen as a ﬁducial observer; they are related to the rapidity parameters αA by νA = tanh αA ; nA denotes the spacelike unit vectors along the spatial directions of motion. Without any loss of generality, we assume that the satellites all start moving from the origin of the coordinate system O, so hereafter we set xα A (0) = 0, and hence α A xα A (τ ) = UA τ .
(5.97)
¯ α and a Let us now consider a general spacetime point P¯ with coordinates W α point PA with coordinates xA on the world line of the Ath satellite corresponding to an elapsed amount of proper time τ A . A photon emitted at PA follows a null geodesic, i.e. the straight line xα (λ) = K α λ + xα A,
(5.98)
¯ where λ is an aﬃne parameter. Such a photon will reach P¯ at a certain value λ of the parameter according to ¯ + xα , ¯ α = K αλ W A
(5.99)
¯ ¯ α = −K α λ. UAα τ A − W
(5.100)
implying that
5.7 Applications of the world function: GPS or emission coordinates
89
Taking the norm of both sides, we obtain ¯ 2 − 2τ A (UA · W ¯ ) = 0, − (τ A )2 + W
(5.101)
since K is a null vector. Solving for τ A and selecting the solution corresponding to the past light cone leads to ¯ ) − (UA · W ¯ )2 + W ¯ 2 . (5.102) τ A = −(UA · W These equations give the four proper times τ A (i.e. the GPS or emission coordinates) associated with each satellite in terms of the Cartesian coordinates of the ¯ 0, . . . , W ¯ 3 ). general point P¯ in the spacetime, i.e. τ A = τ A (W Using Eq. (5.102), one can evaluate the inverse of the transformed metric ∂τ A ∂τ B αβ A B A B ¯ α ∂W ¯ β ≡ η (dτ )α (dτ )β = dτ · dτ , ∂W ¯ α also satisﬁes the properties where the dual frame (dτ A )α = ∂τ A /∂ W g AB = η αβ
¯ α = τ A, (dτ A )α W
(dτ A )α UAα = 1.
Similarly one can introduce the frame vectors α α ¯α ∂ ∂ ∂W A A = , (dτ ) = δB . α ∂τ A ∂τ A ∂τ B
(5.103)
(5.104)
(5.105)
It is then easy to show that the condition g AA = dτ A · dτ A = 0 is fulﬁlled. In ¯ α one obtains fact, by diﬀerentiating both sides of Eq. (5.101) with respect to W (dτ A )α =
¯ α − τ A UAα W ¯ ), τ A + (UA · W
(5.106)
which implies that g AA = (dτ A )α (dτ A )α =
¯ 2 − 2τ A (UA · W ¯) −(τ A )2 + W = 0. A 2 ¯ [τ + (UA · W )]
(5.107)
The metric coeﬃcients gAB = (∂/∂τ A ) · (∂/∂τ B ) = η αβ (∂/∂τ A )α (∂/∂τ B )β can be easily obtained as well by expressing the Cartesian coordinates of P¯ in terms ¯α=W ¯ α (τ 1 , . . . , τ 4 ). of the emission coordinates τ A , i.e. W To accomplish this, it is enough to invert the transformation (5.102). However, in order to outline a general procedure, we start by considering the equation for ¯ α given in the past light cone of the general spacetime point P¯ with coordinates W terms of the world function, which in the case of ﬂat spacetime is simply given by 1 β β α ηαβ (xα (5.108) A − xB )(xA − xB ). 2 The condition ensuring that the past light cone of P¯ cuts the emitter world lines is given by ¯ ) = 0, ¯ 0, x0 < W (5.109) Ωﬂat (xA , W Ωﬂat (xA , xB ) =
A
90
The world function
for each satellite labeled by the index A. This gives rise to a system of four ¯ α of the event P¯ of the quadratic equations in the four unknown coordinates W form (5.101) for each A = 1, . . . , 4. To solve this system, start for example by subtracting the last equation from the ﬁrst three equations to obtain the following system: ¯ ) − Ωﬂat (x4 , W ¯ ) = 0 = −2W ¯ · (xi − x4 ) − (τ i )2 + (τ 4 )2 , Ωﬂat (xi , W ¯ ) = 0 = W ¯ 2 − 2W ¯ · x4 − (τ 4 )2 , (5.110) Ωﬂat (x4 , W with i = 1, 2, 3, consisting of three linear equations and only one quadratic equa¯ 1, W ¯ 2, W ¯3 tion. Thus we can ﬁrst solve the linear equations for the coordinates W 0 ¯ in terms of W , which then can be determined from the last quadratic equation. As a result, the coordinates of the event P¯ are fully determined in terms of the satellite proper times τ A and the known parameters characterizing their world lines. An explicit result can easily be obtained for a ﬁxed kinematical conﬁguration of satellites. As an example one can take one of the satellites at rest at the origin O and the other three in motion along each of the three spatial directions: U1 = cosh α1 ∂t + sinh α1 ∂x , U2 = cosh α2 ∂t + sinh α2 ∂y , U3 = cosh α3 ∂t + sinh α3 ∂z , U4 = ∂t ,
(5.111)
where αi , i = 1, 2, 3, are the rapidities. This choice is the one adopted in Bini et al. (2008) to construct emission coordinates in curved spacetime (but for the metric associated with Fermi coordinates around the world line of an accelerated observer) covering a spacetime region around the Earth.
6 Local measurements
The mathematical tools introduced in the previous chapters are essential for dealing with measurements in curved spacetimes. Here we shall conﬁne our attention to local measurements only, i.e. to those whose measurement domain does not involve spacetime curvature explicitly. Given an observer u, the tensorial projection operators P (u) and T (u) allow one to deﬁne the observer’s restspace and time dimension in neighborhoods of any of his points which are suﬃciently small to allow one to approximate the measurement domain as a point. The above operators, in fact, arise naturally as the inﬁnitesimal limit of the nonlocal procedure for the measurements of space distances and time intervals, as we will show.
6.1 Measurements of time intervals and space distances Let γ be the world line of a physical observer; the parameter s on it is taken to be the proper time, so the tangent vector ﬁeld γ˙ is normalized as γ˙ · γ˙ = −1.
(6.1)
Here we shall analyze the concepts of spatial and temporal distances between two events relative to a given observer, referring closely to the analog in Euclidean geometry. Consider an event P not belonging to γ but suﬃciently close to it that a normal neighborhood exists which contains the intersections A1 and A2 of γ with the generators of the light cone in P. Referring to Fig. 6.1 we see that all points on γ between A1 and A2 can be connected to P by a unique nontimelike geodesic which we shall denote by ζs , with parameter σ. Let A1 = γ(sA1 ) and A2 = γ(sA2 ). Then from (5.18) the world function Ω(σA , σP ; ζs ) along the geodesic ζs connecting a general point A = γ(s) with P is given by 1 Ω(σA , σP ; ζs ) = (σP −σA )2 ζ˙s · ζ˙s ; (6.2) 2 A
92
Local measurements
A2
ζs P A(s)
A1
γ Fig. 6.1. A curve γ is connected to the point P by a light ray emitted at A1 towards P and recorded at A2 after being reﬂected at P. The entire process takes place in a normal neighborhood UN of the spacetime.
the geodesics ζs are assumed aﬃne parameterized so (ζ˙s · ζ˙s ) is constant on them. Because P is kept ﬁxed and A is any point on γ : sA1 ≤ s ≤ sA2 , the world function in (6.2) is only a function of s; hence we write it in general as Ω(σ(s), σP ; ζs ) ≡ Ω(s),
(6.3)
Ω(sA1 ) = Ω(sA2 ) = 0.
(6.4)
with the constraints
Since γ is a smooth curve, there exists a value sA0 : sA1 <sA0 <sA2 at which Ω(s) has an extreme value, namely at A0 = γ(sA0 ): " dΩ "" = 0. (6.5) ds "A0 If P is suﬃciently close to γ then the point lent to
A0
is unique. At
Ωα0 γ˙ α0 = 0,
A0 ,
(6.5) is equiva(6.6)
where γ˙ α0 ≡ γ˙ α (sA0 ) and Ωα0 is the derivative at A0 of the world function along the geodesic ζs A0 joining A0 to P and given from (5.29) as Ωα0 = −(σP − σA0 )ξα0 ,
ξ ≡ ζ˙s A0 ,
(6.7)
where ξα0 = ξα (σA0 ). From (6.4) and (6.6) we deduce that γ˙ α0 ξ α0 = 0.
(6.8)
6.1 Measurements of time intervals and space distances
93
We deﬁne as a measurement of the spatial distance between the observer on γ and the event P the length of the spacelike geodesic segment on ζs A0 which strikes γ orthogonally at A0 ; that is (Synge, 1960), 1 1/2 L(σA0 , σP ; ζs A0 ) ≡ L(P, γ) = 2Ω(sA0 ) 2 = σP − σA0  (ξ · ξ)A0 .
(6.9)
The event A0 on γ is termed simultaneous to P with respect to the observer on γ. We now need to express L in terms of directly measurable quantities. Let sA1 and sA2 be the parameters of the events A1 and A2 on γ in which a light signal is emitted towards P and received after reﬂection at P, respectively. The quantity δTγ ≡ (sA2 − sA1 )
(6.10)
is a physical time directly readable on the observer’s clock. The world function Ω(s) can be expanded in a power series about sA0 as " ∞ 1 dn Ω "" Ω(s) = Ω(sA0 ) + (s − sA0 )n . (6.11) " n! dsn " n=1
sA
0
˙ α = 0) then If we require that the observer’s world line be a geodesic (¨ γ α ≡ a(γ) the derivatives of the world function can be written in general as dn Ω = Ωα1 ...αn γ˙ α1 . . . γ˙ αn . dsn
(6.12)
Limiting ourselves to second derivatives only, we have that the coeﬃcients Ωα0 β0 are given by (5.36). Assuming that the points A0 and P are close enough, we can consider the above coeﬃcients to ﬁrst order in the curvature only. Therefore Ωα0 β0 ≈ gα0 β0 +
1 Sαβγδ ξ γ ξ δ A0 (σP − σA0 )2 , 2
(6.13)
where 2 Sαβγδ = − Rα(γβδ) . 3
(6.14)
Noticing that along γ we have " dΩ "" = 0, ds "A0
(6.15)
Eq. (6.11) can be written as 1 Ω(s) = Ω(sA0 ) − (s − sA0 )2 2 1 α α β γ δ 2 × 1 + (σP − σA0 )ξα a − Sαβγδ γ˙ γ˙ ξ ξ (σP − σA0 ) 2 A0 + O(Riem2 ).
(6.16)
94
Local measurements
Imposing conditions (6.4) and requiring a(γ) ˙ = 0, we can deduce from (6.16) the values sA1 and sA2 corresponding to the events A1 and A2 . We then obtain 1 2 α β γ δ 2 (sA1 /A2 − sA0 ) 1 − Sαβγδ u u ξ ξ (σP − σA0 ) ≈ 2Ω(sA0 ), (6.17) 2 where we set u ≡ γ˙ to stress the role of the observer who makes the measurements played by the vector ﬁeld tangent to γ. Equations (6.17) admit the solutions 1 1 sA1 ≈ sA0 − [2Ω(sA0 )] 2 1 + Sαβγδ uα uβ ξ γ ξ δ (σP − σA0 )2 , 4 A0 sA2 ≈ sA0 + [2Ω(sA0 )]
1 2
1 1 + Sαβγδ uα uβ ξ γ ξ δ (σP − σA0 )2 4
(6.18)
. A0
From the above formulas and deﬁnitions we ﬁnally have 1 δTγ ≈ 2L(P, γ) + Sαβγδ uα uβ ξ γ ξ δ A L3 (P, γ), (6.19) 0 2 where we have reparameterized ζsA so that ξ · ξ = 1. Equation (6.19) gives 0 the ﬁrstorder curvature contribution to the relationship between the round trip time of a bouncing signal and the geometrical distance between γ and P. Formal solutions of (6.19) are given by 1 1/3 −1 χ A − 4χ−1/3 , 6 √ 1 1 1/3 −1 −1/3 χ A + 2χ , = − L(P, γ)1 ± i 3 2 12
L(P, γ)1 = L(P, γ)2,3
(6.20) (6.21)
where 1 Rαγβδ uα uβ ξ γ ξ δ A , 0 3 √ 32 2 2 + 27(δTγ ) . χ = 12A 9(δTγ ) + 3 A
A=−
(6.22) (6.23)
Clearly, δTγ can be read directly on the observer’s clock and the quantities A and χ can be deduced by a suitable spacetime modeling once the observer u is ﬁxed. Solutions (6.20), with (6.22) and (6.23), may ﬁnd application to the Global Positioning System (GPS), with general relativistic corrections to ﬁrst order in the curvature. Let us now uncover the role that the projection operators P (u) and T (u) have in deﬁning the measurements of a time interval and a spatial distance, respectively, between two inﬁnitesimally close events and relative to a given observer. Working in the inﬁnitesimal domain we can neglect the curvature in (6.16) and limit ourselves to terms of the second order in Δσ and Δs. Choose a point A on γ very close to A0 (here γ is again a general timelike curve); Eq. (6.16) gives 2Ω(sA0 ) = 2Ω(sA ) + (sA0 − sA )2 + O(Δσ + Δs3 ).
(6.24)
6.2 Measurements of angles
95
From (6.2) we have 2Ω(sA ) = (σP − σA )2 (ξ α ξα )A ,
(6.25)
so Eqs. (6.1) and (6.5) imply that − (σP − σA )(uα ξα )A =
" 2 dΩ "" d Ω = (sA − sA0 ) + O(Δs2 ) ds "A ds2 A0
= −(sA − sA0 ) + O(Δσ + Δs2 ).
(6.26)
Hence (6.24) becomes 2Ω(sA0 ) = (σP − σA )2 gαβ (sA )ξ α (σA )ξ β (σA ) + (σP − σA )2 (uα ξα )2A + O(Δσ + Δs3 ) = (σP − σA )2 (gαβ + uα uβ )ξ α ξ β A + O(Δσ + Δs3 ).
(6.27)
Recalling that δxα , δσ→0 δσ
ξ α = lim we deduce, at a point
P
suﬃciently close to γ, and from (6.9), that
1 δL(P, γ) = P (u)αβ δxα δxβ 2 + O(δx2 ),
(6.28)
where P (u)αβ = gαβ + uα uβ and δxα denotes the coordinate diﬀerence between A and P. From (6.26) we interpret the time interval between the event A and the event A0 on γ which is simultaneous to P as the temporal separation between the events A and P relative to the observer on γ; it is given by δT (A0 , A) = −(uα δxα )A + O(δx2 ).
(6.29)
Comparing (6.28) and (6.29) with the deﬁnition of the projection operators P (u) and T (u) one justiﬁes their interpretation. Relations (6.28) and (6.29) are very useful since they allow one to express the invariant measurements of the spatial distance and the time interval between any two events suﬃciently close in terms of coordinates and vector components.
6.2 Measurements of angles Angles can be measured with great accuracy; therefore their measurements enter in many problems as observables in terms of which one can ﬁx boundary conditions. Clearly an angle is a spatial quantity; hence its measurement must be carried out in the rest space of the observer. Let the observer be represented by his world line γ and denote this by u ≡ γ. ˙ Let us evaluate the angle between any two null directions stemming from a given point on γ.
96
Local measurements
At the spacetime point where the measurement takes place, a null vector k admits the following decomposition: k = −(k · u)u + k⊥ = −(k · u)[u + νˆ(k, u)],
νˆ(k, u) · νˆ(k, u) = 1,
(6.30)
where k⊥ = P (u)k = k⊥ ˆ ν (k, u) and νˆ(k, u) is the unitary spatial vector tangent to the local line of sight towards P. If a signal is sent from A to another point, say Q, and with a photon gun similar to the previous one, we identify this direction = P (u)k , where k is the vector tangent to the null ray with a vector at A: k⊥ from A to Q. We then deﬁne the angle Θ(k,k ) between these two directions at A by cos Θ(k,k ) =
k⊥ · k⊥ ˆ(k , u) · νˆ(k, u).  = ν k⊥  k⊥
(6.31)
Of course the above formula can be applied to timelike and spacelike directions as well.
6.3 Measurements of spatial velocities The instantaneous spatial velocity of a test particle with 4velocity U relative to a given observer u has been introduced in (3.109) as the magnitude of the spacelike 4vector, ν(U, u)α = −(U σ uσ )−1 U α − uα .
(6.32)
Let us now show why this is interpreted as the instantaneous spatial velocity of the particle U with respect to the observer u. Since we are conﬁning our attention to the inﬁnitesimal domain, we shall deal with local measurements only. Let γ be the world line of the particle with tangent ﬁeld U ≡ γ˙ and assume that the curve γ strikes the world line γ of u at a point A . At this point the particle and the observer coincide so we can ﬁx their proper times to coincide as well. Consider then a later moment when the particle is at a point P on its world line, still very close to A . Once the particle has reached the point P, the observer u on γ will judge that it has covered a spatial distance 1/2 (6.33) δL(P, γ) = P (u)αβ δxα δxβ equal to the length of the (unique) spacelike geodesic segment connecting P to the point A0 on γ which is simultaneous with P with respect to u. A correct way to measure the instantaneous velocity of recession (or approach) of the particle U with respect to the observer u is to track the particle with a light ray (see de Felice and Clarke, 1990, for details). Let A1 be the point of γ which could be connected to P by a light ray; clearly sA <sA1 <sA0 . From the previous discussion it follows that the quantities δxα in (6.33) are the coordinate diﬀerences between P and any point A on γ between A1 and A0 . Clearly the 4vector δxα is deﬁned at the point A on γ. The approximation of conﬁning ourselves to the inﬁnitesimal domain allows one to identify A and A1 with A .
6.4 Composition of velocities
97
Hence, once the particle has reached the point P, the observer u will judge that the particle traveling from A to P took a time, as read on his own clock, equal to δT (A0 , A ) ≈ −(uα δxα ).
(6.34)
This interval of time actually measures the time interval on γ between A , considered as a point of γ, and A0 . By deﬁnition, the instantaneous spatial velocity of U relative to u is the quantity (P (u)αβ dxα dxβ )1/2 δL(P, γ) = − . δx→0 δT (A0 , A ) (dxρ uρ )
ν(U, u) ≡ lim
(6.35)
The spatial instantaneous velocity 4vector can be written as ν(U, u) = [ν(U, u)α ν(U, u)α ]1/2 νˆ(U, u) ≡ ν(U, u)ˆ ν (U, u),
(6.36)
with ν(U, u)α given by (6.32) and νˆ(U, u) being the unitary vector introduced in (3.110). Clearly the 4vector ν(U, u) belongs to LRSu at the point A of γ. Owing to the symmetry of the projection operators, we have ν(U, u) = ν(u, U ) ≡ ν ;
(6.37)
however, vector ν(u, U ) belongs to LRSU still at the point A but considered as a point of γ . From (6.37) it follows as an obvious consequence that the Lorentz factor is γ(U, u) = γ(u, U ) = −(U α uα ).
(6.38)
6.4 Composition of velocities In the theory of relativity velocities add up according to a law which prevents them from becoming larger than the velocity of light, whatever observer one refers to. Let U denote the 4velocity of a test particle and u and u ¯ those of two diﬀerent observers in relative motion. Conﬁning our attention to the inﬁnitesimal domain, we have U = γ(U, u)[u + ν(U, u)] = γ(U, u ¯)[¯ u + ν(U, u ¯)]
(6.39)
u ¯ = γ(¯ u, u)[u + ν(¯ u, u)].
(6.40)
γ(U, u ¯) [¯ u + ν(U, u ¯)] = [u + ν(U, u)]. γ(U, u)
(6.41)
and
From (6.39) it follows that
98
Local measurements
Hence, contracting with u ¯ and recalling that u ¯·u ¯ = −1 = u · u,
ν(U, u ¯) · u ¯ = 0,
ν(¯ u, u) · u = 0,
we obtain identically γ(¯ u, u)[1 − ν(U, u) · ν(¯ u, u)] =
γ(U, u ¯) . γ(U, u)
(6.42)
Moreover from (6.40) and the ﬁrst relation in (6.39) we obtain U − u = ν(U, u), γ(U, u) u ¯ − u = ν(¯ u, u). γ(¯ u, u)
(6.43)
Subtracting Eqs. (6.43) from one another we ﬁnd u ¯ U − = ν(U, u) − ν(¯ u, u), γ(U, u) γ(¯ u, u)
(6.44)
and projecting orthogonally to u ¯, P (¯ u)
U γ(U, u ¯) = ν(U, u ¯) = P (¯ u)[ν(U, u) − ν(¯ u, u)]. γ(U, u) γ(U, u)
(6.45)
Using the identity (6.42), we can now write γ(¯ u, u)ν(U, u ¯) = P (¯ u)
ν(U, u) − ν(¯ u, u) . 1 − ν(U, u) · ν(¯ u, u)
(6.46)
The quantity in the square brackets lives in LRSu ; hence it can also be written as ν(U, u) − ν(¯ u, u) . P (u) 1 − ν(U, u) · ν(¯ u, u)
(6.47)
Therefore relation (6.46) can be written as γ(¯ u, u)ν(U, u ¯) = P (¯ u, u)
ν(U, u) − ν(¯ u, u) , 1 − ν(U, u) · ν(¯ u, u)
(6.48)
where P (¯ u, u) = P (¯ u)P (u) : LRSu → LRSu¯ , or as P (¯ u, u)−1 γ(¯ u, u)ν(U, u ¯) =
ν(U, u) − ν(¯ u, u) , 1 − ν(U, u) · ν(¯ u, u)
where P (¯ u, u)−1 : LRSu¯ → LRSu ; this is the velocity composition law.
(6.49)
6.6 Measurements of frequencies
99
6.5 Measurements of energy and momentum Consider the 4momentum P = μ0 U of a pointlike test particle with mass μ0 ; let us see what information an observer u reads out of its spatial and temporal splittings. In terms of these, the 4momentum can be written as Pα = P (u)α β Pβ + T (u)α β Pβ = p(U, u)α + uα E(U, u),
(6.50)
where p(U, u) is the spatial 4momentum and E(U, u) its total energy,1 both relative to u. Let us now justify this interpretation. From (6.39) we have, using a coordinatefree notation, p(U, u) = P (u)P = P + u(u · P ) = μ0 [U + (u · U )u].
(6.51)
Recalling that u · U = −γ(U, u) and using (6.39), we have p(U, u) = −μ0 ν(U, u) (u · U ) = μ0 γν(U, u) = μ0 γν νˆ(U, u).
(6.52)
From (6.36), the magnitude of p(U, u) is given by p(U, u) = μ0 γν.
(6.53)
E(U, u) = −(u · P ) = −μ0 (u · U ) = γμ0 ;
(6.54)
Similarly we have
hence, the last two relations justify the physical interpretation of p(U, u) and E(U, u).
6.6 Measurements of frequencies Consider a photon with 4momentum k. With respect to an observer u, the 4vector k admits the decomposition k = T (u)k + P (u)k = ω(k, u)u + k⊥ = ω(k, u)[u + νˆ(k, u)],
(6.55)
where ω(k, u) = −u · k represents the frequency of the photon as measured by the observer u; k⊥ = P (u)k is the relative (spatial) momentum and νˆ(k, u) = ω(k, u)−1 k⊥ is a unitary spacelike vector which identiﬁes the local line of sight of the observer u. Whenever an observer absorbs a light signal, he measures its frequency and polarization. This operation takes place within a suﬃciently small measurement’s domain that we can limit our considerations to the local 1
Here E(U, u) is in units of the velocity of light c to ensure the dimensions of a momentum.
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Local measurements
inertial frame where special relativity holds. The surface of discontinuity of an electromagnetic ﬁeld is described in general by an equation Φ(x) = 0,
(6.56)
where Φ = Φ(x) is a diﬀerentiable function of the coordinates known as the eikonal of the wave; it satisﬁes the equation g αβ kα kβ = 0
(6.57)
(the eikonal equation), where kα = ∂α Φ. In the observer’s rest frame, reinterpreting Φ as the phase function in the geometrical optics approximation, the instantaneous frequency of the wave is given by ω(k, u) = −
dΦ , dτu
(6.58)
where τu is the proper time of the observer u. From the properties of Φ, (6.58) is more conveniently written as ω(k, u) = −(∂α Φ)
dxα = −kα uα . dτu
(6.59)
This is the invariant characterization of the frequency of a light signal as measured by the observer u. Let us now consider two observers with 4velocities u and u , tangent to the curves γ and γ respectively. Let the observers be far apart from each other and exchange a light signal. At the event of emission “e” on γ, the observer u measures a frequency ω(k, u)e = −(uα k α )e ; the same signal will be detected at the event “o” on γ , with frequency ω (k, u )o = −(uβ k β )o . The frequency ω(k, u)e at the point of emission should be evaluated at the point of observation on γ . This evaluation is made possible by the properties of the parallel transport along a null geodesic. Along the null geodesic Υ joining “e” to “o” and having tangent vector k, the following relations hold: uα kˇα )o = (ˇ uα kα )o , (uα kα )e = (ˇ
(6.60)
where u ˇ and kˇ are the parallel propagated vectors along Υ with the further property kα = kˇα . The ratio between the emitted and observed frequencies at the point of observation is called the frequency shift and is denoted by ˇ ω(k, u ˇ) (ˇ uα k α )o (1 + z)o = = α . (6.61) ω (k, u ) o (uα k )o When (1 + z)o < 1, we have a blueshift, while when (1 + z)o > 1 we have a redshift, both referred to u . If the frequency shift of the exchanged signal is due to relative motion, then the shift is known as a Doppler shift. This allows an indirect measurement of the relative velocity, termed the Doppler velocity. The variation of the frequency
6.6 Measurements of frequencies
101
is directly observable and is given, in the observer’s rest frame at any event on γ , by ˇ −1/2 ω(k, u ˇ) 1 − ν(ˇ u, u ) cos Θ(k,ˇu) . (6.62) = 1 − ν(ˇ u, u )2 ω (k, u ) o Here ν(ˇ u, u ) is the magnitude of the instantaneous velocity of u ˇ with respect to u at the point of observation. Similarly, Θ(k,ˇu) is the angle between the spatial direction of the light signal and that of the observer who emits it, but referred to the rest frame of the observer u at o. Obviously the spatial direction of motion of the emitter evaluated in the rest frame of the observer u is provided by the parallel transported vector u ˇ at the point of observation on γ . Recalling (6.60), Eq. (6.61) becomes, at “o” on γ , ˇ ω(k, u ˇ) kα u ˇα = ω (k, u ) o kβ u β o =
ˇβ − (uα k α )(uβ u ˇβ ) P (u )αβ k α u (uρ k ρ )
=−
(P (u )αβ k α u ˇβ )(P (u )ρσ u ˇρ u ˇσ )1/2 − (uβ u ˇβ ), (6.63) μ ν 1/2 (P (u )μν k k ) (P (u )πτ u ˇπ u ˇτ )1/2
recalling that (uβ k β )γ(s) < 0. From (6.31), however, this becomes ˇ ω(k, u ˇ) 1/2 = − cos Θ(k,ˇu) (P (u )ρσ u ˇρ u ˇσ ) − (uβ u ˇβ ) ω (k, u ) o (P (u )ρσ u ˇρ u ˇσ )1/2 . ˇβ ) 1 + cos Θ(k,ˇu) = −(uβ u (uβ u ˇβ )
(6.64)
If we deﬁne the spatial velocity of u with respect to u at “o” as the quantity " (P (u )ρσ u ˇρ u ˇσ )1/2 "" (6.65) ν(ˇ u, u ) = − " , (u β u ˇβ ) o we obtain
1 − ν(ˇ u, u )2
−1/2
= −(u β u ˇβ ),
(6.66)
so (6.62) is recovered. Let us now better specify the measurement of frequencies and their comparison. In order to characterize the photon’s frequency as the frequencyattheemission, one has to consider the atom as an observer with 4velocity u who reads the frequency as ωe = −(uα kα )e . In this expression the eﬀect of the background geometry enters through the deﬁnition of the observer, the null geodesic k, and the scalar product itself. Let the emitted photon be observed by a distant observer with the 4velocity u . The frequencyattheobservation is deﬁned as ω ≡ −(uα kα )o , where k is the same null geodesic which describes the emitted photon. At the point of observation one has in general a diﬀerent spacetime
102
Local measurements
geometry, and the proper time of the atoms will run diﬀerently than that of the atoms at the point of emission. In order to make a comparison between the emitted and the observed frequencies, how does the observer at the observation point know about the frequencyattheemission? This information is carried by the emitted photon and deposited in a spectral line of the electromagnetic spectrum once it reaches the observer u at the observation point. In this way the emitted frequency is directly observed at the observation point. Formally this information transfer is assured by parallel transport of the frequencyattheemission along uα kˇα )o . Clearly, being the null geodesic, giving rise to the quantity (ˇ ωe )o = −(ˇ the scalar product invariant under parallel transport, we have (ˇ ωe )o = ωe . This frequency may be compared with the frequency the photon would have had if the same atomic transition which occurred at the emission point did occur at the observation point with the local background geometry and with respect to the observer u . The latter frequency is given by ωo .
6.7 Measurements of acceleration Consider the world line of a test particle with tangent vector ﬁeld U , and let a(U ) = ∇U U be its 4acceleration due to the presence of an external nongravitational force per unit of mass f (U ), so that a(U ) = f (U ).
(6.67)
If a family of observers u is deﬁned all along the world line of the particle then both its 4acceleration and the external force can be expressed in terms of measurements made by u. The local space and time splitting of a(U ) relative to u is given by2 a(U ) = P (u)a(U ) + T (u)a(U ) = P (u)a(U ) − u(u · a(U )).
(6.68)
From the property a(U ) · U = 0 we deduce that a(U ) = P (U )a(U ); hence (6.68) can be written as a(U ) = P (u)P (U )a(U ) − u(u · a(U )) = P (u, U )a(U ) − u(u · a(U )).
(6.69)
Using again the orthogonality between a(U ) and U and the splitting U = γ(u + ν(U, u)), we have u · a(U ) = −ν(U, u) · a(U ); hence a(U ) = P (u, U )a(U ) − u(ν(U, u) · P (u, U )a(U )). 2
(6.70)
To simplify notation we often use juxtaposition of tensors to mean inner product. For example we write P (u, U )a(U ) instead of P (u, U ) a(U ).
6.7 Measurements of acceleration
103
Therefore P (u, U )a(U ) = P (u)∇U U = P (u) {γ∇U [u + ν(U, u)] + [u + ν(U, u)]∇U γ} = γP (u)∇U [u + ν(U, u)] + ν(U, u)∇U γ,
(6.71)
that is D(γν(U, u)) Du + P (u) dτU dτU D(fw,U,u) u D(fw,U,u) p(U, u) =γ + , dτU dτU
P (u, U )a(U ) = γP (u)
(6.72)
where p(U, u) = γν(U, u) is the relative linear momentum of the particle per unit mass, with respect to u. We have already deﬁned in (3.156) the gravitational force (per unit mass) as Du (G) = −F(fw,U,u) ; (6.73) P (u) dτU hence we have in the local rest frame of u: (G)
P (u, U )a(U ) = −γF(fw,U,u) +
D(fw,U,u) p(U, u) . dτU
(6.74)
Let us deﬁne P (u, U )f (U ) ≡ γF (U, u),
(6.75)
then recalling (6.67) we have (G)
− γF(fw,U,u) +
D(fw,U,u) p(U, u) = γF (U, u), dτU
(6.76)
so that, from (3.151), we obtain D(fw,U,u) p(U, u) (G) = F (U, u) + F(fw,U,u) . dτ(U,u)
(6.77)
This is the force equation in a Newtonianlike form. Finally, scalar multiplication of both sides of (6.77) by ν(U, u) gives ν(U, u) ·
D(fw,U,u) p(U, u) (G) = ν(U, u) · [F (U, u) + F(fw,U,u) ]. dτ(U,u)
(6.78)
Recalling that p(U, u) = γν(U, u), we have D(fw,U,u) (γν(U, u)) dτ(U,u) D(fw,U,u) ν(U, u) dγ 2 = ν + γ ν(U, u) · dτ(U,u) dτ(U,u) dγ 1 dγ ν2 + 2 = , = dτ(U,u) γ dτ(U,u)
ν(U, u) ·
(6.79)
104
Local measurements
where we have used the relation D(fw,U,u) ν(U, u) dγ . = γ 3 ν(U, u) · dτ(U,u) dτ(U,u)
(6.80)
Therefore, we have dE(U, u) (G) = ν(U, u) · [F (U, u) + F(fw,U,u) ], dτ(U,u)
(6.81)
where E(U, u) = γ(U, u) ≡ γ is the energy of the particle per unit of mass. Equation (6.81) is the power equation in a Newtonianlike form.
Longitudinaltranverse splitting of the force equation The measurement of the acceleration of a test particle in the presence of an external force permitted a Newtonianlike representation of the equations of motion, as shown by (6.77) and (6.81). One can complete this analysis by considering a further splitting of these equations along directions parallel (longitudinal) and orthogonal (transverse) to that of the velocity of the particle relative to the observer u. A key tool for this study is the use of the relative FrenetSerret frames introduced in Chapter 4. The transverse splitting of the relative acceleration deﬁnes the relative centripetal acceleration. This splitting is accomplished by projecting (6.77) onto the relative FrenetSerret frame {ˆ ν (U, u), ηˆ(fw,U,u) , βˆ(fw,U,u) }, as discussed in Eqs. (4.39), (4.40), and (4.41). Let p(U, u) = γν(U, u) be the magnitude of the speciﬁc (i.e. per unit mass) spatial momentum of U as seen by u, so that p(U, u) = p(U, u)ˆ ν (U, u),
(6.82)
while the gamma factor is the corresponding speciﬁc energy E(U, u) = γ. They satisfy the identity 2
E(U, u) − p(U, u)2 = 1.
(6.83)
Then (6.77) takes the form D(fw,U,u) νˆ(U, u) dp(U, u) νˆ(U, u) + p(U, u) dτ(U,u) τ(U,u)
(G)
F (U, u) + F(fw,U,u) =
dp(U, u) νˆ(U, u) dτ(U,u)
=
+ γν(U, u)2 k(fw,U,u) ηˆ(fw,U,u) , where (4.39) has been used.
(6.84)
6.7 Measurements of acceleration
105
The relative FrenetSerret components of this equation, i.e. the projections on the axes of the triad {ˆ ν (U, u), ηˆ(fw,U,u) , βˆ(fw,U,u) }, are given by dE(U, u) (G) = [F (U, u) + F(fw,U,u) ] · νˆ(U, u), d (U,u) (G)
γν(U, u)2 k(fw,U,u) = [F (U, u) + F(fw,U,u) ] · ηˆ(fw,U,u) , (G) 0 = [F (U, u) + F(fw,U,u) ] · βˆ(fw,U,u) ,
(6.85)
where d (U,u) = ντ(U,u) and E(U, u) dE(U, u) 1 dE(U, u) dE(U, u) dp(U, u) = = = . dτ(U,u) p(U, u) dτ(U,u) ν dτ(U,u) d (U,u)
(6.86)
This equality follows from the identity (6.83), and has been used to express the longitudinal acceleration term in (6.84) as shown by (6.85)1 . Note that if U is tangent to a geodesic, F (U, u) = 0 and Eqs. (6.85) imply dE(U, u) (G) = F(fw,U,u) · νˆ(U, u), d (U,u) (G)
γν(U, u)2 k(fw,U,u) = F(fw,U,u) · ηˆ(fw,U,u) , (G)
0 = F(fw,U,u) · βˆ(fw,U,u) .
(6.87)
On the other hand, the force equation may also be considered from the point ˆ U ), N ˆ(fw,u,U ) , Bˆ(fw,u,U ) } introof view of the comoving FrenetSerret frame {V(u, duced in Chapter 4, Eqs. (4.46). By applying the derivative D/dτU to the representation of u, ˆ U )) u = γ(U + ν(u, U )) = γ(U − ν V(u,
(6.88)
and solving for a(U ) = f (U ), one obtains DU DU = P (U ) dτU dτU D −1 γ u − ν(u, U ) = P (U ) dτU D ˆ D(γ −1 u) ν V(u, U ) + P (U ) = P (U ) dτU dτU Du Dγ P (U )u + γ −1 P (U ) = −γ −2 dτU dτU ˆ U) Dν ˆ DV(u, + . V(u, U ) + νP (U ) dτU dτU
a(U ) =
(6.89)
Using (6.88), we now ﬁnd that ˆ U ), P (U )u = −γν V(u,
(6.90)
106
Local measurements
so that, recalling (3.156) for the deﬁnition of the gravitational force, we have for the previous relation dν ˆ U )) − γ −1 P (U, u)F (G) (−γν V(u, (fw,U,u) dτU ˆ U) DV(u, dν ˆ V(u, U ) + νP (U ) + dτU dτU ˆ U) dν ˆ DV(u, (G) = (1 + γ 2 ν 2 ) V(u, U ) − γ −1 P (U, u)F(fw,U,u) + νP (U ) dτU dτU ˆ DV(u, U ) dν ˆ (G) V(u, U ) − γ −1 P (U, u)F(fw,U,u) + νP (U ) = γ2 dτU dτU ˆ U) DV(u, d(γν) ˆ (G) = γ −1 V(u, U ) − γ −1 P (U, u)F(fw,U,u) + νP (U ) dτU dτU dp(u, U ) ˆ U ) + γν 2 K(fw,u,U ) N ˆ(fw,u,U ) = γ −1 V(u, dτU
f (U ) = −γν
− γ −1 P (U, u)F(fw,U,u) . (G)
(6.91)
Rearranging terms in this equation leads to the suggestive form dp(u, U ) ˆ (G) V(u, U ) = f (U ) + F(fw,u,U ) dτ(U,u) ˆ(fw,u,U ) , − γν 2 K(fw,u,U ) N
(6.92)
where F(fw,u,U ) = γ −1 P (U, u)F(fw,U,u) = −γ −1 P (U ) (G)
(G)
Du Du = −P (U ) . dτU dτ(U,u)
(6.93)
The last term on the righthand side of (6.92), being proportional to the derivative ˆ U ) of u with respect to U , is called the generalized of the unit spatial velocity V(u, centrifugal force; it is measured by the particle U itself. Its nonrelativistic limit in ﬂat spacetime leads to the familiar centrifugal force relative to the usual family of inertial observers. The comoving relative FrenetSerret decomposition of (6.92) is dE(U, u) (G) ˆ U ), = [f (U ) + F(fw,u,U ) ] · V(u, d (U,u) (G) ˆ(fw,u,U ) , γν 2 K(fw,u,U ) = [f (U ) + F(fw,u,U ) ] · N (G) 0 = [f (U ) + F(fw,u,U ) ] · Bˆ(fw,u,U ) .
(6.94)
These equations are valid as long as ν belongs to the open interval (0, 1) and with a slight modiﬁcation when ν = 1. When ν = 0, the spatial arc length parameterization d (U,u) = γνdτU is singular because the particle path in LRSU collapses to a point. However, since Eqs. (6.94) are just projections along the comoving FrenetSerret triad of the spacetime force equation, they continue to hold in
6.7 Measurements of acceleration
107
the limit ν → 0. One may use the terminology comoving relatively straight and comoving relatively ﬂat for those world lines for which the comoving relative curvature and torsion respectively vanish. By solving Eq. (6.84) for the gravitational force dp(U, u) (G) νˆ(U, u) (6.95) F(fw,U,u) = γν 2 ηˆ(fw,U,u) − F (U, u) + dτ(U,u) and inserting the result into Eq. (4.47), that is ˆ(fw,u,U ) = γk(fw,U,u) ηˆ(fw,U,u) K(fw,u,U ) N (G)
+ νˆ(U, u) ×u [ˆ ν (U, u) ×u F(fw,U,u) ],
(6.96)
one ﬁnds the relation ˆ(fw,u,U ) = γ −1 k(fw,U,u) ηˆ(fw,U,u) K(fw,u,U ) N − νˆ(U,u) ×u [ˆ ν(U,u) ×u F (U, u)].
(6.97)
When the curve is a geodesic, that is when F (U, u) = 0, the two curvatures K(fw,u,U ) and k(fw,U,u) only diﬀer by a gamma factor and the two directions ˆ(fw,u,U ) and ηˆ(fw,U,u) coincide. In fact, from (6.97) we ﬁnd N ˆ(fw,u,U ) = γ −1 k(fw,U,u) ηˆ(fw,U,u) , K(fw,u,U ) N
(6.98)
K(fw,u,U ) = γ −1 k(fw,U,u) .
(6.99)
so that
Thus, for a geodesic, the notions of relative FermiWalker straight (as stated following (4.42)) and comoving relatively straight world lines agree with each other. A relationship between the relative and comoving torsion may be established by diﬀerentiating Eq. (4.47) along U and using the last comoving relative FrenetSerret relation (4.46). After some algebra one ﬁnds (G)
2 2 γνK(fw,u,U ˆ(fw,U,u) ] ) T(fw,u,U ) = γ νk(fw,U,u) τ(fw,U,u) [k(fw,U,u) − F(fw,U,u) · η
d(γk(fw,U,u) ) (G) [F(fw,U,u) · βˆ(fw,U,u) ] dτU (G) ˆ(fw,u,U ) · ∇U [ˆ + K(fw,u,U ) N ν (U, u) ×u F
+
(fw,U,u) ].
(6.100)
In the special case of geodesics, this reduces to T(fw,u,U ) = τ(fw,U,u) ,
(6.101)
so the notions of relative FermiWalker ﬂatness and comoving relative FermiWalker ﬂatness also agree.
108
Local measurements
6.8 Acceleration change under observer transformations Consider two accelerated timelike world lines with unit tangent vectors U and u. We shall express the acceleration of U in terms of that of u. By deﬁnition we have a(U ) = P (U )∇U U = γP (U )[∇U u + ∇U ν(U, u)] = γP (U )[∇U u + P (u)∇U ν(U, u) − u(u · ∇U ν(U, u))] = γP (U )[∇U u + P (u)∇U ν(U, u) + u(ν(U, u) · ∇U u)].
(6.102)
Recalling (3.156) and (3.161), namely Du (G) = −F(fw,U,u) , dτU
P (u)
D ν(U, u) = γa(fw,U,u) , dτU
(6.103)
relation (6.102) becomes (G) (G) a(U ) = γP (U ) −F(fw,U,u) + γa(fw,U,u) − u(ν(U, u) · F(fw,U,u) ) (G)
= −γP (U )[P (u) + u ⊗ ν(U, u)] F(fw,U,u) + γ 2 P (U, u)a(fw,U,u) . (6.104) But, from (3.121)4 it follows that P (u) + u ⊗ ν(U, u) = P (u, U )−1 ,
(6.105)
and hence the ﬁnal result from (3.156) is given by a(U ) = −γP (u, U )−1 F(fw,U,u) + γ 2 P (U, u)a(fw,U,u) = γ 2 P (u, U )−1 [a(u) + ω(u) ×u ν(U, u) + θ(u) ν(U, u)] + P (U, u)a(fw,U,u) . (G)
(6.106)
6.9 Kinematical tensor change under observer transformations We now consider two timelike congruences of curves, CU and Cu ; our aim is to express the kinematical tensor of one congruence in terms of the other. From the deﬁnition, the kinematical tensor of CU , k(U ) = −[P (U )∇U ], can be written as k(U )α β = −P (U )α σ P (U )δ β [∇U ]σ δ = −P (U )α σ P (U )δ β ∇δ U σ = −γP (U )α σ P (U )δ β ∇δ uσ − γ[∇(U )ν(U, u)]α β .
(6.107)
6.9 Kinematical tensor change under observer transformations
109
We evaluate the term P (U )∇u as follows: [P (U )∇u]α β = P (U )α σ P (U )δ β ∇δ uσ = P (U )α σ P (U )δ β [−a(u)σ uδ − k(u)σ δ ].
(6.108)
Taking into account that 0 = P (U )α β U β = γP (U )α β [uβ + ν(U, u)β ],
(6.109)
P (U )α β uβ = −P (U )α β ν(U, u)β .
(6.110)
we have
Therefore [P (U )∇u]α β = P (U )α σ P (U )δ β [a(u)σ ν(U, u)δ − k(u)σ δ ] = P (U, u)α σ P (U, u)δ β [a(u)σ ν(U, u)δ − k(u)σ δ ] ,
(6.111)
or in indexfree notation P (U )∇u = −P (U, u)[k(u) − a(u) ⊗ ν(U, u)].
(6.112)
We can then complete our task, obtaining k(U ) = γP (U, u)[k(u) − a(u) ⊗ ν(U, u)] − γ∇(U )ν(U, u).
(6.113)
Since ω(U ) = ALT [k(U ) ], we have the transformation law for the vorticity 2form, # $ ω(U ) = γALT P (U, u)[k(u) − a(u) ⊗ ν(U, u) ] 1 + γd(U )ν(U, u) 2 1 = γP (U, u)[ω(u) − a(u) ∧ ν(U, u)] 2 1 (6.114) + γd(U )ν(U, u) . 2 We shall now derive the analogous expression for the vorticity vector, namely 1 ω(U ) = γ 2 P (u, U )−1 [ω(u) + ν(U, u) ×u a(u)] 2 1 + γ curlU ν(U, u). 2 Let us write (6.114) in the form
where
(6.115)
1 ω(U ) − γd(U )ν(U, u) ≡ P (U, u)X, 2
(6.116)
1 X = γ ω(u) − a(u) ∧ ν(U, u) . 2
(6.117)
110
Local measurements
Next, contract both sides of (6.116) with (1/2)η(U ). The lefthand side becomes 1 γ η(U )αβγ ω(U )αβ − 2∇(U )[α ν(U, u)β] , (6.118) 2 2 that is ω(U )γ −
γ [curlU ν(U, u)]γ . 2
(6.119)
To deduce the eﬀect of contracting the righthand side of (6.116) by (1/2)η(U ), namely 1 η(U )P (U, u)X, 2 let us ﬁrst recall, from (3.20), that η(U )βγδ = U α ηαβγδ = −2U α u[α η(u)β]γδ + u[γ η(u)δ]αβ = γ [η(u)βγδ + ν α (uβ η(u)αγδ + uγ η(u)αδβ − uδ η(u)αγβ )] .
(6.120)
Then let us recall that P (U, u)α μ = P (u)α μ + γUα ν μ
(6.121)
implies, for any antisymmetric 2tensor X ∈ LRSu ⊗ LRSu , the following relation: [P (U, u)X]αβ = P (U, u)α μ P (U, u)β ν Xμν = Xαβ + γUβ Xασ ν σ − γUα Xβσ ν σ = [X + γU ∧ (X
ν)]αβ .
(6.122)
From the latter equation we deduce that η(U )γαβ [P (U, u)X]αβ = η(U )γαβ Xαβ ,
(6.123)
because terms along U vanish after contraction with η(U ). Using (6.120) we now ﬁnd that η(U )γαβ [P (U, u)X]αβ = γ[η(u)γαβ − νμ uγ η(u)μβα ]Xαβ ,
(6.124)
that is ∗(U )
[P (U, u)X] = γ[ ∗(u) X + u ⊗ (ν = γ[P (u) + u ⊗ ν] −1
= γP (u, U ) Since we have ∗(u)
Eq. (6.115) follows.
∗(u)
∗(u) ∗(u)
X)]
X
X.
1 X = γ ω(u) − a(u) ×u ν(U, u) , 2
(6.125)
(6.126)
6.10 Measurements of electric and magnetic ﬁelds
111
Similarly one can obtain the expansion tensor of CU in terms of the kinematical properties of Cu . Since θ(U ) = −SYM [k(U ) ] we have # $ − θ(U ) = γSYM P (U, u)[k(u) − a(u) ⊗ ν(U, u) ] − γSYM[∇(U )ν(U, u) ] 1 = −γP (U, u) θ(u) + [a(u) ⊗ ν(U, u) + ν(U, u) ⊗ a(u)] 2 − γSYM[∇(U )ν(U, u) ].
(6.127)
6.10 Measurements of electric and magnetic fields Assume that an observer u moves through an electromagnetic ﬁeld described by the Faraday 2form Fαβ . The observer learns of this electromagnetic ﬁeld by studying the behavior of a charged particle. Let U be the 4velocity of the particle; its trajectory is not a geodesic because of the Lorentz force, hence it has a 4acceleration e α β F βU , (6.128) a(U )α = μ0 where e is the particle’s electric charge and μ0 its mass. Our aim is to show how the measured force on the charge is deﬁned in terms of the charge and the fourdimensional quantities which characterize the observer and the electromagnetic ﬁeld. Because Fαβ is a 2form, we can apply the general splitting formula (3.14), valid for a pform. The result is: Fαβ = 2u[α E(u)β] + [∗(u) B(u)]αβ ,
(6.129)
where we have introduced the electric part of F , E(u)β ≡ F β ρ uρ ,
(6.130)
and the magnetic part, B(u)α ≡ ∗Fρ α uρ =
1 ραμν η Fμν uρ . 2
(6.131)
In coordinatefree notation the deﬁnition of such ﬁelds is E(u) = F
u,
B(u) ≡ ∗(u) [P (u)F ].
Let us decompose the force term f (U ) = (e/μ0 )F parallel component relative to u, namely
(6.132)
U into a transverse and a
f (U ) = P (u) f (U ) + T (u) f (U ).
(6.133)
112
Local measurements
The transverse force can be written in coordinate components as e P (u)αβ Fβμ U μ P (u)α β f (U )β = μ0 e = P (u)αβ [uβ E(u)μ − uμ E(u)β μ0 + η(u)βμ σ B(u)σ ] γ(uμ + ν(U, u)μ ) e ησπλ α uσ U λ B(u)π + E(u)α (uρ U ρ ) . =− μ0 As a result the transverse force term becomes e γ [E(u) + ν(U, u) ×u B(u)] . P (u)f (U ) = μ0
(6.134)
(6.135)
The parallel force component T (u)f (U ) can be written more conveniently as e (6.136) T (u)α β f (U )β = −uα uβ f (U )β = − uα uβ F β σ U σ . μ0 Recalling that U σ = γ[uσ + ν(U, u)σ ],
(6.137)
T (u)α β f (U )β = uα ν(U, u)β E(U )β .
(6.138)
we can write (6.136) as
The magnitude of this quantity measures the power of the electromagnetic action on the particle as seen by u. From the representation of F we have, relative to both the observers u and U , Fαβ = 2u[α E(u)β] + [∗(u) B(u)]αβ = 2U[α E(U )β] + [∗(U ) B(U )]αβ ;
(6.139)
hence, contracting with U and using the deﬁnitions so far introduced, we obtain E(U ) = γP (u, U )−1 [E(u) + ν(U, u) ×u B(u)] , B(U ) = γP (u, U )−1 [B(u) − ν(U, u) ×u E(u)] .
(6.140)
If we split the ﬁelds along directions parallel and transverse to that of the relative velocity νˆ(U, u) we obtain ν (U, u) + E ⊥ (u), E(u) = E (u)ˆ
(6.141)
and similarly for B(u). Hence (6.140) take the more familiar form [B(lrs) (u, U ) E(U )] = E (u), ⊥
[B(lrs) (u, U ) E(U )] = γ[E ⊥ (u) + ν(U, u) ×u B ⊥ (u)],
(6.142)
and [B(lrs) (u, U ) B(U )] = B (u), ⊥
[B(lrs) (u, U ) B(U )] = γ[ B ⊥ (u) − ν(U, u) ×u E ⊥ (u)].
(6.143)
6.11 Local properties of an electromagnetic ﬁeld
113
Recalling that the physical meaning of a 4vector is encoded in its magnitude, we can deduce that the magnitude of P (u)f (U ) is just the magnitude of the Lorentz force, and so the moduli of the electric and the magnetic parts describe the electric ﬁeld intensity and the magnetic induction. By introducing the complex vector ﬁeld Z(u) = E(u) − iB(u) (Landau and Lifshitz, 1975), Eqs. (6.142) and (6.143) can be written together as [B(lrs) (u, U ) Z(U )] = Z (u), ⊥
[B(lrs) (u, U ) Z(U )] = cosh α Z ⊥ (u) + i sinh α [ˆ ν (U, u) ×u Z ⊥ (u)],
(6.144)
where ν(U, u) = tanh α. In terms of components with respect to an observeradapted frame {eα } (e0 = u, with e3 along νˆ(U, u)), this complex vector representation reﬂects the isomorphism between the group of proper orthochronous Lorentz transformations and SO(3, C), the group of proper orthogonal transformations. In the 2plane orthogonal to e3 within LRSu one has
[B(lrs) (u, U ) Z(U )]⊥ 1 ⊥ [B(lrs) (u, U ) Z(U )] 2
=
−i sinh α cosh α
cosh α i sinh α
Z ⊥ (u)1 Z ⊥ (u)2
.
(6.145)
6.11 Local properties of an electromagnetic field An interesting application is the splitting of the energymomentum tensor of an electromagnetic ﬁeld. We have Tαβ
1 = 4π
Fαρ Fβ
ρ
1 2 TF [F ] αβ . 4π
(6.146)
1 (E(u)2 + B(u)2 ); 8π
(6.147)
1 − gαβ Fρσ F ρσ 4
=
An arbitrary observer u will measure: (i) an energy density E(u) = Tαβ uα uβ =
(ii) a momentum density (Poynting vector) P(u)α = −P (u)α β T β ρ uρ =
1 [E(u) ×u B(u)]α ; 4π
(6.148)
(iii) a uniform pressure p(u) =
1 1 Tr T (u) = E(u). 3 3
(6.149)
114
Local measurements Observers U(em) who measure a vanishing Poynting vector
Given an electromagnetic ﬁeld and a general observer u, the Poynting vector and the energy density are given by i ¯ [Z(u) ×u Z(u)], 2 1 1 4πE(u) = [E(u)2 + B(u)2 ] = Z(u)2 . 2 2
4πP(u) = E(u) ×u B(u) =
(6.150) (6.151)
It is well known (see exercise 20.6 of Misner, Thorne, and Wheeler, 1973) that observers exist who see a vanishing Poynting vector or, equivalently, electric and magnetic ﬁelds parallel to each other. Let U be an observer in relative motion with respect to u with U = γ[u + ν νˆ(U, u)] = u cosh α + νˆ(U, u) sinh α .
(6.152)
With respect to U , the magnitude of the Poynting vector is given by 4πP(U ) = E(U ) ×U B(U ) " i " ¯ ×u Z(u)) cosh 2α + i Z(u)2 sinh 2α " = " νˆ(U, u) · (Z(u) 2 1 = Z(u)2  cosh 2α tanh 2α(em) − sinh 2α 2  sinh 2(α − α(em) ) , (6.153) = 4πE(u) cosh 2α(em) where we have deﬁned ¯ νˆ(U, u) · (Z(u) × Z(u)) 2 Z(u) νˆ(U, u) · (E(u) ×u B(u)) =2 . E(u)2 + B(u)2
tanh 2α(em) = i
(6.154)
When α = α(em) , i.e. selecting U as a particular observer U(em) , we have P(U(em) ) = 0. With respect to U(em) , the electromagnetic energy density takes a minimum value. In fact, the electromagnetic energy density measured by U is in general 1 Z(u)2 [cosh 2α − sinh 2α tanh 2α(em) ] 2 cosh 2(α − α(em) ) . = 4πE(u) cosh 2α(em)
4πE(U ) =
(6.155)
Therefore, when α = α(em) , E(U(em) ) takes a minimum value equal to E(U(em) ) =
E(u) . cosh 2α(em)
(6.156)
6.13 Gravitoelectromagnetism
115
6.12 Timeplusspace (1 + 3) form of Maxwell’s equations The electromagnetic or Faraday 2form F is the exterior derivative of a 4potential 1form F = dA = u ∧ E(u) + ∗(u)B(u) .
(6.157)
The splitting of Maxwell’s equations d2 A = 0,
∗ ∗
d F = 4πJ
(6.158)
leads to divu B(u) + 2ω(u) ·u E(u) = 0,
(6.159)
curlu E(u) + a(u) ×u E(u) = −[£(u)u + Θ(u)]B(u),
(6.160)
divu E(u) − 2ω(u) ·u B(u) = 4πρ(u),
(6.161)
curlu B(u) + a(u) ×u B(u) − [£(u)u + Θ(u)]E(u) = 4πJ(u),
(6.162)
where J = ρ(u)u+J(u) is the splitting of the 4current. This shows that Maxwell’s equations in their traditional form hold only in an inertial frame where ω(u), a(u), and θ(u) vanish.
6.13 Gravitoelectromagnetism Consider a unit mass charged particle in motion in a given spacetime. The equation of motion (6.77), D(fw,U,u) p(U, u) (G) = F (U, u) + F(fw,U,u) , dτ(U,u)
(6.163)
becomes D(fw,U,u) p(U, u) = e[E(u) + ν(U, u) ×u B(u)] + dτ(U,u) − γ[a(u) + ω(u) ×u ν(U, u) + θ(u) ν(U, u)],
(6.164)
where F (U, u) has been replaced by the Lorentz force (6.135) (rescaled by a γ factor because on the lefthand side the temporal derivative involves the relative standard time and not the proper time as a parameter along the particle’s world (G) line), and the gravitational force F(fw,U,u) is given by (3.156), namely (G)
F(fw,U,u) = −γ[a(u) + ω(u) ×u ν(U, u) + θ(u) ν(U, u)].
(6.165)
Comparing these two forces leads to the identiﬁcation of a gravitoelectric ﬁeld Eg (u) = −a(u)
(6.166)
116
Local measurements
and a gravitomagnetic ﬁeld Bg (u) = ω(u),
(6.167)
so that (G)
F(fw,U,u) = γ[Eg (u) + ν(U, u) ×u Bg (u) + θ(u) ν(U, u)].
(6.168)
Therefore, apart from the gamma factor in the gravitational force and the contribution of the expansion tensor θ(u), the Lorentz force is closely similar to the gravitational force. This analogy is what we call gravitoelectromagnetism, even though the gravitational and electromagnetic interactions remain distinct. It is worth noting that (1) no linear approximation of the gravitational ﬁeld has been made here in order to introduce gravitoelectromagnetism (Jantzen, Carini, and Bini, 1992); (2) the analogy between gravity and electromagnetism, developed here starting from the analysis of test particle motion, could be pursued similarly studying ﬂuid or ﬁeld motions.
6.14 Physical properties of fluids A physical observer who studies the behavior of a relativistic ﬂuid which is in general away from thermodynamic equilibrium must specify in his own rest frame the parameters which invariantly characterize the ﬂuid and determine its evolution. For this purpose one would need a full thermodynamic treatment (Israel, 1963), which is well beyond the scope of this book. We shall instead outline some of the results in the case of a simple ﬂuid, i.e. when the deviations from local equilibrium are small and the selfgravity of the ﬂuid is neglected. If T is the energymomentum tensor of the ﬂuid and u is the vector ﬁeld tangent to a congruence of observers, the following decomposition holds for each ﬂuid element (Ellis, 1971; Ellis and van Elst, 1998): T = T (u) + u ⊗ q(u) + q(u) ⊗ u + ρ(u)u ⊗ u,
(6.169)
T (u)αβ ≡ P (u)α ρ P (u)β σ Tρσ ,
(6.170)
q(u)α ≡ −P (u)α Tρσ u ,
(6.171)
where
ρ
ρ(u) ≡ Tρσ u u . ρ σ
σ
(6.172)
An insight into the physical interpretation of these quantities arises from the definition of the 4momentum of an extended body. If we choose a spacelike hyperspace Σ in such a way that its unit normal is parallel to u, then the quantities Pˆ α = −T α β uβ
(6.173)
6.14 Physical properties of ﬂuids
117
turn out to be the components of the 4momentum density of the ﬂuid. With respect to the observer u this 4vector can then be decomposed as Pˆ α = P (u)α β Pˆ β + T (u)α β Pˆ β = −P (u)α β T β μ uμ + uα uβ T β μ uμ = q(u)α + ρ(u)uα .
(6.174)
The ﬁrst term q(u)α represents the threedimensional energy ﬂux density and describes not only the linear momentum of the ﬂuid elements but thermoconduction processes such as convection, radiation and heat transfer (Landau and Lifshitz, 1959; Novikov and Thorne, 1973). The second term ρ(u)uα describes an energy density current, so its magnitude ρ(u) is just the energy density of the ﬂuid relative to u. The remaining transverse quantity T (u) in (6.170), being a symmetric tensor in the threedimensional LRSu , can be written as the sum of a tracefree tensor and a trace, as follows: T (u)αβ = [T (u)]TF αβ +
1 [Tr T (u)] P (u)αβ , 3
(6.175)
where [T (u)]TF is the tracefree part of T (u) and Tr T (u) is its trace. The quantity [T (u)]TF is termed the viscous stress tensor and describes nonisotropic dissipative processes; to the assumed accuracy (namely small deviations from local equilibrium) the viscous stresses enter the energymomentum tensor as linear perturbations to the equilibrium conﬁguration and are given in terms of the ﬂuid shear, with a coeﬃcient called the shear viscosity. The trace Tr T (u) is only related to the uniform properties of the ﬂuid; it can be written as ˜ Tr T (u) = 3 p(u) + φ(u) , (6.176) ˜ where p(u) is the hydrostatic pressure and φ(u) is a contribution to the pressure arising from the appearance of a volume (or bulk) viscosity. Finally, we deﬁne as comoving with the ﬂuid that observer U with respect to whom the quantities q(U )α describe only processes of thermoconduction, the ﬂuid elements having zero momentum (Landau and Lifshitz, 1959). This observer’s fourvelocity will be hereafter be identiﬁed as the 4velocity of the ﬂuid; the other physical quantities, density ρ(U ), pressure p(U ), and internal mechanical stresses T (U ), will be denoted by ρ0 , p0 , and T0 , respectively. In the case of a perfect ﬂuid, namely when [T (U )]TF = 0,
φ˜ = 0,
q(U ) = 0,
the energymomentum tensor becomes Tαβ = [p(U ) + ρ(U )]Uα Uβ + gαβ p(U ).
(6.177)
118
Local measurements
A perfect ﬂuid is termed dust if, in the comoving frame, p(U ) = 0; hence it is described by the energymomentum tensor Tαβ = ρ(U )Uα Uβ .
(6.178)
In what follows we will discuss separately the cases of ordinary ﬂuids (i.e. those without thermal stresses) and ﬂuids with thermal stress, discussing both absolute and relative dynamics with respect to a general observer.
Ordinary ﬂuids: absolute dynamics A ﬂuid is termed ordinary when its stresses in the comoving frame, hereafter referred to as proper stresses, are purely mechanical. In this case its energymomentum tensor can be written as T = ρ0 U ⊗ U + T0 ,
T0
U = 0,
ρ0 = μ0 ˆ0 ,
(6.179)
where U is the 4velocity ﬁeld of the ﬂuid elements, ρ0 includes the matter energy 0 ), and T0 ≡ T (U ) is the proper mechanical stress (μ0 ) and internal energy (ˆ tensor. In the presence of matter and eventually other external ﬁelds the evolution equations of the ﬂuid are div T = μ0 f,
(6.180)
where the spacetime divergence operation div is deﬁned in (2.111) and f represents the action of any external ﬁeld. Let us consider the splitting of (6.180) in the comoving frame of the ﬂuid. First of all we have ∇α T αβ = ∇α (ρ0 U β )U α + ρ0 U β ∇α U α + [div T0 ]β = μ0 f β ,
(6.181)
∇U (ρ0 U ) + ρ0 U Θ(U ) = μ0 fm ,
(6.182)
that is
where Θ(U ) = ∇α U α and fm ≡ f −
1 div T0 μ0
(6.183)
is the total mechanical force, including both internal and external actions given respectively by (1/μ0 )div T0 and f . Equation (6.182) can then be written in the form ρ0 a(U ) + U div (ρ0 U ) = μ0 fm .
(6.184)
Projecting (6.184) orthogonally to U then gives (e)
(i)
ρ0 a(U ) = μ0 P (U )fm ≡ μ0 (f0 + f0 ),
(6.185)
6.14 Physical properties of ﬂuids
119
where the contributions from external and internal forces have been made explicit according to the following deﬁnitions: (e)
f0
= P (U )f
(6.186)
and (i)
1 P (U )[div T0 ] μ0 1 = − [divU T0 + a(U ) T0 ] , μ0
f0 ≡ −
(6.187)
where we have introduced the spatial divergence operator (see (3.37)) [divU T0 ]α = ∇(U )μ T0μα .
(6.188)
The following notation will prove useful: w0 = −f · U
Proper power of external forces per unit of mass;
(i)
w0 = −U · div T0 = Tr[T0
θ(U )]
Power of internal forces per unit proper volume;
(i)
W0 = μ0 w0 − w0 = −μ0 (fm · U )
Total power per unit of proper volume.
(6.189)
Multiplying Eq. (6.184) by U , one ﬁnds the energy equation div (ρ0 U ) = W0 .
(6.190)
Excluding processes of matter creation or annihilation, one must add to this equation the conservation of the proper mass of the ﬂuid, 0 = div (μ0 U ) = ∇U μ0 + μ0 Θ(U ).
(6.191)
In this case, using the relation ρ0 = μ0 ˆ0 , the energy equation (6.190) becomes ∇U ˆ0 =
1 W0 ; μ0
(6.192)
therefore, recalling (6.189), we ﬁnd Dˆ 0 1 (i) = w0 − w , dτU μ0 0
(6.193)
which represents the First Law of Thermodynamics in the rest frame of the ﬂuid. Summarizing, the fundamental equations in the rest frame of the ﬂuid are given by (i)
(e)
ρ0 a(U ) = μ0 (f0 + f0 )
Equation of motion;
1 (i) Dˆ 0 = w0 − w dτU μ0 0
Energy equation.
(6.194)
120
Local measurements
The equation of motion can also be cast in the following form. According to (6.185), (i)
μ0 f0 = −divU T0 − a(U ) T0 ,
(6.195)
(i)
that is, f0 contains the ﬂuid acceleration a(U ). One can collect terms involving acceleration, rewriting the equation of motion in the form (e)
[ρ0 P (U ) − T0 ] a(U ) = μ0 f0 − divU T0 ,
(6.196)
similar to the second law of test particle dynamics.
Ordinary ﬂuids: relative dynamics We now study ﬂuid dynamics as seen by an observer u not comoving with the ﬂuid. Splitting the 4velocity ﬁeld of the ﬂuid in the standard way, U = γ(U, u)[u + ν(U, u)],
(6.197)
the evolution equations with respect to u are obtained by the spatial and temporal projections of (6.194) with respect to u. Projecting (6.185) ﬁrst orthogonally to U and then orthogonally to u gives ρ0 P (u, U )a(U ) = μ0 P (u, U )fm .
(6.198)
Using relation (6.106), namely (G)
P (u, U )a(U ) = −γF(fw,U,u) + γ 2 P (u, U, u)a(fw,U,u) ,
(6.199)
P (u, U, u) ≡ P (u, U ) P (U, u),
(6.200)
with
one ﬁnds (G)
ρ0 γ 2 P (u, U, u)a(fw,U,u) = ρ0 γF(fw,U,u) + μ0 P (u, U )fm .
(6.201)
Let us now deﬁne, following Ferrarese and Bini (2007), the relative energy and mass densities μ ˆ = ρ0 γ 2 ,
μ = μ0 γ 2 ,
(6.202)
where the presence of the square of the gamma factor has a simple explanation in terms of the Lorentz transformation from the comoving frame U to the observer’s frame u: one γ comes from the energy transformation and the other from the volume transformation. Equation (6.201) then becomes μ ˆa(fw,U,u) = P (u, U, u)−1 [ρ0 γF(fw,U,u) + μ0 P (u, U )fm ] (G)
= P (u, U, u)−1 [ρ0 γF(fw,U,u) + μ0 P (u)fm − γνW0 ]. (G)
(6.203)
6.14 Physical properties of ﬂuids
121
Let us now introduce the total relative force (tot)
(G)
μF(fw,U,u) = μ0 P (u)fm + ρ0 γF(fw,U,u) ,
(6.204)
which contains the contributions of external, internal, and gravitational forces. The equation of motion becomes μ ˆa(fw,U,u) = μP (u, U, u)−1 F(fw,U,u) − (tot)
W0 ν γ
= μP (u, U, u)−1 F(fw,U,u) − W ν, (tot)
(6.205)
where W0 /γ ≡ W represents the total relative power, namely (i)
W =
μ0 w0 − w0 = μw ˜ − w(i) , γ
where we have introduced, from (6.202), the quantities w ˜≡
w0 , γ3
(i)
w(i) ≡
w0 , γ
(6.206)
which represent the relative power of the external forces per unit of mass (w) ˜ and of the internal forces (w(i) ), respectively. Therefore W (tot) (tot) (6.207) ˆ0 a(fw,U,u) = F(fw,U,u) − ν(ν · F(fw,U,u) ) − ν , μ dˆ W = γ2 . dτ(U,u) μ
(6.208)
Here dτ(U,u) = γdτU is the relative standard time, ˆ = ˆ0 , μ ˆ = ˆμ, and we have used the relation P (u, U, u)−1 = P (U, u)−1 P (u, U )−1 = P (u) − ν(U, u) ⊗ ν(U, u).
(6.209)
Transformation law for proper mechanical stresses The transformation of the spatial (with respect to U ) and symmetric tensor T0 = T (U ) along u and onto LRSu proceeds in a standard way. Let us denote T (u, U ) = P (u, U )T (U ),
(6.210)
where the projection is understood to be on each index. Acting on (6.210) with the operator P (u, U )−1 and recalling that P (u, U )−1 P (u, U ) = P (U ),
(6.211)
T (U ) = P (u, U )−1 T (u, U ).
(6.212)
we obtain
122
Local measurements
If u denotes another family of observers, then T (u , U ) = P (u , U )T (U );
(6.213)
hence the transformation law for mechanical stresses follows: T (u , U ) = [P (u , U )P (u, U )−1 ]T (u, U ).
(6.214)
As an example let us evaluate the relative power of internal forces w(i) . From its deﬁnition (see (6.189)) we have (i)
w(i) =
1 w0 = [T (U )αβ ∇β Uα ]. γ γ
(6.215)
Recalling that Uα = γ [uα + ν(U, u)α ], it follows that w(i) = T (U )αβ ∇β (uα + να ).
(6.216)
Using the transformation law for mechanical stresses (6.214) one then ﬁnds w(i) = T (U )αβ ∇β (uα + να ) = T (u, U )μν [P (u, U )−1 ]α μ [P (u, U )−1 ]β ν [−uβ a(u)α − k(u)αβ + ∇β να ] = T (u, U )μσ [∇σ νμ + νσ ∇u νμ ] + Tr [T (u, U ) θ(u)] + [ν
T (U, u)] · [−γ −1 F(fw,U,u) − 2ν (G)
θ(u) − ν(ν · a(u))],
(6.217)
where the representation P (u, U )−1 = P (u) + u ⊗ ν(U, u)
(6.218)
of the mixed projector has been used.
Example: the perfect ﬂuid In this case T0 = p0 P (U ), where P (U ) is the projection operator orthogonal to U . Using mass conservation we have (i)
w0 = p0 Θ(U ).
(6.219)
According to the notation previously introduced (see Eqs. (6.189)), we also have div T0 = ∇(U )p0 + p0 U Θ(U ) + p0 a(U ),
(6.220)
μ0 fm = μ0 f − ∇(U )p0 − p0 U Θ(U ) − p0 a(U ),
(6.221)
μ0 P (U )fm = μ0 [f − U w0 ] − [∇(U )p0 + p0 a(U )],
(6.222)
and the evolution equations reduce to (ρ0 + p0 )a(U ) = −∇(U )p0 + μ0 [f − U w0 ], dρ0 = −(ρ0 + p0 )Θ(U ) + μ0 w0 . dτU
(6.223)
6.14 Physical properties of ﬂuids
123
In terms of the internal energy, the First Law of Thermodynamics becomes p0 Dˆ 0 = w0 − Θ(U ). dτU μ0 One may also introduce the proper entropy s0 per unit of proper mass, TK
ds0 = μ0 w0 , dτU
(6.224)
where TK is the temperature. In the absence of external forces (f = 0 and hence w0 = 0) one has the conservation of the entropy density s0 along the ﬂow lines of U , Ds0 = 0, dτU and the acceleration of the ﬂuid lines reduces to a(U ) = −
1 ∇(U )p0 . ρ0 + p0
The relative point of view can be obtained directly from (6.205): a(fw,U,u) = −
1 [P (U, u)∇p0 − ν(ν · P (U, u)∇p0 )] ρ0 + p0
+ γ −1 [F(fw,U,u) − ν(ν · F(fw,U,u) )]. (G)
(G)
(6.225)
Hydrodynamics with thermal ﬂux: absolute formulation We can generalize our previous results to the case of a ﬂuid with thermal ﬂux. We shall here illustrate only the general procedure, omitting unnecessary detail. The energymomentum tensor can be written as in (6.179) but having in addition thermal stresses Q0 ≡ Q(U ): T = ρ0 U ⊗ U + S0 , S0 =
T0 + Q0 , %&'( %&'( mechanical thermal
where Q0 = U ⊗ q0 + q0 ⊗ U, with q0 = q0 (U ) and ALT S0 = 0, ALT X0 = 0, X0
U = 0, q0
U = 0.
124
Local measurements
Let us ﬁrst consider the splitting of the equations of motion div T = μ0 f in the comoving frame of the ﬂuid. Projection along U and onto LRSU gives (i)
(e)
(th)
ρ0 a(U ) = μ0 P (U )(fm + fth ) = μ0 P (U )(f0 + f0 + f0
),
W0 Q0 dˆ 0 = + (qc )0 ≡ , dτU μ0 μ0
(6.226)
where (qc )0 is the thermal conduction power in the comoving frame. We now have a thermal force and a thermal power, μ0 P (U )fth = −P (U )div Q,
(6.227)
and (i)
Q0 = −[μ0 f − div S] · U = (μ0 q0tot − w0 ) = W0 + μ0 (qc )0 ,
(6.228)
with q0tot = w0 + (qc )0 , μ0 (qc )0 = U · div Q ≡ −[∇ · q0 + a(U ) · q0 ], (i)
w0 = −U · div T .
(6.229)
Hydrodynamics with thermal ﬂux: relative formulation Let us project the evolution equations of the proper frame along and orthogonally to u. We ﬁnd Q0 tot , (6.230) −ν μ ˆa(fw,U,u) = μP (u, U, u)−1 F(fw,U,u) γ where tot μF(fw,U,u) = μ0 P (u)[fm + fth ].
(6.231)
From the energy theorem we ﬁnd instead Q dˆ = γ2 , dτ(U,u) μ
(6.232)
where Q = Q0 /γ.
Example: the viscous ﬂuid of LandauLifshitz The viscous ﬂuid of LandauLifshitz (see Misner, Thorne, and Wheeler, 1973, p. 567) is characterized by the following mechanical stress tensor: T0 = [p0 − ζΘ(U )]P (U ) − 2ησ(U ),
(6.233)
6.14 Physical properties of ﬂuids
125
where σ(U ) = θ(U ) − 13 Θ(U )P (U ) is the tracefree part of the expansion ﬁeld. We have μ0 (qc )0 = −[divU q0 + 2a(U ) · q0 ], (i) w0
(6.234)
= p0 Θ(U ) − [ζΘ(U ) + 2ηTr(σ(U ) )]. 2
2
(6.235)
Also in this case one introduces the scalar entropy s, related to the temperature by the First Law of Thermodynamics (generalized to include the eﬀects of the thermal action), (i)
TK
p0 Θ(U ) − w0 Ds = q0tot + ≡ w0 +(qc )0 +[ζΘ(U )2 +2ηTr(σ(U )2 )], dτU μ0
(6.236)
and the entropy 4vector (see Exercise 22.7 of Misner, Thorne, and Wheeler, 1973) q0 . (6.237) S = μ0 sU + TK Taking into account the mass conservation law, ∇U (μ0 U ) = 0, one ﬁnds TK ∇ · S = μ0 w0 + [ζΘ(U )2 + 2ηTr(σ(U )2 )] − q0 · [∇ ln TK + a(U )].
(6.238)
In the absence of external forces we have f = 0 and w0 = 0; therefore the above expressions are considerably simpliﬁed.
7 Nonlocal measurements
The Principle of Equivalence states that gravitational and inertial accelerations cannot be distinguished from each other if one neglects the curvature of the background geometry. But even if curvature is taken into account, the choice of the observer together with a frame adapted to him/her pollutes the measurements with inertial contributions which are entangled with the curvature in a nonseparable way. The curvature is responsible for a relative acceleration among freely falling particles. This acceleration induces a ﬁeld of strains which can be measured with a suitable experimental device; however, a relative acceleration also arises from orbital constraints if the bodies are not in free fall. In this case a relativistically complete and correct description of the relative strains must also take into account the properties of the observer and of his frame. A measurement which carries the signature of the spacetime curvature within its measurement domain is termed nonlocal. Here we shall ﬁrst identify the components of the spacetime curvature relative to a given frame and then discuss various ways to measure them.
7.1 Measurement of the spacetime curvature Let u be a vector ﬁeld whose integral curves form a congruence Cu which we assume to be representative of a family of observers. With respect to the latter, then, the spatial and temporal splitting of the Riemann tensor identiﬁes the following three spatial ﬁelds: E(u)αβ = Rαμβν uμ uν , H(u)αβ = −R ∗ αμβν uμ uν , F(u)αβ = [∗ R∗ ]αμβν uμ uν ,
(7.1)
where E(u)[αβ] = 0 = F(u)[αβ] and H(u)α α = 0. The ﬁrst two quantities are termed, respectively, the electric and magnetic parts of the Riemann tensor. The
7.1 Measurement of the spacetime curvature
127
20 independent components of the Riemann tensor are then summarized by the 6 independent components of the electric part (spatial and symmetric tensor), the 8 independent components of the magnetic part (spatial and tracefree tensor), and the 6 independent components of the mixed part (spatial and symmetric tensor). Consider a frame {eα } (not necessarily orthonormal) adapted to the observer u so that u = e0 with ea a basis in LRSu . Then all the above spatial quantities can be written as E(u)ab = Ra0b0 , 1 η(u)cd b Ra0cd , 2 1 = η(u)a cd η(u)b ef Rcdef , 4
∗ H(u)ab = −Ra0b0 =
F(u)ab = [∗ R∗ ]a0b0
(7.2)
and can be inverted to give Ra0 cd = H(u)ab η(u)bcd , Rabcd = η(u)abr η(u)cds F(u)rs .
(7.3)
Using these relations one has also the frame components of the Ricci tensor Rα β = Rμα μβ , R0 0 = −E(u)c c , R0 a = η(u)abc H(u)bc , Ra b = −E(u)a b − F(u)a b + δ a b F(u)c c ,
(7.4)
R = R0 0 + Ra a = −2(E(u)c c − F(u)c c ).
(7.5)
so that
As shown in (2.66), the Riemann tensor can be written in terms of the Weyl tensor, the Ricci tensor, and the scalar curvature; in four dimensions we have Rαβ γδ = C αβ γδ + δ α [γ Sδ] β − δ β [γ Sδ] α = C αβ γδ + 2δ [α [γ Sδ] β] ,
(7.6)
where 1 (7.7) Sαβ = Rαβ − R gαβ . 6 Clearly, in vacuum (Rαβ = 0, R = 0), the Weyl and Riemann tensors coincide. Similarly to the Riemann tensor, the splitting of the Weyl tensor identiﬁes the following two spatial ﬁelds, because of the identity ∗ C ∗ = −C (or ∗ C = C ∗ ): E(u)αβ = Cαμβν uμ uν ,
H(u)αβ = −C ∗ αμβν uμ uν .
(7.8)
Tensors E(u) and H(u) are termed, respectively, the electric and magnetic parts of the Weyl tensor and are related to E(u), H(u), and F(u) by the following relations:
128
Nonlocal measurements 1 [E(u) − F(u)](TF) , 2 H(u) = SYM H(u), E(u) =
(7.9)
from (7.6). Written in terms of components with respect to a frame adapted to u, we have E(u)a b =
1 1 E(u)a b − F(u)a b − δ a b (E(u)c c − F(u)c c ) , 2 3
H(u)ab = H(u)(ab) .
(7.10)
In Chapter 3 we evaluated the components of the Riemann tensor in an adapted frame; in fact expressions (3.92), (3.97), and (3.98) are equivalent to E(u)a b = [∇(u)b + a(u)b ]a(u)a + ∇(u)(fw) k(u)a b − [k(u)2 ]a b , H(u)ab = − ∇(u)[c k(u)a d] η(u)bcd + 2a(u)a ω(u)b , 1 F(u)a b = [θ(u)2 − Θ(u)θ(u)]a b − δ a b [Trθ(u)2 − Θ(u)2 ] 2 + 3ω(u)a ω(u)b − G(sym) a b ,
(7.11)
where 1 G(sym) a b = R(sym) a b − δ a b R(sym) , 2
(7.12)
with R(sym) a b = R(sym) ca cb and R(sym) = R(sym) a a from (3.105). Furthermore, E(u)c c = [∇(u)c + a(u)c ]a(u)c − ∇(u)(fw) Θ(u) + 2ω(u)c ω(u)c − Tr θ(u)2 , H(u)c c = 2[∇(u)c − a(u)c ]ω(u)c ≡ 0, 1 1 F(u)c c = − [Tr θ(u)2 − Θ(u)2 ] + 3ω(u)c ω(u)c + R(sym) , 2 2
(7.13)
where the trace of H(u) vanishes identically from (3.82). From (7.11) and recalling the deﬁnition of the symmetric curl (see Eqs. (3.40) and (3.41)), we have H(u)ab = H(u)(ab) = −[Scurlu k(u)]ab − 2a(u)(a ω(u)b) = −[Scurlu ω(u)]ab + [Scurlu θ(u)]ab − 2a(u)(a ω(u)b) $TF # = [Scurlu θ(u)]ab − 2a(u)(a ω(u)b) − ∇(u)(a ω(u)b)
(7.14)
and E(u)ab =
1 [∇(u)(a a(u)b) + a(u)a a(u)b − ∇(u)(fw) θ(u)ab 2 + 2ω(u)a ω(u)b − 2[θ(u)2 ]ab + Θ(u)θ(u)ab − R(sym)ab ]TF (7.15)
7.2 Vacuum Einstein’s equations in 1+3 form
129
If u is expansionfree, the electric and magnetic parts of the Weyl tensor reduce to 1 [∇(u)(a a(u)b) + a(u)a a(u)b + 2ω(u)a ω(u)b − R(sym)ab ]TF 2 = −[2a(u)(a ω(u)b) + ∇(u)(a ω(u)b) ]TF (7.16)
E(u)ab = H(u)ab
7.2 Vacuum Einstein’s equations in 1+3 form Einstein’s equations without the cosmological constant are given by 1 Rαβ − Rgαβ = 8πTαβ . 2
(7.17)
In the absence of matter energy sources (Tαβ = 0) they become Rαβ = 0. Using (7.4) we can write these equations in terms of the kinematical parameters of the observer congruence Cu and their derivatives. From (7.2)1 and with respect to a frame adapted to u, we have E(u)c c = 0 H(u)[ab] = 0 E(u)a b + F(u)a b = δ a b F(u)c c
(R0 0 = 0), (R0 a = 0), (Ra b = 0).
(7.18)
Using the tracefree property of E(u) in the last equation we ﬁnd F(u)c c = 0 so that from the above equations we deduce that E(u)a b = −F(u)a b .
(7.19)
Moreover, from (7.13)1 , the condition R0 0 = 0 gives [∇(u)c + a(u)c ]a(u)c − ∇(u)(fw) Θ(u) + 2ω(u)c ω(u)c − Trθ(u)2 = 0.
(7.20)
From (7.4)2 , the components R0 a = 0 can be written in the form ηcab H(u)ab = 0,
(7.21)
which is equivalent to − ∇(u)a [θ(u)a c − Θ(u)δ a c ] − [curlu ω(u)]c − 2[a(u) ×u ω(u)]c = 0.
(7.22)
Finally, Ra b = 0 gives − [∇(u)(lie) +Θ(u)]θ(u)a b − [∇(u)(b + a(u)(b ]a(u)a) + R(sym) a b − 2[ω(u)a ω(u)b − δ a b ω(u)c ω(u)c ] = 0.
(7.23)
130
Nonlocal measurements
This set of equations takes a simpliﬁed form when the observer congruence is Bornrigid, i.e. θ(u) = 0. In this case we have [∇(u)c+ a(u)c ]a(u)c + 2ω(u)c ω(u)c = 0,
(7.24)
[curlu ω(u)]c + 2[a(u) ×u ω(u)]c = 0,
(7.25)
R(sym)ab − [∇(u)(b + a(u)(b ]a(u)a) − 2[ω(u)a ω(u)b − P (u)ab ω(u)c ω(u)c ] = 0.
(7.26)
Contracting the indices in (7.26) yields R(sym) − ∇(u)b a(u)b − a(u)a a(u)a + 4ω(u)a ω(u)a = 0
(7.27)
which, using (7.24), leads to R(sym) + 6ω(u)a ω(u)a = 0.
(7.28)
7.3 Divergence of the Weyl tensor in 1+3 form The divergence of the Weyl tensor ∇δ C δ αβγ in an adapted frame (1 + 3 form) is represented by the following independent ﬁelds: ∇δ C δ 0a0 ,
∇δ C δ (ab)0 ,
∇δ ∗ C δ 0a0 ,
∇δ ∗ C δ (ab)0 .
(7.29)
Other components give nonindependent relations due to the tracefree property of the Weyl tensor and the Ricci identity. It is suﬃcient to deduce the 1 + 3 form of the ﬁrst pair of the above relations because the remaining two are obtained by the substitution E(u) → H(u) and H(u) → −E(u), similar to what is done for the electromagnetic ﬁeld. We have ∇δ C δ 0a0 = eδ C δ 0a0 + Γδ μδ C μ 0a0 − Γσ 0δ C δ σa0 − Γσ aδ C δ 0σ0 − Γσ 0δ C δ 0aσ .
(7.30)
Substituting the values (3.71) of the connection coeﬃcients of an adapted frame and using deﬁnitions (7.8), this equation can be cast in the form # $ (7.31) ∇δ C δ 0a0 = divu E(u) − 2∗(u) [θ(u) H(u)] − 2H(u) ω(u) . a
Similarly we have ∇δ C δ (ab)0 = ∇(u)(fw) E(u)ab − [Scurlu H(u)]ab + 2Θ(u)E(u)ab − 2[a(u) ×u H(u)]ab − [ω(u) ×u E(u)]ab − 3[SYM(θ(u)
E(u))]TF ab .
(7.32)
Summarizing, the set of equations representing the divergence of the Weyl tensor is the following ∇δ C δ 0a0 = {divu E(u) − 2∗(u) [θ(u) H(u)] − 2H(u) ω(u)}a , ∇δ ∗ C δ 0a0 = {divu H(u) + 2∗(u) [θ(u) E(u)] + 2E(u) ω(u)}a ,
7.4 Electric and magnetic parts of the Weyl tensor
131
∇δ C δ (ab)0 = ∇(u)(fw) E(u)ab − [Scurlu H(u)]ab + 2Θ(u)E(u)ab − 2[a(u) ×u H(u)]ab − [ω(u) ×u E(u)]ab − 3[SYM(θ(u) E(u))]TF ab , ∇δ ∗ C δ (ab)0 = ∇(u)(fw) H(u)ab + [Scurlu E(u)]ab + 2Θ(u)H(u)ab + 2[a(u) ×u E(u)]ab − [ω(u) ×u H(u)]ab − 3[SYM(θ(u) H(u))]TF ab .
(7.33)
In vacuum (Tα β = 0) and with respect to an observer u described by an expansionfree (θ(u) = 0) congruence of integral curves, and from (7.8), we can write Einstein’s equations in Maxwelllike form (compare with (6.159)–(6.162)): 0 = divu E(u) − 2H(u) ω(u), 0 = divu H(u) + 2E(u) ω(u), 0 = ∇(u)(fw) E(u) − [Scurlu H(u)] − 2[a(u) ×u H(u)] − [ω(u) ×u E(u)], 0 = ∇(u)(fw) H(u) + [Scurlu E(u)] + 2[a(u) ×u E(u)] − [ω(u) ×u H(u)]. (7.34)
7.4 Electric and magnetic parts of the Weyl tensor Let U and u be two diﬀerent families of observers related by a boost U = γ(U, u)[u + ν(U, u)].
(7.35)
Both of these observers can be used to split the Weyl tensor (and similarly the Riemann tensor) in terms of associated electric and magnetic parts, E(u)αβ = Cαμβν uμ uν ,
∗ H(u)αβ = −Cαμβν uμ uν , ∗ H(U )αβ = −Cαμβν U μU ν .
(7.36)
E(U )αβ = γ(U, u)2 Cαμβν [uμ + ν(U, u)μ ][uν + ν(U, u)ν ],
(7.37)
E(U )αβ = Cαμβν U μ U ν , Using (7.35) we have, for example,
that is, abbreviating γ(U, u) = γ and ν(U, u) = ν, E(U )αβ = γ 2 [E(u)αβ + Cαμβν uμ ν ν + Cαμβν ν μ uν + Cαμβν ν μ ν ν ].
(7.38)
Let eα denote a frame adapted to u (namely u = e0 and ea with a = 1, 2, 3 spanning the local rest space of u). We then have E(U )ab = γ 2 [E(u)ab + Ca0bc ν c + Cacb0 ν c + Cacbd ν c ν d ] = γ 2 [E(u)ab + H(u)af η(u)f bc ν c + H(u)bf η(u)f ac ν c − E(u)f g η(u)f ac η(u)g bd ν c ν d ], E(U )00 = γ 2 [C0b0c ν b ν c ] = γ 2 [E(u)bc ν b ν c ], E(U )0b = γ 2 [C0cb0 ν c + C0cbd ν c ν d ] = γ 2 [−E(u)cb ν c − H(u)cf η(u)f bd ν c ν d ].
(7.39)
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Nonlocal measurements
Therefore, in compact form, we have [P (u, U )E(U )]αβ = γ 2 [E(u)αβ + 2H(u)(αf  η(u)f β)c ν c − E(u)f g η(u)f αc η(u)g βd ν c ν d ]
(7.40)
and similarly [P (u, U )H(U )]αβ = γ 2 [H(u)αβ − 2E(u)(αf  η(u)f β)c ν c − H(u)f g η(u)f αc η(u)g βd ν c ν d ].
(7.41)
7.5 The BelRobinson tensor In the theory of general relativity, gravitation is described as the curvature of the background geometry; hence any manifestation of pure gravity is presented in terms of the Riemann tensor. Neglecting the contribution to gravity by the local distribution of matterenergy (that is, setting Rαβ = 0), a possible generalization of the energymomentum tensor to the gravitational ﬁeld is the BelRobinson tensor (Bel, 1958), deﬁned by Tαβ γδ =
1 (Cαρβσ C γρδσ + ∗Cαρβσ ∗C γρδσ ). 2
(7.42)
In standard terminology, we use superenergy density and superPoynting vector (Maartens and Bassett, 1998) in reference to the analogous contractions in the electromagnetic case relative to a general observer u, namely E (g) (u) = Tαβγδ uα uβ uγ uδ 1 = [E(u)2 + H(u)2 ], 2 P (g) (u)α = Tαβγδ uβ uγ uδ = [E(u) ×u H(u)]α ,
(7.43)
where the notation (3.39) has been used. Similarly to what was done for the electromagnetic ﬁeld in Sections 6.11 and 6.12 with the electric and magnetic parts of the Weyl tensor, we can introduce the complex (symmetric tracefree) spatial tensor ﬁeld Z (g) (u) = E(u) − iH(u), abbreviated to Z(u) in this section, and evaluate the eﬀect of a boost in a general direction U , diﬀerent from u, in terms of the orthogonal decompositions with respect to the relative spatial velocity of the observers u and U . Z(u) can be decomposed into a scalar Z (u), a vector Z ⊥ (u), and a tensor Z ⊥⊥ (u), the latter two being orthogonal to νˆ(U, u), i.e. ν (U, u) ⊗ νˆ(U, u) + Z ⊥ (u) ⊗ νˆ(U, u) Z(u) = Z (u)ˆ + νˆ(U, u) ⊗ Z ⊥ (u) + Z ⊥⊥ (u),
(7.44)
where Z ⊥⊥ (u) can then be further decomposed into its puretrace part involving Tr Z ⊥⊥ (u) = −Z (u) and its tracefree part Z ⊥⊥(TF) (u).
7.5 The BelRobinson tensor
133
Let U be a family of observers; by deﬁnition we have Z(U ) = E(U ) − iH(U ).
(7.45)
Moreover, Eqs. (7.40) and (7.41) imply that [P (u, U )Z(U )]αβ = γ 2 [Z(u)αβ + 2iZ(u)(αf  η(u)f β)c ν c − Z(u)f g η(u)f αc η(u)g βd ν c ν d ].
(7.46)
Alternatively, one can boost Z(U ) onto the local rest space of u; in this case we have B(lrs) (u, U )Z(U ) = B(lrs)u (u, U )P (u, U )Z(U ),
(7.47)
where the map B(lrs)u (u, U ) has been introduced in (3.134), conveniently rewritten here as α 1 α B(lrs) u (u, U ) β = P (u) β − 1 − νˆα νˆβ γ 1 ν ⊗ νˆ]α β , = [P (u) − νˆ ⊗ νˆ]α β + [ˆ (7.48) γ with the ﬁrst term in the square brackets projecting orthogonal to νˆ in the local rest space of u. In terms of components, (7.47) can be written as [B(lrs) (u, U )Z(U )]αβ = B(lrs)u (u, U )α μ B(lrs)u (u, U )β ν [P (u, U )Z(U )]μν .
(7.49)
To complete the analogy with the corresponding transformation laws for electromagnetic ﬁelds in (6.144), we need to replace the magnitude of ν with the rapidity parameter ν = tanh α. Considering components parallel and perpendicular to the direction of the velocity, one then ﬁnds [B(lrs) (u, U )Z(U )] = Z (u), [B(lrs) (u, U )Z(U )]⊥ = cosh α Z ⊥ (u) + i sinh αˆ ν (U, u) ×u Z ⊥ (u), ⊥⊥(TF)
[B(lrs) (u, U )Z(U )]
= cosh 2α Z
⊥⊥(TF)
(7.50)
(u)
− i sinh 2α νˆ(U, u) ×u Z ⊥⊥(TF) (u). Note that the transformation law for the vector Z ⊥ (u) is exactly the same as in (6.144)2 for the corresponding electromagnetic vector Z (em)⊥ (u), while the one for the tensor Z(U )⊥⊥ is formally the same apart from a sign change and having 2α in place of α. The condition that the transformed value of Z ⊥ (U ) is zero requires the impossible condition tanh α = ±1, which means that either Z ⊥ (U ) is initially zero or no observer U can be found for which it becomes zero.
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Nonlocal measurements
In terms of Z, the superenergy density and the superPoynting vector are obtained from the BelRobinson tensor as 1 1 ¯ Tr [E(u) · E(u) + H(u) · H(u)] = Tr [Z(u) · Z(u)], 2 2 i ¯ ×u Z(u)]α , P (g) (u)α = [E(u) ×u H(u)]α = [Z(u) (7.51) 2 E (g) (u) =
and their decomposition is 1 E (g) (u) = Tr Z¯ ⊥⊥(TF) (u) · Z ⊥⊥(TF) (u) 2 3 + Z¯ ⊥ (u) · Z ⊥ (u) + Z (u)2 , 4 i ¯ ⊥⊥(TF) Z (u) ×u Z ⊥⊥(TF) (u) + Z¯ ⊥ (u) ×u Z ⊥ (u) P (g) (u) = 2 − νˆ(u, U ) ×u m Z¯ (u)Z ⊥ (u) (7.52) + Z¯ ⊥⊥(TF) (u) Z ⊥ (u) . There are two complementary cases in which the eﬀective transformation reduces to a single electromagneticlike transformation which one can use to transform the superPoynting vector to zero: either (i) Z ⊥ (u) = 0 or (ii) Z ⊥ (u) is the only nonvanishing part of Z(u). If one starts from a Weyl principal frame (Stephani et al., 2003) in which Z(u) takes its canonical Petrovtype form, and considers boosts along one of the spatial frame vectors, condition (i) is equivalent to requiring that Z(u) have block diagonal form with respect to the chosen frame, which is possible for all Petrov types except III, while condition (ii) describes exactly type III. In that case, however, Z ⊥ (u) corresponds to a null electromagnetic ﬁeld and so one cannot transform the superPoynting vector to zero. In the ﬁrst case, the superquantities simplify to 3 1 E (g) (u) = Tr Z¯ ⊥⊥(TF) (u) · Z ⊥⊥(TF) (u) + Z (u)2 , 2 4 i ¯ ⊥⊥(TF) (g) ⊥⊥(TF) Z (u) ×u Z (u) . (7.53) P (u) = 2 Evaluating the magnitude of the transformed supermomentum tensor leads to " " i (g) cosh 4α νˆ(U, u) · Z¯ ⊥⊥(TF) (u) ×u Z ⊥⊥(TF) (u) P (U ) = ""− 2 " " (7.54) + i sinh 4α Tr Z¯ ⊥⊥(TF) (u) · Z ⊥⊥(TF) (u) " . As long as Z ⊥⊥(TF) (u) = 0, one can deﬁne νˆ(U, u) · Z¯ ⊥⊥(TF) (u) ×u Z ⊥⊥(TF) (u) tanh 4α(g) = i , Tr Z¯ ⊥⊥(TF) (u) · Z ⊥⊥(TF) (u)
(7.55)
7.6 Measurement of the electric part of the Riemann tensor
135
but for α to be real and ﬁnite, independent of the particular value of νˆ(U, u), the inequality Z¯ ⊥⊥(TF) (u) ×u Z ⊥⊥(TF) (u) 0 (i.e. dμ/dr < 0) if β > 0, and dθ/dr < 0 (i.e. dμ/dr > 0) if β < 0. Therefore, when β > 0 the photon must encounter a turning point at μ = μ+ : μ starts from 0, goes up to μ+ , then goes down to μobs (which is ≤ μ+ ). When β < 0, the photon must not encounter a turning point at μ = μ+ : μ starts from 0 and monotonically increases to μobs . The total integration over μ along the path of the photon from the emitting source to the observer is thus given by ⎧ 3 μ+ 3 μ+ dμ ⎪ + μobs √ (β > 0), ⎪ Θμ (μ) ⎨ 0 (9.126) Iμ = 3 ⎪ 3 3 ⎪ ⎩ μobs √ dμ = μ+ + μ+ √ dμ (β < 0). 0 0 μ obs
Θμ (μ)
Θμ (μ)
By using (9.116) we get Iμ =
1 a2 (μ2+ + μ2− )
K(mμ ) + β cn
−1
" μobs "" mμ , μ+ "
(9.127)
where β = 1 if β > 0, 0 if β = 0, and −1 if β < 0, and K(m) is the complete elliptic integral of the ﬁrst kind. Now let us consider the integration over r. Since the observer is at inﬁnity, the photon reaching him must have been moving in the allowed region deﬁned by r ≥ r1 when R(r) = 0 has four real roots (case A), or the allowed region deﬁned by r ≥ r3 when R(r) = 0 has two complex roots and two real roots (case B). There are then two possibilities for the photon during its trip: it has encountered a turning point at r = rt (rt = r1 in case A, rt = r3 in case B), or it has not encountered any turning point in r. Deﬁne ! ∞ ! rem dr dr
, Irem ≡ . (9.128) I∞ ≡ R(r) R(r) rt rt Obviously, according to (9.113), a necessary and suﬃcient condition for the occurrence of a turning point in r on the path of the photon is that I∞ < Iμ . Therefore, the total integration over r along the path of the photon from the emitting source to the observer is ⎧ ⎪ ⎨ I∞ + Irem (I∞ < Iμ ), Ir = (9.129) ⎪ 3 ∞ √dr ⎩ (I ≥ I ). ∞ μ rem R(r)
By deﬁnition, I∞ , Irem , and Ir are all positive. According to (9.113), we must have Ir = Iμ for the orbit of a photon. The relevant cases to be considered are the following.
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Measurements in physically relevant spacetimes
Case A: R(r) = 0 has four real roots. When r1 = r2 , by using Eq. (9.118) to evaluate integrals in (9.128) with rt = r1 , we get " " 2 r − r " 2 4 I∞ =
sn−1 "m4 , r − r (r1 − r3 )(r2 − r4 ) 1 4" Irem =
2
(r1 − r3 )(r2 − r4 ) " (r2 − r4 )(rem − r1 ) "" −1 × sn "m4 . (r1 − r4 )(rem − r2 ) "
(9.130)
Substitute these expressions into (9.129), then let Ir = Iμ , and ﬁnally solve for rem : rem =
r1 (r2 − r4 ) − r2 (r1 − r4 )sn2 (ξ4 m4 ) , (r2 − r4 ) − (r1 − r4 )sn2 (ξ4 m4 )
where ξ4 =
1 (Iμ − I∞ ) (r1 − r3 )(r2 − r4 ). 2
(9.131)
(9.132)
Since sn2 (−ξ4 m4 ) = sn2 (ξ4 m4 ), the solution given by (9.131) applies whether Iμ − I∞ is positive or negative, i.e. whether or not there is a turning point in r along the path of the photon. Case B: R(r) = 0 has two complex roots and two real roots. By using (9.121) to evaluate integrals in (9.128) with rt = r3 , we get " 1 (A − B)rem + r3 B − r4 A "" −1 cn Irem = √ "m2 , (A + B)rem − r3 B − r4 A " AB " " 1 A − B " I∞ = √ cn−1 (9.133) "m2 , " A + B AB where A and B are given by (9.122). Substitute these expressions into (9.129), then let Ir = Iμ , and ﬁnally solve for rem : rem = where
r4 A − r3 B − (r4 A + r3 B)cn(ξ2 m2 ) , (A − B) − (A + B)cn(ξ2 m2 ) √ ξ2 = (Iμ − I∞ ) AB.
(9.134)
(9.135)
Since cn(−ξ2 m2 ) = cn(ξ2 m2 ), the solution given by Eq. (9.134) applies whether Iμ − I∞ is positive or negative, i.e. whether or not there is a turning point in r along the path of the photon.
9.8 Highprecision astrometry
225
Firstorder image In this case the photon trajectory crosses the equatorial plane once. The integration over μ along the path of the photon from the emitting source to the observer is given by ⎧ 3 3 μ+ 3 μobs dμ −μ+ ⎪ √ + − − ⎪ ⎪ 0 −μ+ μ+ Θμ (μ) ⎪ ⎪ ⎪ ⎪ ⎪ 3 μ+ 3 μ+ 3 μ+ ⎪ dμ ⎪ ⎪ ⎪ = − 0 + 2 −μ+ + μobs √Θμ (μ) (β > 0), ⎨ Iμ = (9.136) 3 ⎪ 3 μobs dμ ⎪ −μ+ ⎪ − 0 + −μ+ √ ⎪ ⎪ Θμ (μ) ⎪ ⎪ ⎪ ⎪ 3 ⎪ 3 3μ ⎪ μ μ ⎪ ⎩ = − 0 + + 2 −μ+ − μ + √ dμ (β < 0). +
obs
Θμ (μ)
By using (9.116) we get Iμ =
1 a2 (μ2+ + μ2− )
3K(mμ ) + β cn−1
" μobs "" mμ . μ+ "
(9.137)
Now let us consider the integration over r. The discussion holding for the case of the direct image applies also in this case. The solution is thus given by (9.131) and (9.134), with Iμ given by (9.137).
9.8 Highprecision astrometry Modern space technology allows one to measure with high accuracy the general relativistic corrections to light trajectories due to the background curvature. Astrometric satellites like GAIA and SIM, for example, are expected to provide measurements of the position and motion of a star in our galaxy with an accuracy of the order of O(3), recalling that we set O(n) ≡ O(1/cn ). With such accuracy, we must model and interpret the satellite observations in a general relativistic context. In particular the measurement conditions as well as the determination of the satellite rest frame must be modeled with equal accuracy. This frame consists of a clock and a triad of orthonormal axes which is adapted to the satellite attitude. We shall give here two examples of frames which best describe the satellite’s measuring conditions. First we construct a Fermi frame which can be operationally ﬁxed by a set of three mutually orthogonal gyroscopes; then we ﬁnd a frame which mathematically models a given satellite attitude. Since we have in mind applications to satellite missions, we ﬁx the background geometry as that of the Solar System, assuming that it is the only source of gravity. Moreover we assume that it generates a weak gravitational ﬁeld, so we shall retain only terms of ﬁrst order in the gravitational constant G and consider these terms only up to order O(3).
226
Measurements in physically relevant spacetimes
The rest frame of a satellite consists of a clock which measures the satellite proper time and a triad of orthonormal axes. The latter are described by 4vectors whose components are referred to a coordinate system which in general is not connected to the satellite itself. Moreover, they are deﬁned up to spatial rotations; there are inﬁnitely many possible orientations of the spatial triad which can be ﬁxed to a satellite, so our task is to identify those which correspond to actual attitudes. The spacetime metric is given by ds2 ≡ gαβ dxα dxβ = [ηαβ + hαβ + O(h2 )]dxα dxβ ,
(9.138)
where O(h2 ) denotes nonlinear terms in h, the coordinates are x0 = t, x1 = x, x2 = y, x3 = z, the origin being ﬁxed at the center of mass of the Solar System, and ηαβ is the Minkowski metric, so that the metric components are g00 = −1 + h00 + O(4), 2
g0a = h0a + O(5), 3
gab = 1 + h00 δab + O(4). 2
(9.139)
Here h00 = 2U , where U is the gravitational potential generated by the bodies of 2 the Solar System, and the subscripts indicate the order of 1/c (e.g. h0a ∼ O(3)). 3 Following Bini and de Felice (2003) and Bini, Crosta, and de Felice (2003), we shall not specify the metric coeﬃcients, in order to ensure generality. Let us ﬁx the satellite’s trajectory by the timelike, unitary 4vector u α (u u α = −1), given by u = Ts (∂t + β1 ∂x + β2 ∂y + β3 ∂z ),
(9.140)
where the ∂α are the coordinate basis vectors relative to the coordinate system, and βi are the coordinate components of the satellite 3velocity with respect to a local static observer, recalling that we use subscripts here to refer to contravariant components, so as not to confuse them with power indices. Finally we deﬁne Ts = 1 + (U + 12 β 2 ) and β 2 = β12 + β22 + β32 .
Fermi frame A Fermi frame adapted to a given observer can be obtained from any frame adapted to that observer, provided we know its Fermi coeﬃcients. The latter tell how much the spatial axes of the given triad must rotate in order to be reduced to a Fermi frame. We shall apply this procedure to a tetrad adapted to u . Although we use the simplest possible frame, we ﬁnd that an algebraic solution for a Fermi frame is possible only if we conﬁne ourselves to terms of the order O(2) and set β3 = 0. To this order a tetrad solution is given by Bini and de Felice (2003) as
9.8 Highprecision astrometry 1 2 = 1 + U + β ∂t + β[cos ζ(t)∂x + sin ζ(t)∂y ], 2 = (1 − U )[sin ζ(t)∂x − cos ζ(t)∂y ], 1 = β∂t + β 2 − U + 1 (cos ζ(t)∂x + sin ζ(t)∂y ), 2 = (1 − U )∂z ,
227
λˆ0 λˆ1 λˆ2 λˆ3
(9.141)
where we have set β1 = β cos ζ(t), β2 = β sin ζ(t), and β3 = 0, ζ being the angular velocity of orbital revolution of the given observer. This tetrad is not a Fermi tetrad because one Fermi coeﬃcient is diﬀerent from zero, namely ˙ C(fw)ˆ2ˆ1 = −ζ,
(9.142)
a dot meaning derivative with respect to the coordinate time. Subtracting the Fermi rotation at each time, the triad {λaˆ } reduces to a Fermi triad: 1 Rˆ1 = −β sin ζ(t)∂t − β 2 sin 2ζ(t)∂x 4 2 β 2 sin ζ(t) − U + 1 ∂y , − 2 β2 cos2 ζ(t) ∂x Rˆ2 = β cos ζ(t)∂t + 1 − U + 2 1 + sin 2ζ(t)β 2 ∂y , 4 Rˆ3 = (1 − U )∂z .
(9.143)
It is easy to verify that all the Fermi coeﬃcients of the triad (9.143) vanish identically. The operational setting of a Fermi triad by means of three mutually orthogonal gyroscopes has a degree of arbitrariness which arises from the freedom one has in ﬁxing a spatial triad; a Fermi triad, in fact, is deﬁned up to a constant rotation. In most cases it is more convenient to deﬁne a frame which is ﬁxed to the satellite and is constrained according to criteria of best eﬃciency for the mission goal.
Attitude frame Let us ﬁnd now a frame adapted to the satellite’s attitude, keeping the approximation to the order O(3). We ﬁrst ﬁx a coordinate system centered at the baricenter of the Solar System, with the spatial axes pointing to distant sources; the latter identify a global Cartesianlike spatial coordinate representation (x, y, z) with respect to which the spacetime metric takes the form (9.138). The world line of an observer at rest with respect to the chosen coordinate grid is given by the unit 4vector u = (gtt )−1/2 ∂t ≈ (1 + U )∂t ,
(9.144)
228
Measurements in physically relevant spacetimes
where t is a coordinate time. The observer u, together with the spatial axes as speciﬁed, is a static observer, termed baricentric, and the parameter on its world line is the baricentric proper time. Obviously, at each point in spacetime there exists a static observer u who carries a triad of spatial and mutually orthogonal unitary vectors which point to the same distant sources as for the baricentric frame. As shown in Bini, Crosta, and de Felice (2003), the spatial triad of a static observer at each spacetime point is given to O(3) by the following vectors: λˆ1 = h01 ∂t + (1 − U )∂x , λˆ2 = h02 ∂t + (1 − U )∂y ,
(9.145)
λˆ3 = h03 ∂t + (1 − U )∂z . We need to identify the spatial direction to the geometrical center of the Sun as seen from within the satellite. To this end we ﬁrst identify this direction with respect to the local static observer which is deﬁned at each point of the satellite’s trajectory, then we boost the corresponding triad to adapt it to the motion of the satellite. Let x0 (t), y0 (t), z0 (t) be the coordinates of the satellite’s center of mass with respect to the baricenter of the Solar System, and x (t), y (t), z (t) those of the Sun at the same coordinate time t. Here the time dependence is assumed to be known. The relative spatial position of the Sun with respect to the satellite at the time t is then x = x − x0 , = y − y0 , y z
(9.146)
= z − z0 .
With respect to a local static observer, the Sun direction is ﬁxed, rotating the triad (9.145) by an angle φs around λˆ3 and then by an angle θs around the vector image of λˆ2 under the above φs rotation, where φs = tan−1
y x
,
θs = tan−1
z 2 x2 + y
.
(9.147)
Thus we have the new triad adapted to the observer u, λ ˆ, ˆ = R2 (θs )R3 (φs )λa sa where
⎛
cos θs R2 (θs ) = ⎝ 0 −sin θs
⎞ 0 sin θs 1 0 ⎠ 0 cos θs
(9.148)
(9.149)
9.8 Highprecision astrometry and
⎛
cos φs ⎝ R3 (φs ) = −sin φs 0
sin φs cos φs 0
⎞ 0 0⎠ . 1
229
(9.150)
It should be noted here that, since the Sun is an extended body, its geometrical center may be diﬃcult to determine with great precision. The uncertainty in this measurement may aﬀect the precision in ﬁxing the angles φs and θs from on board the satellite, contributing to the determination of the error box. From Eqs. (9.148)–(9.150), the explicit expressions for the coordinate components of the vectors of the new triad are λ 1 = [cos θs (cos φs h01 + sin φs h02 ) + sin θs h03 ]∂t sˆ + cos φs cos θs (1 − U )∂x + sin φs cos θs (1 − U )∂y + sin θs (1 − U )∂z ,
(9.151)
λ 2 = −(sin φs h01 + cos φs h02 )∂t sˆ − sin φs (1 − U )∂x − cos φs (1 − U )∂y ,
(9.152)
λ 3 = −[sin θs (cos φs h01 + sin φs h02 ) − cos θs h03 ]∂t sˆ − cos θs sin θs (1 − U )∂x − sin φs sin θs (1 − U )∂y + cos θs (1 − U )∂z .
(9.153)
It is easy to verify that the set {u, λ ˆ } forms an orthonormal tetrad; moreover, sa it is equally straightforward to see that cos θs λ 2 = sˆ
d λˆ , dφs s 1
λ 3 = sˆ
d λˆ . dθs s 1
(9.154)
All quantities in equations (9.151)–(9.153) are deﬁned at the position (x0 , y0 , z0 ) of the satellite at time t. Let us remember that our aim here is to identify a tetrad frame which is adapted to the satellite and whose spatial triad mirrors its attitude. Recalling that the satellite 4velocity is given by u as in (9.140), we boost the vectors of the triad {λ ˆ } along the satellite’s relative motion to obtain the following boosted sa triad (see (3.143)): γ α α σ ρ σ ν (ν λ , (9.155) λˆ = P (u ) σ λ a) ˆ − sa s ρˆ bsa γ+1 a ˆ=1,2,3 where P (u )α σ = δσα + uα uσ
(9.156)
230
Measurements in physically relevant spacetimes
is the operator which projects onto the restspace of u , ν α ≡ ν(u , u)α is the relative spatial velocity of u with respect to the local static observer u, deﬁned as να =
1 α (u − γuα ), γ
(9.157)
and γ ≡ γ(u , u) = −u uα is the relative Lorentz factor. The vector λˆ1 identiﬁes bs the direction to the Sun as seen from within the satellite. The other vectors of the boosted triad are related to λˆ1 by the simple relations α
bs
λˆ bs2
=
d λˆ , dφs bs1
λˆ bs3
=
d λˆ . dθs bs1
(9.158)
The tetrad {λˆ0 ≡ u , λaˆ } will be referred to as the Sunlocked frame. The relation bs bs between the components ν(u , u)α of the spatial velocity ν(u , u) and the components βi appearing in (9.140) is easily established from (9.140) itself and (9.157), and is given by 1 iα Ts βi δ + δ 0α − uα γ . (9.159) ν(u , u)α = γ The explicit expressions for the components of the vectors λaˆ can be found in bs the cited literature (Bini, Crosta, and de Felice, 2003; de Felice and Preti, 2008).
10 Measurements of spinning bodies
A test gyroscope is a pointlike massive particle having an additional “structure” termed spin. A spinning body is not, strictly speaking, pointlike because its average size cannot be less than the ratio between its spin and its mass, in geometrized units. This guarantees that no point of the spinning body moves with respect to any observer at a velocity larger than c. Rotation is a common feature in the universe so knowing the dynamics of rotating bodies is almost essential in modern physics. Although in most cases the assumption of no rotation is necessary to make the equations tractable, the existence of a strictly nonrotating system should be considered a rare event, and a measurement which revealed one would be of great interest. This is the case for the massive black holes which appear to exist in the nuclei of most galaxies. Detailed measurements are aimed at detecting their intrinsic angular momentum from the behavior of the surrounding medium. A black hole, however, could also be seen directly if we were able to detect the gravitational radiation it would emit after being perturbed by an external ﬁeld. A direct measurement of gravitational waves is still out of reach for earthbound detectors; nevertheless they are still extensively searched for since they are the ultimate resource to investigate the nature of spacetime. Because of the vast literature available on this topic, in what follows we shall limit ourselves to the interaction of gravitational waves with a spinning body, with the aim of searching for new ways to detect them.
10.1 Behavior of spin in general spacetimes As already stated, we shall distinguish between a pointlike gyroscope and an extended spinning body. In the ﬁrst case, let us consider the world line γ of an observer U who carries the gyroscope. We denote the spin of the gyroscope by a vector S deﬁned along γ, everywhere orthogonal to its tangent vector U and undergoing FermiWalker transport:
232
Measurements of spinning bodies D(fw,U ) S = ∇U S − (a(U ) · S)U = 0, dτU
S · U = 0.
(10.1)
In the second case, the behavior of an extended spinning body is described by the MathissonPapapetrou equations (Mathisson, 1937; Papapetrou, 1951), later generalized by Dixon (1970a; 1970b; 1979): D μ 1 P = − Rμ αβγ U α S βγ ≡ F (spin)μ , dτU 2 D μν S = 2P [μ U ν] , dτU
(10.2) (10.3)
where U is the unit timelike vector tangent to the line representative of the body (hereafter, the center of mass line), and Pμ and S μν are its momentum 4vector and antisymmetric spin tensor, satisfying the additional conditions S μν Pν = 0.
(10.4)
Because of the spin–curvature coupling appearing in (10.2), U is in general accelerated, with a(U ) = f (U ); therefore, following the notation of Section 6.7 (see (6.75)), we set P (u, U )f (U ) = γF (U, u).
(10.5)
In the limit of small spin relative to the length scale of the background curvature, Eqs. (10.2)–(10.4) imply (10.1), so the two descriptions agree. In what follows we shall consider ﬁrst the motion of a test gyroscope.
Test gyroscope: the projected spin vector Let u be a vector ﬁeld tangent to a congruence Cu of curves which cross the world line of U . With respect to Cu the spin vector S admits the representation S = Σ(U, u) + [ν(U, u) · Σ(U, u)]u,
Σ(U, u) = P (u, U )S.
(10.6)
Since S is FermiWalker transported along U , its magnitude is constant, whereas the projected vector ˆ Σ(U, u) = Σ(U, u)Σ(U, u)
(10.7)
varies. The evolution of Σ(U, u) along U , as measured by the observer u, can be obtained as follows. Starting from (10.1) and using (10.6), we ﬁnd D Du DS = Σ(U, u) + (ν(U, u) · Σ(U, u)) dτU dτU dτU D D +u ν(U, u) · Σ(U, u) + ν(U, u) · Σ(U, u) , dτU dτU
(10.8)
10.1 Behavior of spin in general spacetimes
233
so that, acting on both sides with P (u), we ﬁnd, from (10.1), (a(U ) · S)γν =
D(fw,U,u) (G) Σ(U, u) − (ν(U, u) · Σ(U, u))F(fw,U,u) , dτU
(10.9)
recalling the deﬁnition (3.156) of the FermiWalker gravitational force Du (G) = −F(fw,U,u) . dτU
(10.10)
Let us evaluate the term (a(U ) · S). We have a(U ) · S = a(U ) · Σ(U, u) + (ν(U, u) · Σ(U, u))u · a(U ) = γF (U, u) · Σ(U, u) + (ν(U, u) · Σ(U, u))u · ∇U U.
(10.11)
But u · ∇U U = −
dγ (G) + γν · F(fw,U,u) = −γF (U, u) · ν, dτU
(10.12)
where we have used Eq. (6.85), that is, dγ (G) = γ[F (U, u) + F(fw,U,u) ] · ν(U, u). dτU
(10.13)
Hence a(U ) · S = γ [F (U, u) · Σ(U, u) − (ν(U, u) · Σ(U, u))F (U, u) · ν] .
(10.14)
Using this in (10.9) and setting ν(U, u) = ν and F (U, u) = F to simplify notation, we obtain D(fw,U,u) (G) Σ(U, u) = (ν · Σ(U, u))F(fw,U,u) + (a(U ) · S)γν dτU (G)
= (ν · Σ(U, u))F(fw,U,u) + (F · Σ(U, u) − (ν · Σ(U, u))F · ν)γ 2 ν.
(10.15)
Let us now consider the evolution of the magnitude of Σ(U, u), that is Σ(U, u), ˆ ˆ separately. Using (10.7) we ﬁnd and of its unit direction Σ(U, u) ≡ Σ d ˆ γ 2 χ + F (G) ˆ , ln Σ(U, u) = (ν · Σ) · Σ (10.16) (fw,U,u) dτU where ˆ − (ν · Σ)(F ˆ χ=F ·Σ · ν). Using the relative standard time parameterization, we deduce D(fw,U,u) ˆ ˆ ×u (ν ×u Σ) ˆ Σ = γχ Σ dτ(U,u) 1 ˆ ˆ ×u (F (G) ˆ . · ν) Σ ×u Σ) + (Σ (fw,U,u) γ
(10.17)
(10.18)
234
Measurements of spinning bodies
We can also cast (10.18) in the form D(fw,U,u) ˆ (project) ˆ ×u Σ, Σ = ζ(fw) dτ(U,u)
(10.19)
where (project)
ζ(fw)
ˆ Σ ˆ ×u F (G) ˆ ×u ν) + 1 (ν · Σ) = γχ(Σ (fw,U,u) γ ˆ (G) ˆ ×u γχν + 1 (ν · Σ)F =Σ (fw,U,u) γ
(10.20)
is the precession rate of the gyroscope as it would be locally measured by u.
Boosted spin vector Let us examine now the complementary problem of describing the precession of a gyroscope as it would be measured by the observer U who carries it, relative to a frame, say {e(u)a }, adapted to the observers of the congruence Cu . According to U , the axes e(u)a are moving axes; hence their orientation in LRSU is deﬁned by boosting them onto LRSU itself, that is E(U )a = B(lrs) (U, u)e(u)a .
(10.21)
Our aim here is to study the precession rate of the spin vector S with respect to the axes E(U )a . However, since the boost is an isometry, the precession of S with respect to the axes E(U )a is the same as the precession of the boosted vector S(U, u) = B(lrs) (u, U )S,
(10.22)
momentarily at rest with respect to u, as measured with respect to the axes e(u)a . Using the representation (3.143) of the boost, we have S(U, u) = Σ(U, u) −
γ (ν(U, u) · Σ(U, u))ν(U, u). 1+γ
(10.23)
The evolution of S(U, u) along U is a consequence of the similar result for Σ(U, u) and can be cast in the form D(fw,U,u) (boost) S(U, u) = ζ(fw) ×u S(U, u), dτ(U,u) (boost)
where ζ(fw)
(10.24)
is made up of two terms: (boost)
ζ(fw)
= ζ(Thomas) + ζ(geo) ,
(10.25)
where ζ(Thomas) = −
γ ν(U, u) ×u F (U, u), 1+γ
(10.26)
10.1 Behavior of spin in general spacetimes
235
and ζ(geo) =
1 (G) ν(U, u) ×u F(fw,U,u) . 1+γ
(10.27)
The Thomas precession arises from the acceleration of the world line of the observer who carries the gyroscope and is due to the nongravitational force F (U, u) deﬁned in (6.75). The geodesic precession is due to the gravitational (G) force F(fw,U,u) only. Let us derive (10.24). Start from (10.23) and replace Σ(U, u) using (10.6) to obtain [ν(U, u) · Σ(U, u)] (U + u) 1+γ γ[ν(U, u) · S(U, u)] =S− (U + u), 1+γ
S(U, u) = S −
(10.28)
because ν(U, u) · Σ(U, u) = γ[ν(U, u) · S(U, u)].
(10.29)
Let f denote the quantity f=
γ[ν(U, u) · S(U, u)] . 1+γ
(10.30)
Diﬀerentiate both sides of (10.28) along U : DS df D D S(U, u) = − (u + U ) − a(U ) + u f dτU dτU dτU dτU df = (a(U ) · S)U − (u + U ) dτU (G) − a(U ) − F(fw,U,u) f,
(10.31)
where we have used the properties of S of being orthogonal to U and of being FermiWalker transported along it, that is D(fw,U ) D S= S − (a(U ) · S)U = 0. dτU dτU
(10.32)
From (10.28) we have a(U ) · S = a(U ) · S(U, u) + (u · a(U ))f.
(10.33)
But u · a(U ) = u · ∇U U = −
dγ (G) + γν(U, u) · F(fw,U,u) , dτU
(10.34)
so that Eq. (10.13) gives u · a(U ) = −γν(U, u) · F (U, u).
(10.35)
236
Measurements of spinning bodies
Therefore D S(U, u) = [a(U ) · S(U, u) − f γν(U, u) · F (U, u)] U dτU df (G) − (u + U ) − a(U ) − F(fw,U,u) f dτU = γ [F (U, u) · S(U, u) − f ν(U, u) · F (U, u)] U df (G) − (u + U ) − a(U ) − F(fw,U,u) f. dτU
(10.36)
First project both sides of (10.36) orthogonally to u, obtaining D(fw,U,u) S(U, u) = γ 2 [F (U, u) · S(U, u) dτU − f ν(U, u) · F (U, u)] ν(U, u) df (G) − γν + −γF + F(fw,U,u) f. dτU
(10.37)
Then take the scalar product of both sides of (10.36) with u, to give u·
D S(U, u) = −γ 2 (F (U, u) · S(U, u) − f ν(U, u) · F (U, u)) dτU df + (γ + 1) − (u · a(U )) f, dτU
(10.38)
that is, since u · S(U, u) = 0, (G)
F(fw,U,u) · S(U, u) = −γ 2 (F (U, u) · S(U, u) − f ν(U, u) · F (U, u)) df (γ + 1) + dτU + f γν(U, u) · F (U, u) = −γ 2 F (U, u) · S(U, u) + γ(f ν(U, u) · F (U, u))(1 + γ) df (γ + 1). + dτU From this it follows that df (G) = (1 + γ)−1 F(fw,U,u) + γ 2 F (U, u) · S(U, u) dτU − γf ν(U, u) · F (U, u).
(10.39)
(10.40)
Substituting this expression into (10.37), we obtain the ﬁnal result, D(fw,U,u) γ [ν(U, u) ⊗ S(U, u) − ν(U, u) · S(U, u)P (u)] S(U, u) = dτU 1+γ (G) γF (U, u) − F(fw,U,u) , (10.41) which is equivalent to (10.24).
10.2 Motion of a test gyroscope in a weak gravitational ﬁeld
237
If the frame {e(u)a } is orthonormal, then we can further manipulate (10.24). In this case we have D(fw,U,u) dS(U, u)aˆ S(U, u)aˆ = dτ(U,u) dτ(U,u) ˆ + C(fw) aˆ cˆ + Γaˆ cˆˆb ν(U, u)b S(U, u)cˆ.
(10.42)
Introducing the quantities 1 ˆ a ˆ = − η(u)aˆbˆc C(fw)ˆbˆc ζ(fw) 2
(10.43)
and the spatial curvature angular velocity 1 ˆ dˆ a ˆ = − η(u)aˆbˆc Γ[ˆcd ζ(sc) ˆˆ b] ν(U, u) , 2
(10.44)
already deﬁned in (4.18) and (4.20) respectively, we have D(fw,U,u) dS(U, u)a S(U, u)aˆ = + [(ζ(fw) + ζ(sc) ) ×u S(U, u)]aˆ , dτ(U,u) dτ(U,u)
(10.45)
and hence aˆ dS(U, u)aˆ (boost) = ζ(fw) − ζ(fw) − ζ(sc) ×u S(U, u) dτ(U,u) aˆ ≡ ζ (tot) ×u S(U, u) ,
(10.46)
or equivalently dS(U, u)aˆ ˆ = η(u)aˆ ˆbˆc ζ (tot) b S(U, u)cˆ. dτ(U,u)
(10.47)
Note that a change of parameter along U implies a rescaling of the precession angular velocity. For instance, the above formula can also be written as dS(U, u)aˆ ˆ = η(u)aˆ ˆbˆc [γζ (tot) ]b S(U, u)cˆ, dτU
(10.48)
when the world line U is parameterized by the proper time τU . In this case the eﬀective total angular velocity is (tot)
ζU
= γζ (tot) .
(10.49)
10.2 Motion of a test gyroscope in a weak gravitational field Let us consider the postNewtonian (PN) treatment of a weak gravitational ﬁeld within general relativity and evaluate the orders of magnitude of the quantities we shall meet in our treatment in terms of powers of 1/c, c being the velocity of light in vacuum. In a PN coordinate system, say (t, xa ) (a = 1, 2, 3), the spacetime metric can be written as
238
Measurements of spinning bodies ds2 = −(1 − 2Φ)dt2 + 2Φi dtdxi + (1 + 2Φ)δij dxi dxj + O(4),
(10.50)
where Φ = O(2) and Φi = O(3). We use the threedimensional notation = Φ1 ∂x + Φ2 ∂y + Φ3 ∂z Φ
(10.51)
and the ordinary Euclidean space operations for grad, curl, div, and vector product, unless otherwise speciﬁed. Let u be the family of observers at rest with respect to the coordinate grid 1 u= √ ∂t = (1 + Φ)∂t + O(4), −gtt
(10.52)
u = (−1 + Φ)dt + Φa dxa + O(4).
(10.53)
so that
Let us ﬁx an orthonormal frame adapted to u as follows: eaˆ = Φa ∂t + (1 − Φ)∂a ,
ω aˆ = (1 + Φ)δbaˆ dxb .
(10.54)
The observers u have acceleration a(u) = −gradu Φ + O(4) ;
(10.55)
they also have vorticity 1 + O(4) curlu Φ 2
(10.56)
θ(u)ab = ∂t Φ δab + O(4).
(10.57)
ω(u) = and expansion1
A direct calculation shows that ∇u eaˆ = −∂a Φ u + ω(u) ×u eaˆ + O(4) = a(u)a u + ω(u) ×u eaˆ + O(4),
(10.58)
and hence ˆ
C(fw) b aˆ eˆb = ω(u) ×u eaˆ ,
(10.59)
ω(u) = ζ(fw) .
(10.60)
which implies
Similarly, the nonvanishing spatial Ricci rotation coeﬃcients ω cˆ ∇eˆb eaˆ = Γcˆaˆˆb 1
(10.61)
We note that for these three ﬁelds, that is acceleration, vorticity, and expansion, the coordinate and frame components coincide.
10.3 Motion of a test gyroscope in Schwarzschild spacetime
239
are as follows: Γxˆ yˆyˆ = Γxˆ zˆzˆ = −Γyˆxˆyˆ = −Γzˆxˆzˆ = −∂x Φ, Γyˆxˆxˆ = Γyˆzˆzˆ = −Γxˆ yˆxˆ = −Γzˆyˆzˆ = −∂y Φ, Γzˆxˆxˆ = Γzˆyˆyˆ = −Γxˆ zˆxˆ = −Γyˆzˆyˆ = −∂z Φ.
(10.62)
a ˆ
If U = γ[u + ν(U, u) eaˆ ] is the 4velocity of the observer carrying the gyroscope and we assume that it satisﬁes the slow motion condition ν(U, u) 1, then we can evaluate the spatial curvature angular velocity ˆ
ˆ
ˆ
f , ν cˆΓd aˆcˆ = −d aˆfˆζ(sc)
(10.63)
ζ(sc) = −ν ×u gradu Φ.
(10.64)
and ﬁnd that
Finally, we have 1 1 1 1 = − ν ×u F − ν ×u a(u) = − ν ×u F + ν ×u gradu Φ, 2 2 2 2 so that the precession angular velocity of (10.46) becomes (boost)
ζ(fw)
(10.65)
1 1 ζ (tot) = − ν ×u F + ν ×u gradu Φ 2 2 1 + ν ×u gradu Φ − curlu Φ 2 3 1 = − ν ×u F + ν ×u gradu Φ 2 2 1 (10.66) − curlu Φ. 2 When a gyroscope moves along a geodesic (F = 0), Eq. (10.66) is known as the Schiﬀ formula; that is, ζ (tot) =
3 1 ν ×u gradu Φ − curlu Φ. 2 2
(10.67)
The part of ζ (tot) which is contributed by the intrinsic spin of the gravity source is responsible for a relativistic eﬀect implied by the LenseThirring metric, and known as the LenseThirring eﬀect.
10.3 Motion of a test gyroscope in Schwarzschild spacetime We shall now study the motion of a test gyroscope in Schwarzschild spacetime. Our aim here is to deduce the precession of a gyroscope carried by an observer U , but with respect to a frame e(m)a given by (8.18), adapted to the static observers. In this way we can specialize the general formulas given above. The evolution of the boosted spin vector S(U, m) along U can be cast in the form D(fw,U,m) (boost) S(U, m) = ζ(fw) (U, m) ×m S(U, m), (10.68) dτ(U,m)
240
Measurements of spinning bodies
where (boost)
ζ(fw)
(boost)
ζ(fw)
= ζ(Thomas) + ζ(geo) γ (G) ν(U, m) ×m −F (U, m) + γ −1 F(fw,U,m) = 1+γ γ ν(U, m) ×m [−F (U, m) − a(m)] . = 1+γ
(10.69)
ˆ has a component only along the latitudinal frame direction θ, (boost) θˆ
ζ(fw) with
=
γ ˆ ν(U, m)φ −F (U, m)rˆ − a(m)rˆ , 1+γ
γ M M 2 2 = , − rζ − rζ r2 N r2 −1/2 2M M M rˆ = , a(m) = 1 − 2 r r N r2
(10.70)
F (U, m)rˆ = Γ
with
N=
We then have (boost) θˆ
ζ(fw)
=
2M 1− r
1/2 =
r . R
(10.71)
(10.72)
γ M M γ ˆ 2 ν(U, m)φ − − − rζ 1+γ N r2 N r2 ˆ
= −γ
ˆ
ν(U, m)φ M rγ 2 ζ 2 ν(U, m)φ . + N r2 1+γ N
(10.73)
Recalling that ˆ
ν(U, m)φ =
rζ , N
(10.74)
we have (boost) θˆ
ζ(fw)
= −γ
Mζ r2 γ 2 ζ 2 ζ + . rN 2 1 + γ N2
(10.75)
Moreover, γ2 − 1 γ 2 ν(U, m)2 r2 γ 2 ζ 2 1 = = γ − 1. = 2 1+γ N 1+γ 1+γ Therefore (boost) θˆ
ζ(fw)
Mζ M + ζ(γ − 1) = ζγ − + 1 −ζ rN 2 rN 2 3M ζγ − ζ. = 2 1− N r
(10.76)
= −γ
(10.77)
10.4 Motion of a spinning body in Schwarzschild spacetime
241
Taking into account now that ζ(fw) = 0 and that, from (8.48) with θ = π/2, ζ(sc) = −ζeθˆ, we have the Shiﬀ formula in Schwarzschild spacetime, 3M ζγ (boost) θˆ θˆ (tot) θˆ . ζ = ζ(fw) − ζ(sc) = 2 1 − N r
(10.78)
(10.79)
Finally, from (8.69), we see that ˆ
ζ (tot) θ =
τ1 (U ) . γ
(10.80)
As a convention the physical (orthonormal) component along −∂θ , perpendicular to the equatorial plane, is taken to be along the positive zaxis, denoted by zˆ. Therefore, ζ (tot) zˆ = −
τ1 (U ) . γ
(10.81)
g
This result agrees with the expression for ζ given in Section 9.2. In fact, assuming U is a geodesic parameterized by the proper time and taking into account the discussion at the end of Section 10.1 we have g 3M (tot) −1 , (10.82) ζ ≡ ζU zˆ = −τ1 (U ) = Γ2 ζ r where the expression (8.69) for τ1 (U ) has been used. Finally, introducing the p
proper frequency ζ = Γζ yields g
p
ζ = Γζ
3M −1 . r
(10.83)
10.4 Motion of a spinning body in Schwarzschild spacetime The equations of motion are given by (10.2) and (10.3) coupled to the supplementary condition (10.4), where, as stated, Pμ is the total 4momentum of the particle and S μν is its (antisymmetric) spin tensor. In those equations, U is the timelike unit tangent vector of the centerofmass line used to make the multipole reduction. Equation (10.2) is also referred to as the Riemann force equation. Let the Schwarzschild metric be written in sphericaltype coordinates (8.1), and introduce an orthonormal frame adapted to the static observers: etˆ = (1 − 2M/r)−1/2 ∂t , 1 eθˆ = ∂θ , r
erˆ = (1 − 2M/r)1/2 ∂r , 1 ∂φ , eφˆ = r sin θ
(10.84)
242
Measurements of spinning bodies
with dual ˆ
ω rˆ = (1 − 2M/r)−1/2 dr,
ˆ
ω φ = r sin θ dφ.
ω t = (1 − 2M/r)1/2 dt,
ˆ
ω θ = r dθ,
(10.85)
Let us assume that U is tangent to a (timelike) spatially circular orbit, which we here denote as a U orbit, with γ = (1 − ν 2 )−1/2 .
U = Γ[∂t + ζ∂φ ] = γ[etˆ + νeφˆ],
(10.86)
ζ is the angular velocity of the orbital revolution as it would be measured at inﬁnity, ν is the magnitude of the local proper linear velocity measured in the frame (10.84), and Γ is a normalization factor given by −1/2 Γ = −gtt − ζ 2 gφφ (10.87) in order to ensure that U · U = −1. The angular velocity ζ is related to ν by gtt ζ= − ν. (10.88) gφφ Here ζ and therefore also ν are assumed to be constant along the U orbit. We limit our analysis to the equatorial plane (θ = π/2) of the Schwarzschild solution; again, as a convention, the physical (orthonormal) component along −∂θ , perpendicular to the equatorial plane, will be taken to be along the positive zaxis and will be indicated by zˆ, as necessary. Among the circular orbits analyzed in detail in Section 8.2, particular attention is devoted to the timelike spatially circular geodesics which corotate (ζ+ ) and counterrotate (ζ− ) relative to a preassigned positive sense of variation of the azimuthal coordinate φ. They have respective angular velocities ζ± ≡ ±ζK = ±(M/r3 )1/2 , so that U± = Γ± [∂t + ζ± ∂φ ] = γK [etˆ ± νK eφˆ], where
νK
M = r − 2M
1/2
,
γK
r − 2M = r − 3M
(10.89)
1/2 ,
(10.90)
and with the timelike condition νK < 1 satisﬁed if r > 3M. Here γK is the local (geodesic) Lorentz factor relative to the frame (10.84). Let us now introduce the Lie relative curvature (8.66) of each U orbit, 1/2 2M 1 ζK √ 1− k(lie) ≡ k(lie) rˆ = −∂rˆ ln gφφ = − =− , (10.91) r r νK as well as a FrenetSerret intrinsic frame along U , deﬁned by Etˆ = U,
Erˆ = erˆ,
Ezˆ = ezˆ,
Eφˆ = γ[νetˆ + eφˆ],
(10.92)
10.4 Motion of a spinning body in Schwarzschild spacetime
243
satisfying the following system of evolution equations: DErˆ = κU + τ1 Eφˆ, dτU
DU ≡ a(U ) = κErˆ, dτU DEφˆ dτU
DEzˆ = 0, dτU
= −τ1 Erˆ,
(10.93)
where 2 κ = k(lie) γ 2 [ν 2 − νK ],
τ1 = −
1 dκ γ2 = −k (lie) 2 ν; 2γ 2 dν γK
(10.94)
in this case the second torsion τ2 is identically zero. The dual of (10.92) is given by ˆ
Ωt = −U,
Ωrˆ = ω rˆ,
ˆ
ˆ
ˆ
Ωφ = γ[−νω t + ω φ ].
Ωzˆ = ω zˆ,
(10.95)
To study the motion of spinning test particles in circular orbits let us consider ﬁrst the evolution equation for the spin tensor (10.3). For simplicity, we search for solutions of the Riemann force equation (10.2) describing frame components of the spin tensor which are constant along the orbit. By contracting both sides of (10.3) with Uν , one obtains the following expression for the total 4momentum: P μ = −(U · P )U μ − Uν
DS μν ≡ mU μ + Psμ , dτU
(10.96)
where m is the particle’s bare mass, i.e. the mass it would have were it not spinning, and Ps = U DS/dτU is a 4vector orthogonal to U . As a consequence of (10.96), Eq. (10.3) is equivalent to P (U )μα P (U )νβ
DS αβ = 0, dτU
(10.97)
where P (U )μα = δαμ + U μ Uα projects into the local rest space of U ; this implies Stˆφˆ = 0,
Srˆθˆ = 0,
Stˆθˆ + Sφˆθˆ
ν 2 = 0. νK
(10.98)
From (10.92)–(10.95) it follows that DS ˆ = ms [Ωφ ∧ U ] ; dτU
(10.99)
hence Ps can be written as ˆ
Ps = ms Ωφ , where ms ≡ Ps  = γ
ζK 2 −νK Srˆφˆ + νStˆrˆ . νK
(10.100)
(10.101)
244
Measurements of spinning bodies
From (10.96) and (10.100), and provided m + νms = 0, the total 4momentum P can be written in the form P = μ Up , with Up = γp [etˆ + νp eφˆ],
νp =
ν + ms /m , 1 + νms /m
μ=
γ (m + νms ), γp
(10.102)
where γp = (1 − νp2 )−1/2 . Up is a timelike unit vector; hence μ has the property of a physical mass. The ﬁrst part of (10.102) tells us that, as the particle’s center of mass moves along the U orbit, the momentum P is instantaneously (i.e. at each point of the U orbit) parallel to a unit vector which is tangent to a spatially circular orbit, hereafter denoted as a Up orbit, which intersects the U orbit at each of its points. Although U  and Up orbits have the same spatial projections into the U quotient space, there exists in the spacetime one Up orbit for each point of the U orbit where the two intersect. Let us now consider the equation of motion (10.2). The spinforce is equal to ν 2 2Stˆrˆ + νSrˆφˆ erˆ − γ 2 Sθˆφˆeθˆ, (10.103) F (spin) = γ ζK r while the term on the lefthand side of Eq. (10.2) can be written, from (10.96) and (10.100), as DEφˆ DP = ma(U ) + ms , dτU dτU = (mκ − ms τ1 )erˆ,
(10.104)
where κ and τ1 are given in (10.94) and the quantities μ, m, and ms are constant along the world line of U . The acceleration κ of the U orbit vanishes if the latter is a geodesic (ν = νK ); the term −ms τ1 erˆ instead is a spin–rotation coupling force that was predicted by Mashhoon (1988; 1995; 1999), a sort of centrifugal force which of course vanishes when the particle is at rest (ν = 0), as can be seen in (10.94)2 . Since DP/dτU is directed radially, as (10.104) shows, Eq. (10.2) requires that Sθˆφˆ = 0 (and therefore also Stˆθˆ = 0 from (10.98)); hence (10.2) can be written as (spin)
mκ − ms τ1 − Frˆ
= 0,
(10.105)
γν 2S . + ν ζ + νS ˆ ˆ K K r ˆ t r ˆ φ 2 γK
(10.106)
or, more explicitly, 2 0 = mγ[ν 2 − νK ] + ms
Summarizing, from the equations of motions (10.2) and (10.3) and before imposing (10.4), the spin tensor turns out to be completely determined by only two components, namely Stˆrˆ and Srˆφˆ, related by (10.106). The spin tensor then takes the form ˆ
ˆ
S = ω rˆ ∧ [Srˆtˆω t + Srˆφˆω φ ].
(10.107)
10.4 Motion of a spinning body in Schwarzschild spacetime
245
It is useful to introduce, together with the quadratic invariant s2 =
1 Sμν S μν = −Srˆ2tˆ + Srˆ2φˆ, 2
(10.108)
a frame adapted to Up , given by E0p = Up ,
E1p = erˆ,
E2p = γp (νp etˆ + eφˆ),
E3p = ezˆ,
(10.109)
whose dual frame is denoted by Ωpˆa . Imposing (10.4), one now gets Srˆtˆ+Srˆφˆνp = 0, or ˆ
S = s ω rˆ ∧ Ωp φ ,
ˆ
ˆ
ˆ
Ωp φ = γp [−νp ω t + ω φ ],
(10.110)
so that (Srˆtˆ, Srˆφˆ) = (−sγp νp , sγp ) and (10.106) reduces to 2 ) + γp 0 = γ(ν 2 − νK
2 γ ζK 2 2 Mˆ s 2 ν(ννp − νK ) + νK (2νp + ν) . νK γK
(10.111)
Solving with respect to sˆ, we obtain sˆ = −
2 νK ν 2 − νK 2 )ν 2 + 2ν 2 ]ν − νν 2 (ν 2 − ν 2 ) . Mγp γζK [(1 − 3νK K p K K
(10.112)
Recalling the deﬁnition (10.101), ms becomes ms ζK 2 = γγp Mˆ s[ννp − νK ], m νK
(10.113)
and using (10.102) for νp , we obtain sˆ = −
νK ν − νp 2 ) ; Mγp γζK (1 − ννp )(ννp − νK
(10.114)
this condition must be considered together with (10.112). Relations (10.112) and (10.114) imply that the spinless case (ˆ s = 0) is compatible only with ν = νp = ±νK . Eliminating sˆ from Eqs. (10.114) and (10.112) and solving with respect to νp , we have νp(±) =
1 νK 2 2 2 4 1/2 }. 2 {3ννK ± [ν (13νK + 4ν ) − 8νK ] 2 2 ν + 2νK
(10.115)
Since the case sˆ 1 is the only physically relevant one, of the two branches of (+) the solution (10.115) we shall consider only the branch νp when ν > 0, and (−) (±) νp when ν < 0. By substituting νp = νp into (10.112), for instance, we obtain a relation between ν and sˆ. The reality condition (10.115) requires that ν take values outside the interval (¯ ν− , ν¯+ ), with √ √ ν¯± = ±νK 2 −13 + 3 33/4 ±0.727νK ;
246
Measurements of spinning bodies
moreover, the timelike condition for νp  < 1 is satisﬁed for all values of ν outside the same interval. A linear relation between ν and sˆ can be obtained in the limit of small sˆ: 3 ν = ±νK − ζK νK Mˆ s + O(ˆ s2 ). 2
(10.116)
From this approximate solution for ν we also have that 3 s + O(ˆ s2 ), νp(±) = ±νK − ζK νK Mˆ 2
(10.117)
and the total 4momentum P is given by (10.102), with s2 ). νp = ν + O(ˆ
(10.118)
The reciprocals of the angular velocities ζ, ζp also coincide to ﬁrst order in sˆ, and are then given by 1 1 3 1 ≡ s , =± ± Mˆ ζ ζ(±,±) ζK 2
(10.119)
where the signs in front of 1/ζK correspond to co/counterrotating orbits, while the signs in front of sˆ refer to a positive or negative spin direction along the zaxis; for instance, the quantity ζ(+,−) denotes the angular velocity of U corresponding to a corotating orbit (+) with spindown (−) alignment, etc.
Clock eﬀect for spinning bodies One can measure the diﬀerence in the arrival times, after one complete revolution, with respect to a static observer, of two oppositely rotating spinning test particles (to ﬁrst order in the spin parameter sˆ) with either spin orientation. From (10.119) we have that the coordinate time diﬀerence is given by 1 1 = 6πM ˆ s, (10.120) Δt(+,+;−,+) = 2π + ζ(+,+) ζ(−,+) and analogously for Δt(+,−;−,−) . A similar result can be obtained, referring to any observer moving in spatially circular orbits, with a slight modiﬁcation of the discussion. This eﬀect creates an interesting parallelism with the clock eﬀect in Kerr spacetime, as we shall discuss next. In the present case, in fact, it is the spin of the particle which creates a nonzero clock eﬀect, while in the Kerr metric this eﬀect is induced by the rotation of the spacetime even to geodesic spinless test particles. The latter have angular velocities 1 ζ(Kerr) ±
=±
1 + a, ζK
(10.121)
10.5 Motion of a test gyroscope in Kerr spacetime
247
where a is the angular momentum per unit mass of the Kerr black hole; hence Kerr’s clock eﬀect is given by 1 1 Δt(+,−) = 2π = 4πa. (10.122) + ζ(Kerr) + ζ(Kerr) − This complementarity suggests a sort of equivalence principle: to ﬁrst order in the spin, a static observer cannot decide whether he measures a time delay of spinning clocks in a nonrotating spacetime or a time delay of nonspinning clocks moving on geodesics in a rotating spacetime.
10.5 Motion of a test gyroscope in Kerr spacetime Consider the gyroscope moving along a spatially circular orbit U on the equatorial plane and a static observer m, at rest with respect to the spatial coordinates, with an adapted frame given by (8.18). We have (boost)
ζ(fw)
= ζ(Thomas) + ζ(geo) γ (G) ν(U, m) ×m −F (U, m) + γ −1 F(fw,U,m) . = 1+γ
(10.123)
Repeating all the calculations done for the Schwarzschild case in Section 10.3, one gets ˆ
(boost) θˆ
ζ (tot) θ = ζ(fw)
ˆ
θ − ζ(sc) =
τ1 (U ) , γ
(10.124)
where τ1 (U ) is now given by Eq. (8.163). It is quite natural on the equatorial ˆ plane to have a vector ezˆ = −eθˆ. This simply changes the sign of ζ (tot) θ , ζ (tot) zˆ = −
τ1 (U ) . γ
(10.125)
g
This result agrees with the expression for ζ given in Section 9.4. Assuming U = UK± to be a geodesic parameterized by the proper time, and taking into account the discussion at the end of Section 10.1, we have g
(tot) zˆ
ζ ≡ ζU
= −τ1 (UK± ) = ∓
M/r3 .
This result gives the precession angle Δφ after a full revolution as τ1 (UK± ) Δφ = ∓2π −1 ΓK± ζK± ⎤ ⎡ 1/2 M 3M ± 2a − 1⎦ , = ∓2π ⎣ 1 − r r3
(10.126)
(10.127)
248
Measurements of spinning bodies
where τ1 (UK± ) is the ﬁrst torsion for geodesics, M , τ1 (UK± ) = ± r3
(10.128)
and ΓK± is the normalization factor (see (8.111) with θ = π/2 and ζ = ζK± ). Equation (10.127) in the linear approximation in a reduces to the wellknown Schiﬀ precession formula.
10.6 Motion of a spinning body in Kerr spacetime Consider the Kerr metric written in standard BoyerLindquist coordinates (8.73) and introduce the ZAMO family of ﬁducial observers, with 4velocity n = N −1 (∂t − N φ ∂φ ),
n = −N dt ;
(10.129)
here N = (−g tt )−1/2 and N φ = gtφ /gφφ are the lapse and shift functions, respectively, explicitly given in (8.100). A suitable orthonormal frame adapted to ZAMOs is given by etˆ = n,
1 erˆ = √ ∂r , grr
1 eθˆ = √ ∂θ , gθθ
eφˆ = √
1 ∂φ , gφφ
(10.130)
with dual ˆ
ω t = N dt, √ ˆ ω θ = gθθ dθ,
ω rˆ =
√
grr dr, √ ω = gφφ (dφ + N φ dt). ˆ φ
(10.131)
As for Schwarzschild spacetime, the physical (orthonormal) component along −∂θ , perpendicular to the equatorial plane, will be taken to be along the positive zaxis and will be indicated by zˆ. The spacetime trajectory described by (10.86) will be termed a U orbit. In terms of (10.129) the line element can be expressed in the form ds2 = −N 2 dt2 + gφφ (dφ + N φ dt)2 + grr dr2 + gθθ dθ2 .
(10.132)
We require here that the center of mass of the spinning body moves on a world line whose spatial projection on the equatorial plane of the Kerr metric is geometrically circular; we shall term this world line the U orbit. Therefore we recall the following: (i) The 4velocity U of uniformly rotating spatially circular orbits can be parameterized either by ζ, the (constant) angular velocity with respect to inﬁnity, or equivalently by ν, the (constant) linear velocity with respect to ZAMOs: U = Γ[∂t + ζ∂φ ] = γ[etˆ + νeφˆ],
γ = (1 − ν 2 )−1/2 .
(10.133)
10.6 Motion of a spinning body in Kerr spacetime
249
Here Γ is a normalization factor which ensures that Uα U α = −1 and is given by −1/2 γ (10.134) Γ = N 2 − gφφ (ζ + N φ )2 = , N and N ζ = −N φ + √ ν. gφφ
(10.135)
(ii) On the equatorial plane of the Kerr solution there exists a large variety of special circular orbits, already examined in Chapter 8. Particular interest is devoted to the corotating (+) and counterrotating (−) timelike spatially circular geodesics whose angular and linear velocities (with respect to ZAMOs) are respectively −1 ζK± ≡ ζ± = a ± (r3 /M)1/2 , √ a2 ∓ 2a Mr + r2
νK± ≡ ν± = √ . (10.136) Δ(a ± r r/M) Other special orbits include geodesic meeting point orbits, with ν(gmp) =
aM(3r2 + a2 ) ν + + ν− = −√ , 2 Δ(r3 − a2 M)
(10.137)
as already stated in (8.155), and those characterized by ν(pt) =
2 (r2 + a2 )2 − 4a2 Mr √ , = −1 −1 ν+ + ν− a Δ(3r2 + a2 )
(10.138)
both of these playing a role in the study of parallel transport of vectors along circular orbits. (iii) It is convenient to introduce the Lie relative curvature of each orbit, √ (r3 − a2 M) Δ √ k(lie) ≡ k(lie) rˆ = −∂rˆ ln gφφ = − 2 3 , (10.139) r (r + a2 r + 2a2 M) as well as a FrenetSerret intrinsic frame along U , both well established in the literature. (iv) It is well known that any circular orbit on the equatorial plane of Kerr spacetime has zero second torsion τ2 , while the geodesic curvature κ and the ﬁrst torsion τ1 are simply related by τ1 = −
1 dκ , 2γ 2 dν
(10.140)
so that κ = k(lie) γ 2 (ν − ν+ )(ν − ν− ), τ1 = k(lie) ν(gmp) γ 2 (ν − ν(crit)+ )(ν − ν(crit)− ),
(10.141)
250
Measurements of spinning bodies where ν(crit)± are the spatial 3velocities associated with extremely accelerated observers. The FrenetSerret frame along U is then given by E0 ≡ U = γ[etˆ + νeφˆ],
E1 = erˆ,
E2 ≡ Eφˆ = γ[νn + eφˆ],
E3 = ezˆ,
(10.142)
satisfying the following system of evolution equations: DE0 = κE1 , dτU DE2 = −τ1 E1 , dτU
DE1 = κE0 + τ1 E2 , dτU DE3 = 0. dτU
(10.143)
To study the behavior of spinning test particles in spatially circular orbits, let us consider ﬁrst the evolution equation for the spin tensor (10.3), assuming that the frame components of the spin tensor are constant along the orbit. Following the analysis for Schwarzschild spacetime, the total 4momentum can be written as DS μν ≡ mU μ + Psμ , (10.144) P μ = −(U · P )U μ − Uν dτU where Ps = U DS/dτU and m is the bare mass of the particle, i.e. the mass it would have in the rest space of U if it were not spinning. From (10.144), Eq. (10.3) implies Stˆφˆ = 0,
Srˆθˆ = 0,
Stˆθˆ + Sφˆθˆ
ν − ν(gmp) = 0. ν(gmp) (ν − ν(pt) )
(10.145)
It is clear from (10.144) that Ps is orthogonal to U ; moreover, it turns out to be aligned also with Eφˆ: Ps = ms Eφˆ,
(10.146)
where ms ≡ Ps  is given by ms = −γk(lie) Stˆrˆ(ν − ν(gmp) ) + Srˆφˆν(gmp) (ν − ν(pt) ) .
(10.147)
From (10.144) and (10.146) the total 4momentum P can be written in the form P = μ Up , with Up = γp [etˆ + νp eφˆ],
νp =
ν + ms /m , 1 + νms /m
μ=
γ (m + νms ) , γp
(10.148)
and γp = (1 − νp2 )−1/2 . The same arguments apply here as in the Schwarzschild case. Since Up is a unit vector, the quantity μ can be interpreted as the total mass of the particle in the rest frame of Up . We see from (10.148) that the total 4momentum P is parallel to the unit tangent of a spatially circular orbit, which we shall denote as a Up orbit. The latter intersects the U orbit at only one point, where it makes sense
10.6 Motion of a spinning body in Kerr spacetime
251
to compare the vectors U and Up and the physical quantities related to them. It is clear that there exists one Up orbit for each point of the U orbit where the two intersect. Hence, along the U orbit, we can only compare at the point of intersection the quantities deﬁned in a frame adapted to U with those deﬁned in a frame adapted to Up . Let us now consider the equation of motion (10.2). Direct calculation shows that the spinforce is equal to ˆ
F (spin) = F (spin) rˆerˆ + F (spin) θ eθˆ, with
(10.149)
# γ M [r2 (2r2 + 5a2 ) + a2 (3a2 − 2Mr) r4 r(r2 + a2 ) + 2a2 M √ − 3a(r2 + a2 ) Δν]Stˆrˆ + {[r2 (r2 + 4a2 ) + a2 (3a2 − 4Mr)]ν $ √ − 3a(r2 + a2 ) Δ}Srˆφˆ} , # Sθˆφˆ γ ˆ √ F (spin) θ = 3 r (r2 + a2 )2 − 4a2 Mr − a(3r2 + a2 ) Δν F (spin) rˆ =
− ν[3a2 r(a2 − 2Mr) + r3 (r2 + 4a2 ) − 2Ma4 ] $ √ + a Δ{−4Ma2 + ν 2 [3r(r2 + a2 ) + 2a2 M]} ,
(10.150)
while the term on the lefthand side of (10.2) can be written as DEφˆ DP = ma(U ) + ms , dτU dτU
(10.151)
where a(U ) = κerˆ,
DEφˆ dτU
= −τ1 erˆ =
1 dκ erˆ, 2γ 2 dν
(10.152)
μ, m, and ms being constant along the U orbit. Since DP/dτU is directed radially, Eq. (10.149) requires that Sθˆφˆ = 0 (and therefore, from (10.145), that Stˆθˆ = 0); Eqn. (10.2) then reduces to (spin)
mκ − ms τ1 − Frˆ
= 0.
(10.153)
Summarizing, from the equations of motion (10.2) and (10.3) we deduce that the spin tensor is completely determined by two components, namely Stˆrˆ and Stˆφˆ, such that ˆ
ˆ
S = ω rˆ ∧ [Srˆtˆω t + Srˆφˆω φ ].
(10.154)
From the relations ˆ
ˆ
ˆ
ˆ
ω t = γ[−U + νΩφ ], ω φ = γ[−νU + Ωφ ],
ˆ
Ωφ = [Eφˆ] ,
(10.155)
252
Measurements of spinning bodies
one obtains the useful relation ˆ S = γ (Srˆtˆ + νSrˆφˆ)U ∧ ω rˆ + (νSrˆtˆ + Srˆφˆ)ω rˆ ∧ Ωφ .
(10.156)
Since the components of S are constant along U , then from the FrenetSerret formalism one ﬁnds DS ˆ = γ[(τ1 + κν)Srˆtˆ + (ντ1 + κ)Srˆφˆ]U ∧ Ωφ , dτU
(10.157)
or, from (10.3), ˆ
ˆ
Ps = −γ[(τ1 + κν)Srˆtˆ + (ντ1 + κ)Srˆφˆ]Ωφ ≡ ms Ωφ .
(10.158)
Let us introduce the quadratic invariant s2 =
1 Sμν S μν = −Srˆ2tˆ + Srˆ2φˆ. 2
(10.159)
Equation (10.3) is identically satisﬁed if the only nonzero components of S are Srˆtˆ and Srˆφˆ. To discuss the physical properties of the particle motion one needs to supplement (10.153) with the algebraic conditions S μν Pν = 0, which are equivalent to (Srˆtˆ, Srˆφˆ) = s(−γp νp , γp ).
(10.160)
Let us summarize the results. The quantity ms in general is given by ms = −sγp γ[−νp (τ1 + κν) + (ντ1 + κ)],
(10.161)
and, once inserted in the equation of motion, it gives (spin)
mκ + sγp γ[−νp (τ1 + κν) + (ντ1 + κ)]τ1 − Frˆ
= 0,
(10.162)
where (spin)
Frˆ with
= sγγp [Aννp + Bν + Cνp + A],
√ 3Ma(r2 + a2 ) Δ , r4 (r3 + a2 r + 2a2 M) M r4 + 3a4 + 4a2 r(r − M) , B= 4 r r3 + a2 r + 2a2 M M C=B+ 3. r
(10.163)
A=−
(10.164)
Thus one can solve Eq. (10.162) for the quantity sˆ = ±ˆ s = ±s/(mM), which denotes the signed magnitude of the spin per unit (bare) mass m of the test particle and M of the black hole, obtaining sˆ = −
κ , Mγγp D
(10.165)
10.6 Motion of a spinning body in Kerr spacetime
253
Table 10.1. The limiting values of ν are listed for particular values of the blackhole rotational parameter a and for a ﬁxed radial distance r/M = 8, with blackhole mass M = 1. a 0 0.2 0.4 0.6 0.8 1
ν¯1
ν¯2
−0.2970 −0.2955 −0.2933 −0.2902 −0.2861 −0.2808
0.2970 0.2979 0.2984 0.2986 0.2984 0.2981
with D = {[−νp (τ1 + κν) + (ντ1 + κ)]τ1 − (Aννp + Bν + Cνp + A)} ,
(10.166)
and with κ and τ1 given by (10.141). Recalling its deﬁnition (10.161), ms becomes ms = −Mˆ sγγp [−νp (τ1 + κν) + (ντ1 + κ)]. m
(10.167)
Using (10.148) for νp , we obtain sˆ =
1 ν − νp , Mγγp (1 − ννp )[−νp (τ1 + κν) + (ντ1 + κ)]
(10.168)
which must be considered together with (10.165); of course, the case sˆ = 0 (absence of spin) is only compatible with geodesic motion: ν ≡ νp = ν± . By eliminating sˆ from (10.165) and (10.168) and solving with respect to νp , we obtain √ 1 (2κτ1 + A)ν 2 + [2(κ2 + τ12 ) + M/r3 ]ν + 2κτ1 − A ± Ψ (±) νp = 2 κ2 ν 2 + (2κτ1 + A)ν + τ12 + C Ψ = [A2 + 4κ(κB + τ1 A)]ν 4 + [8Aκ2 + 2(B + C)(2κτ1 + A)]ν 3 + [4κ2 M/r3 + (B + C)2 + 2A2 ]ν 2 − [8Aκ2 + 2(B + C)(2κτ1 − A)]ν − A(4κτ1 − A) − 4Cκ2 . (±)
(10.169)
Now, by substituting νp = νp for instance into (10.168), we obtain the main relation between sˆ and ν. The reality condition (10.169) requires that ν take values outside the interval (¯ ν1 , ν¯2 ), with ν¯1 and ν¯2 listed in Table 10.1 for particular values of a (r, M ﬁxed); the timelike condition for νp  < 1 is satisﬁed for all values of ν outside the same interval.
254
Measurements of spinning bodies
To ﬁrst order in sˆ we have s2 ), ν = ν± + N sˆ + O(ˆ
√ 3 M(r3 + a2 r + 2a2 M){ar(r − 5M) ± Mr[2a2 − r(r − 3M)]}2
√ √ N = 2 r2 Δ Mr(a ± r r/M)2 √ × { Mr[4a2 (r − 4M) − r(r − 3M)2 ] ± a[4a2 M + r(r − 3M)(r − 7M)]}−1 .
(10.170)
Therefore, from the preceding approximate solution for ν we have that s2 ), νp(±) = ν + O(ˆ
(10.171) (±)
and the total 4momentum P is given by (10.148) with νp = νp . The corresponding angular velocity ζp and its reciprocal are then given by (Abramowicz and Calvani, 1979) √ Nr Δ sˆ + O(ˆ s2 ), ζp = ζ + 3 r + a2 r + 2a2 M √ N 1 r Δ 1 sˆ + O(ˆ s2 ) . = − 2 3 (10.172) ζp ζ ζ± r + a2 r + 2a2 M Clock eﬀect for spinning bodies Circularly rotating spinning bodies, to ﬁrst order in the spin parameter sˆ and the metric parameter a, and therefore neglecting terms containing sˆa, have orbits close to being geodesics (as expected), with 1 ζ(±,±)
=
1 ± Mˆ sJ , ζ±
(10.173)
with J = 3/2. Equation (10.173) deﬁnes such orbits with the various signs corresponding to co/counterrotating orbits with positive/negative spin direction along the z axis. For instance, ζ(+,−) indicates the angular velocity of U corresponding to a corotating orbit (+) with spindown (−) alignment. Therefore one can study the diﬀerences in the arrival times after one complete revolution with respect to a local static observer: 1 1 = 4π (a + Mˆ sJ ) , + Δt(+,+;−,+) = 2π ζ(+,+) ζ(−,+) Δt(+,+;−,−) = Δt(+,−;−,+) = 4πa, sJ ) . Δt(+,−;−,−) = 4π (a − Mˆ
(10.174)
s the clock eﬀect can be In the latter case, it is easy to see that if a = 32 Mˆ made vanishing; this feature can be directly measured (Faruque, 2004). More interesting is the case of corotating spinup against counterrotating spindown or, alternatively, corotating spindown against counterrotating spinup. Now the
10.7 Gravitational waves and the compass of inertia
255
compensating eﬀect makes the spin contribution to the clock eﬀect equal to zero; this case therefore appears indistinguishable from that of spinless particles.
10.7 Gravitational waves and the compass of inertia The existence of gravitational waves as predicted by general relativity has been indirectly ascertained by observation of the binary pulsar PSR 1913+16 (Taylor and Weisberg, 1989). A large body of literature is now available on the properties of gravitational waves, and ways to detect them. Hence we shall conﬁne our attention to a particular aspect of this problem which has been considered only recently, namely the dragging of inertial frames by a gravitational wave (Bini and de Felice, 2000; Sorge, Bini, and de Felice, 2001; Biˇcak, Katz, and LyndenBell, 2008). We then ask the question: what observable eﬀects might be produced by a plane gravitational wave acting on a test gyroscope? It is well known that in the absence of signiﬁcant coupling between the background curvature and the multipole moments of the energymomentum tensor of an extended body, its spin vector is FermiWalker transported along its own trajectory (see de Felice and Clarke, 1990, and references therein); for measurable eﬀects in a gravitational wave background, see also (Cerdonio, Prodi, and Vitale, 1988; Mashhoon, Paik, and Will, 1989; Krori, Chaudhury, and Mahanta, 1990; Fortini and Ortolan, 1992; Herrera, Paiva, and Santos, 2000). The eﬀects of a plane gravitational wave on a frame which is not FermiWalker transported are best appreciated by studying the precession of a gyroscope at rest in that frame (de Felice, 1991). The task, then, is to ﬁnd a frame which is not FermiWalker transported and is also operationally welldeﬁned so that, monitoring the precession of a gyroscope with respect to that frame, we can study the dragging induced on it by a plane gravitational wave. Our main purposes are: (i) to establish the existence of a relativistic eﬀect which is measurable, namely the gyroscopic precession induced by a plane gravitational wave; (ii) to show that a frame can be selected with respect to which the precession is due to one polarization state only. In the latter case we will have constructed a gravitational polarimeter. The metric of a plane monochromatic gravitational wave, elliptically polarized and propagating along a direction which we ﬁx as the x coordinate direction, can be written in transversetraceless (TT) gauge, as in Chapter 8, Eq. (8.195). The timelike geodesics of this metric, deduced in de Felice (1979), are described by (8.196).
Test gyroscopes in motion along a geodesic We consider a test gyroscope moving along a geodesic described by a tangent vector ﬁeld U = U(g) given by (8.196). The spin vector S(U ) satisﬁes the equation
256
Measurements of spinning bodies D(fw,U ) S(U ) = 0. dτU
(10.175)
This property implies that S(U ) does not precess with respect to spatial axes which are FermiWalker transported along the world line of U . If the observer U is comoving with the gyroscope and refers to a general orthonormal spatial frame {e(U )aˆ }a=1,2,3 adapted to his world line, then Eq. (10.175) becomes dS(U )aˆ ˆ b cˆ e(U )aˆ = 0, + aˆ ˆbˆc ζ(fw, S(U ) (10.176) U, e(U )) dτU where ζ(fw, U, e(U )) is the observed rate of spin precession, deﬁned by 1 aˆˆbˆc a ˆ a ˆ ζ(fw, e(U )ˆb · ∇(U )(fw) e(U )cˆ ≡ −∗(U ) C(fw, U, e(U )) = − U, e(U )) . (10.177) 2 Test gyroscopes at rest in the TTgrid of a gravitational wave Let us now consider the case of gyroscopes carried by observers at rest in the TTgrid of a gravitational wave. These observers move along world lines which form a vorticityfree congruence of geodesics and whose tangent ﬁeld is given by u = −dt,
u = ∂t .
(10.178)
One can adapt to this ﬁeld an inﬁnite number of spatial frames by rotating any given one arbitrarily. For example, consider the following adapted frame ˆ {eαˆ } = {eˆ0 = u, eaˆ = e(u)aˆ } with its dual {ω αˆ } = {ω 0 = −u , ω aˆ = ω(u)aˆ }: u = ∂t , e(u)ˆ1 = ∂x , −1/2
e(u)ˆ2 = (1 − h+ )
1 ∂ y 1 + h+ ∂ y , 2
e(u)ˆ3 = (1 − h2+ − h2× )−1/2 [(1 − h+ )−1/2 h× ∂y + (1 − h+ )1/2 ∂z ] 1 (10.179) h× ∂y + 1 − h+ ∂z , 2 −u = dt , ˆ
ω(u)1 = dx ,
h× 1 ω(u) = (1 − h+ ) dy − dz 1 − h+ dy − h× dz , 1 − h+ 2 2 2 1/2 1 − h+ − h× 1 ˆ (10.180) dz 1 + h+ dz. ω(u)3 = 1 − h+ 2 ˆ 2
1/2
This is not a FermiWalker frame; the Fermi rotation coeﬃcients are given by C(fw, u, e(u))ˆbˆa = e(u)ˆb ·u ∇(u)(fw) e(u)aˆ ,
(10.181)
10.7 Gravitational waves and the compass of inertia
257
and therefore, to ﬁrst order in the metric perturbations h, one ﬁnds that the only independent nonzero component is2 ˆ
1 − ζ(fw, u, e(u) ˆ) = C(fw, u, e(u)dˆ)ˆ 3ˆ 2 =− d
[h˙ × (1 − h+ ) + h˙ + h× ] 1 − h˙ × . 2 2 2 2(1 − h+ ) 1 − h+ − h× (10.182)
Clearly Eq. (10.182) shows that the Fermi nature of the frame only depends on the h× component of the wave amplitude. In fact, if we set h× = 0 and consider the weakﬁeld limit, the spatial triad becomes e(u)ˆ1 = ∂x ,
1 e(u)ˆ2 1 + h+ ∂y , 2
1 1 − h+ ∂ z , 2
e(u)ˆ3
(10.183)
showing that these axes do not rotate with respect to the coordinate directions. In this case (h× = 0) the axes e(u)aˆ are FermiWalker transported along u, as expected. In the complementary case (h+ = 0) we have e(u)ˆ1 = ∂x ,
e(u)ˆ2 ∂y ,
e(u)ˆ3 h× ∂y + ∂z ;
(10.184)
hence the wave rotates the e(u)ˆ3 axis and therefore a gyroscope must be seen to precess with respect to it. However, the frame e(u)aˆ , although special in selecting only one state of polarization, cannot be operationally deﬁned in a simple way. Of course there exist inﬁnitely many spatial frames {˜ e(u)aˆ } which are adapted to the observers (10.178) and can be obtained from the one in (10.179) by a spatial rotation R: ˆ
e˜(u)aˆ = e(u)ˆb Rb aˆ .
(10.185)
Two of these are quite natural: a FermiWalker frame and a FrenetSerret one. In the ﬁrst case (FermiWalker) the spatial directions are easily ﬁxed by three mutually orthogonal axes of comoving gyroscopes. Obviously this frame is not suitable for measuring the Fermi rotation itself, as already noted. The second case (FrenetSerret) does not correspond to a uniquely deﬁned frame along u, since its world line is a geodesic. Properly speaking, in this degenerate situation (u geodesic and the 4acceleration a(u) = κ = 0) the ﬁrst vector e1 of a FrenetSerret frame can be chosen orthogonally to u in an arbitrary way (i.e. according to ∞2 diﬀerent possibilities) and then the standard procedure for ﬁxing a frame gives 2
We use a dot notation for the derivative with respect to t.
258
Measurements of spinning bodies De1 = τ1 e2 , dτu De2 = −τ1 e1 + τ2 e3 , dτu De3 = −τ2 e2 , dτu
(10.186)
corresponding to a FrenetSerret rotation angular velocity ω(FS) = τ1 e3 + τ2 e1 .
(10.187)
We recall here that ω(FS) is the negative of the angular velocity of precession of a gyroscope. This arbitrariness can be eliminated by choosing e1 to be aligned with the spin of a gyroscope. Then, τ1 vanishes and the previous relations reduce to De1 = 0, dτu De2 = τ2 e3 , dτu De3 = −τ2 e2 , dτu
(10.188)
where now ω(FS) = τ2 e1 .
(10.189)
This corresponds to a second degenerate condition for the FrenetSerret approach; in fact, as τ1 = 0, the FrenetSerret procedure starts with e2 , which in turn can be any direction in the subspace orthogonal to u and e1 . The form of the metric suggests the choice e1 = ∂x ,
(10.190)
which corresponds to a gyroscope pointing along the direction of propagation of the wave. Thus a parametric form for e2 (and consequently for e3 ) can be given in terms of the frame (10.179) by introducing an angle φ which is an arbitrary function along the world line e2 = cos φ e(u)ˆ2 + sin φ e(u)ˆ3 , e3 = −sin φ e(u)ˆ2 + cos φ e(u)ˆ3 ,
(10.191)
so that τ2 becomes a function of φ. Obviously, particular choices for φ allow τ2 (and ω(FS) ) to depend only on h+ or h× or even to vanish (τ2 = 0), leading back to the FermiWalker case. In general, a straightforward calculation shows that, in the weakﬁeld limit and for arbitrary φ, 1 (10.192) ω(FS) = φ˙ − h˙ × e1 . 2
10.7 Gravitational waves and the compass of inertia
259
Assuming that φ depends only on h+ and h× along the world line, and expanding this for weak ﬁelds, one has
so
φ = φ 0 + φ+ h + + φ× h × ,
(10.193)
1 ˙ ˙ h× e1 . = φ+ h+ + φ× − 2
(10.194)
ω(FS)
Special cases are the following: φ+ = 1,
φ× = 1/2,
φ+ = 0,
φ× = 3/2,
ω(FS) = h˙ + e1 , ω(FS) = h˙ × e1 ,
φ+ = 0,
φ× = 1/2,
ω(FS) = 0 .
It is then clear that according to the way in which one selects the spatial axis e2 , the angular velocity of precession ω(FS) (i.e. the gyro dragged along u) becomes a “ﬁlter” for the gravitational wave polarization.
Test gyroscopes in general geodesic motion Let us now consider the case of a gyroscope in motion along a general geodesic with 4velocity U = U(g) , and assume that u = ∂t is a family of observers deﬁned all along the world line of U . As discussed in detail in Chapter 3, when dealing with diﬀerent families of observers (u and U in this case) the two local rest spaces LRSu and LRSU are naturally connected by two maps: the mixed projection map P (U, u) = P (U )P (u) : LRSu → LRSU and the local rest space boost map B(lrs) (U, u) = P (U )B(U, u)P (u) : LRSu → LRSU , where B(U, u) is deﬁned so that U = γ[u + ν νˆ(U,u) ] = B(U, u)u , where ν ≡ ν(U, u) = ν(u, U ) is the magnitude of the relative spatial velocity, νˆ(U, u) is the unit spatial direction of the velocity of u relative to U , γ = γ(U, u) = γ(u, U ) is the relative Lorentz factor, and the corresponding expression for B(lrs) (U, u) is given by (3.143). In this case we have, from (8.196) and (10.178), 1 + f + E2 , 2E ν = [(1 + f + E 2 )2 − 4E 2 ]1/2 (1 + f + E 2 )−1 ,
γ=
νˆ = [(1 + f + E 2 )2 − 4E 2 ]−1/2 [(1 + f − E 2 ) e(u)ˆ1 + 2Eα(1 − h+ )−1/2 e(u)ˆ2 2E[β(1 − h+ ) + αh× ] + e(u)ˆ3 ]. (1 − h+ )1/2 (1 − h2+ − h2× )1/2
(10.195)
260
Measurements of spinning bodies
In order to study the spin precession as seen by an observer U comoving with the gyro, we must ﬁrst choose axes with respect to which the precession will be measured. We then proceed to ﬁnd a spatial frame which is suitable for actual measurements. It is clear that the observers with 4velocity u can unambiguously determine in their rest frame a spatial direction given by that of the relative velocity ν of the gyroscope, νˆ. Let the direction of propagation of the gravitational wave be the xaxis with unit vector e(u)ˆ1 . Then from these two directions, namely the (instantaneous) relative velocity of the gyro and the wave propagation direction, it is possible to construct the following spatial triad: λ(u)1 = e(u)ˆ1 = ∂x , ν (U, u)2 )2 + (ˆ ν (U, u)3 )2 ]−1/2 [ˆ ν(U,u) ×u e(u)ˆ1 ] , λ(u)2 = [(ˆ ˆ
ˆ
λ(u)3 = λ(u)1 ×u λ(u)2 .
(10.196)
This is a FrenetSerret triad, which can be cast into the form (10.191) if
tan φ = −
1 − h2+ − h2× νˆ(U, u) = − . β(1 − h+ ) + αh× νˆ(U, u)ˆ3 ˆ 2
α
(10.197)
In the weakﬁeld limit (see eq. (10.193)) we obtain φ0 = −arctan
α αβ α2 , φ+ = − 2 , φ = . × β α + β2 α2 + β 2
(10.198)
Since the family of observers u is deﬁned along the world line of the observer U carrying the gyro, the latter can identify spatial directions in his own local rest space simply by boosting the directions of the FrenetSerret generated frame {λ(u)a }. At each event along his world line, the observer U will see the axes λ(u)a deﬁned by (10.196) to be in relative motion, whereas the boost of these axes, λ(U )aˆ = B(lrs) (U, u)λ(u)a γ [ν(U,u) · λ(u)a ](u + U ), = λ(u)a + γ+1
(10.199)
will be identiﬁed as the corresponding axes with the same orientation which are “momentarily at rest” with respect to U . The precession of the spin then corresponds to the spatial dual of the FermiWalker structure functions of λ(U )aˆ , namely C(fw,U,λ(U ))ˆbˆa , according to (10.177). To ﬁrst order in h, let us consider the precession angular velocity of the triad e(U )aˆ which is the boosted FrenetSerret ea given by (10.191). We then have
10.8 Motion of an extended body in a gravitational wave spacetime − ζ(fw, U, e(U )aˆ )
ˆ 1
ˆ
−ζ(fw, U, e(U )aˆ ) 2
ˆ
261
E ˙ 2α2 ˙ h× −1 + 2φ× − + 2φ+ h+ , − 2 (E + 1)2 + α2 + β 2 E(E + 1) − [h˙ + (β cos φ0 + α sin φ0 ) (E + 1)2 + α2 + β 2 + h˙ × (β sin φ0 − α cos φ0 )] ,
−ζ(fw, U, e(U )aˆ ) 3
E(E + 1) [h˙ + (β cos φ0 + α sin φ0 ) (E + 1)2 + α2 + β 2 + h˙ × (β sin φ0 − α cos φ0 )] .
(10.200)
Next we specialize these results to the operationally deﬁned triad λ(U )a , corresponding to the values of φ0 , φ+ , and φ× given in (10.198); in this special case, we ﬁnally have ˆ
2α2 (E + 1)2 E ˙ h× −1 + 2 2 (α + β 2 )[(E + 1)2 + α2 + β 2 ] αβ ˙ h+ , +2 2 α + β2
E(E + 1) α2 + β 2 ˙ α2 − β 2 ˙× , h − + h + (E + 1)2 + α2 + β 2 α2 + β 2
E(E + 1) α2 + β 2 ˙ α2 − β 2 ˙ h (10.201) + h + 2 × . (E + 1)2 + α2 + β 2 α + β2
− ζ(fw, U, λ(U )aˆ ) 1 −
ˆ
−ζ(fw, U, λ(U )aˆ ) 2 ˆ
−ζ(fw, U, λ(U )aˆ ) 3
Moreover, by rescaling the precession angular velocity ζ(fw, U, λ(U )aˆ ) by a γfactor, i.e. γ −1 ζ(fw, U, λ(U )aˆ ) , one refers the same angular velocity to the proper time of the observer u, obtaining a sort of reconstruction made by u of the precession seen by the observer U carrying the gyroscope. From these relations it is clear that the essential features we have found for a gyroscope at rest in the background of the wave are still valid when the gyro itself is moving. Equations (10.200) represent the components of the precession angular velocity as functions of the parameters E, α, β, φ0 , φ+ , and φ× . Special choices of these parameters can simplify the components or specialize them to particular physical conditions.
10.8 Motion of an extended body in a gravitational wave spacetime Consider an extended body as described by the Dixon model, Eqs. (10.2)–(10.4). Introduce the unit tangent vector Up aligned with the 4momentum P , i.e. μ P = mUpμ (with Up · Up = −1), as well as the spin vector S β = 12 ηα βγδ Upα Sγδ .
(10.202)
262
Measurements of spinning bodies
Let the body move in the spacetime (8.195) of a weak gravitational plane wave, propagating along the x direction, with metric functions given by h+ = A+ sin ω(t − x),
h× = A× cos ω(t − x).
(10.203)
The geodesics of this metric have already been presented in Eq. (8.196). For a weak gravitational wave they reduce to the following form: t(λ) = Eλ + t0 + x(λ) − x0 , x(λ) = (μ2 + α2 + β 2 − E 2 )
λ 2E
1 [(α2 − β 2 )A+ cos ω(Eλ + t0 − x0 ) 2ωE 2 − 2αβA× sin ω(Eλ + t0 − x0 )] + x0 , −
y(λ) = αλ + y0 1 [αA+ cos ω(Eλ + t0 − x0 ) − βA× sin ω(Eλ + t0 − x0 )], − ωE 1 [βA+ cos ω(Eλ + t0 − x0 ) z(λ) = βλ + z0 + ωE + αA× sin ω(Eλ + t0 − x0 )], (10.204) where λ is an aﬃne parameter, xα 0 are integration constants, and α, β, and E are conserved Killing quantities; μ2 = 1, 0, −1 correspond to timelike, null, and spacelike geodesics, respectively. It is in general a hard task to solve the whole set of equations (10.2)–(10.4) in complete generality. However, looking for weakﬁeld solutions, it is suﬃcient to search for changes in the mass parameter m of the body, its velocity and total 4momentum, as well as the spin tensor, which are of the same order as the metric functions h+,× , i.e. ˜, ˜ U = U0 + U m = m0 + m, P = P0 + P˜ , S μν = S0μν + S˜μν .
(10.205)
The quantities with subscript “0” are the ﬂat background ones, i.e. those corresponding to the initial values, before the passage of the wave. Let U and Up be written in the form U = γ(∂t + ν˜a ∂a ),
γ = (1 − ν˜2 )−1/2 ,
Up = γp (∂t + ν˜pa ∂a ),
γp = (1 − ν˜p2 )−1/2 .
(10.206)
The zerothorder set of equations turns out to be DP0μ = 0, dτU0
DS0μν [μ ν] = 2P0 U0 . dτU0
(10.207)
10.8 Motion of an extended body in a gravitational wave spacetime
263
Consider the body initially at rest at the origin of the coordinates, i.e. t = τU0 ,
x(τU0 ) = 0,
y(τU0 ) = 0,
z(τU0 ) = 0,
(10.208)
where τU0 denotes the proper time parameter. In this case the unit tangent vector reduces to U0 = ∂t . The set of equations (10.207) is fulﬁlled, for instance, by taking P0 = m0 U0 , implying that the components of the background spin tensor S0μν remain constant along the path. The conditions (10.4) simply give S00a = 0. The remaining components are related to the background spin vector components by S012 = S03 ,
S013 = −S02 ,
S023 = S01 .
(10.209)
The ﬁrstorder system of equations can also be solved straightforwardly. Setting σ α ≡ (S α /m0 )ω and splitting σ α = σ0α + σ ˜ α , according to (10.205) we have (Bini et al., 2009) m ˜ = 0, ν˜p1 = −A× σ02 σ03 , 1 1 2 σ0 A× sin ωτU + σ03 A+ (cos ωτU − 1) + A× σ01 σ03 , 2 2 1 1 σ 3 A× sin ωτU − σ02 A+ (cos ωτU − 1) + A× σ01 σ02 , ν˜p3 = 2 0 2 ν˜p2 = −
1 2 2 [(σ ) − (σ03 )2 ]A+ sin ωτU + σ02 σ03 A× (cos ωτU − 1), 2 0 1 ν˜2 = − A+ σ01 σ02 + A× σ02 sin ωτU 2 1 − A× σ01 σ03 + A+ σ03 (cos ωτU − 1), 2 1 ν˜3 = A+ σ01 σ03 + A× σ03 sin ωτU 2 1 − A× σ01 σ02 + A+ σ02 (cos ωτU − 1), 2
(10.210)
ν˜1 =
(10.211)
and σ ˜ 1 = 0, 1 A× σ03 (cos ωτU − 1) − 2 1 σ ˜ 3 = A× σ02 (cos ωτU − 1) − 2 σ ˜2 =
1 A+ σ02 sin ωτU , 2 1 A+ σ03 sin ωτU , 2
(10.212)
264
Measurements of spinning bodies
so that the spatial orbit is 1 ω x = − [(σ02 )2 − (σ03 )2 ]A+ (cos ωτU − 1) 2 + σ02 σ03 A× (sin ωτU − ωτU ), σ2 ω y = 0 (A+ σ01 + A× )(cos ωτU − 1) 2 σ3 − 0 (A× σ01 + A+ )(sin ωτU − ωτU ), 2 σ03 ω z = − (A+ σ01 + A× )(cos ωτU − 1) 2 σ02 − (A× σ01 + A+ )(sin ωτU − ωτU ), 2 in agreement with the results of Mohseni and Sepangi (2000).
(10.213)
Epilogue
The mathematical structure of general relativity makes its equations quite remote from a direct understanding of their content. Indeed, the combination of a covariant fourdimensional description of the physical laws and the need to cope with the relativity of the observations makes a physical measurement an elaborate procedure. The latter consists of a few basic steps: (i) Identify the covariant equations which describe the phenomenon under investigation. (ii) Identify the observer who makes the measurements. (iii) Choose a frame adapted to that observer, allowing the spacetime to be split into the observer’s space and time. (iv) Decide whether the intended measurement is local or nonlocal with respect to the background curvature. (v) Identify the frame components of those quantities that are the observational targets. (vi) Find a physical interpretation of the above components, following a suitable criterion such as a comparison with what is known from special relativity or from nonrelativistic theories. (vii) Verify the degree of residual ambiguity in the interpretation of the measurements and decide on a strategy to eliminate it. Clearly, each step of the above procedure relies on the previous one, and the very ﬁrst step provides the seed of a measurement despite the mathematical complexity. • Fixing the observer is independent of the coordinate representation; we can have many observers with a given choice of the coordinate grid, but we can also deal with a given observer within many coordinate systems. • Whatever the choice of the coordinate system, one may adapt to a given observer many spatial frames, each providing a diﬀerent perspective. • A measurement requires a mathematical modeling of the target, but also of the measuring conditions which account for the dynamical state of the observer and the level of accuracy of his measuring devices. • The physical interpretation of a measurement requires some previous knowledge of the object of investigation, so that it can be identiﬁed as a generalization from
266
Epilogue
a restricted case as from special to general relativity, from empty to nonempty states, or from static to nonstatic cases, just to mention a few. • Without a comparison cornerstone, the result of a measurement may reveal itself as a new general relativistic eﬀect. • General relativity stems from the Principle of Equivalence, which states a fundamental ambiguity, namely the impossibility of distinguishing true gravity from inertial accelerations. Like a kind of original sin, this ambiguity surfaces in many measuring operations, forcing the observer to envisage sets of independent measurements to cure this inadequacy and reach the required degree of certainty. In this book we have illustrated all of the above steps, with the intention of highlighting the method rather than of considering all possible measurements, most of which are widely discussed in the literature. The notation used is meant to help the reader to recognize at each step who the observer is and what is being observed; clearly this distinction leads easily to confusion when more than two observers are involved. Although we conclude here our bird’seye view of the theory of measurements in general relativity, we hardly consider this subject closed. Indeed much remains to be done to understand how classical measurements in relativity theory would match with quantum measurements in the search for a uniﬁed theory.
Exercises
1. Consider the Euclidean 2sphere V2 with Riemannian metric ds2 = dθ2 + sin2 θdφ2 . (a) Show that the orthonormal frame eˆ1 = ∂θ ,
eˆ2 =
1 ∂φ , sin θ
with dual ˆ
ω 1 = dθ,
ˆ
ω 2 = sin θdφ,
admits a single nonvanishing structure function ˆ
C 2 ˆ1ˆ2 = −cot θ. (b) Show that the only nonvanishing Christoﬀel symbols are Γφ θφ = cot θ,
Γθ φφ = −sin θ cos θ.
(c) Solve the Killing equation ∇(β ξα) = 0 in the above metric and show that the general solution for the vector ξ is given by ξ = c1 ξ1 + c2 ξ2 + c3 ξ3 , where c1 , c2 , c3 are constants and ξ1 = sin φ∂θ + cot θ cos φ∂φ , ξ2 = cos φ∂θ − cot θ sin φ∂φ , ξ3 = ∂φ , are the wellknown angular momentum operators or the generators of the rotation group. (d) Show that the 4acceleration of the curve with unit tangent vector eφˆ = 1/ sin θ∂φ is a(eφˆ) ≡ ∇eφˆ eφˆ = −cot θ∂θ . Is it possible to specify a priori the direction of a(eφˆ)? (e) Evaluate the wedge product a(eφˆ) ∧ eφˆ.
268
Exercises (f) Discuss the parallel transport of a general vector X = X θ ∂θ + X φ ∂φ along a φloop, that is, the curve with parametric equations θ = θ0 = constant,
φ = φ(λ),
i.e. with unit tangent vector eφˆ = (1/ sin θ)∂φ . Hint. Using φ (in place of λ) as a parameter along , the transport equations reduce to the system dX θ − sin θ cos θX φ = 0, dφ
cos θ θ dX φ + X = 0. dφ sin2 θ
It is also convenient to use frame components for X: ˆ
Xθ = Xθ,
ˆ
X φ = sin θX φ .
In fact the above equations reduce to ˆ
dX θ ˆ − cos θX φ = 0, dφ and can be easily solved: ˆ X θ (φ) ˆ X φ (φ) where
R(φ cos θ) =
ˆ
dX φ ˆ + cos θX θ = 0, dφ
= R(φ cos θ)
ˆ
X θ (0) , ˆ X φ (0)
cos(φ cos θ) sin(φ cos θ) . −sin(φ cos θ) cos(φ cos θ)
(g) Study the conditions for holonomy invariance of the transported vector after one loop. Hint. When φ = 2π, for θ = π/2 (equatorial orbit) as well as for θ = 0 we have 1 0 R(2π cos θ) → . 0 1 For a general value of θ the invariance is lost. 2. Repeat the above exercise for the 2pseudosphere with metric ds2 = dθ2 + sinh2 θdφ2 . 3. Consider Schwarzschild spacetime written in standard sphericallike coordinates. Discuss the parallel transport of a vector X = X α ∂α along a circular orbit on the equatorial plane with unit tangent vector U = Γ(∂t + ζ∂φ ).
Exercises
269
4. Considering the setting of the previous problem, discuss the parallel transport of the vector X = X α ∂α along a φloop with parametric equations t = t0 ,
r = r0 ,
θ = π/2,
φ = φ(λ),
and unit tangent vector eφˆ = 1/r0 ∂φ . (a) Discuss the conditions for holonomy invariance after a φloop. (b) Compare with the limiting case ζ → ∞. 5. Consider the Schwarzschild metric. (a) Show that the unit 4velocity vector of an observer at rest with respect to the spatial coordinates is 2M m =− 1− dt. r (b) Show that dm = m ∧ g(m),
g(m) =
r2
M dr. 1 − 2M r
(c) Show that g(m) is a gradient, i.e. that dg(m) = 0, and ﬁnd the potential. (d) Show that £m g(m) = 0. 6. Discuss the geometric properties of the 2metric ds2 = dv 2 − v 2 du2 . 7. Show that the Riemann tensor of a maximally symmetric spacetime is given by R αβ δ , Rαβ γδ = 12 γδ where R is the curvature scalar. 8. Show that, for the unit 4form η, η αβγδ ηαβγδ = −4! 9. Prove that P (u)η = 0, where P (u) is the projection tensor onto LRSu . Hint. A tensor vanishing in a frame vanishes identically. 10. Prove Eq. (2.51). Hint. First permute indices in Eq. (2.45) as follows: eγ (gδβ ) − 2Γ(δβ)γ = 0, eδ (gβγ ) − 2Γ(βγ)δ = 0, eβ (gγδ ) − 2Γ(γδ)β = 0,
270
Exercises then add and subtract these relations side by side so that eγ (gδβ ) − eδ (gβγ ) + eβ (gγδ ) − 2Γ(δβ)γ + 2Γ(βγ)δ − 2Γ(γδ)β = 0, or, equivalently, eγ (gδβ ) − eδ (gβγ ) + eβ (gγδ ) − 2Γδ(βγ) + 2Γβ[γδ] + 2Γγ[βδ] = 0. Noting that C γ αβ = 2Γγ [βα] , one can cast the above relation in the form eγ (gδβ ) − eδ (gβγ ) + eβ (gγδ ) − 2Γδ(βγ) + Cβδγ + Cγδβ = 0, which in turn gives 2Γδ(βγ) = eγ (gδβ ) − eδ (gβγ ) + eβ (gγδ ) + Cβδγ + Cγδβ . From this it follows that Γδβγ = Γδ(βγ) + Γδ[βγ] 1 = [eγ (gδβ ) − eδ (gβγ ) + eβ (gγδ ) 2 + Cβδγ + Cγδβ + Cδγβ ] , as in (2.51).
11. Show that FermiWalker transport preserves orthogonality. 12. Let u be the timelike unit tangent vector to a world line γ parameterized by the proper time τ . Show that D(fw,u) α u = 0. dτ 13. Let u be the timelike unit tangent vector to a world line γ and let X be a spatial vector with respect to u (X · u = 0) FermiWalker dragged along γ. Show that D α X = [a(u) · X]uα . dτ 14. Discuss the extension of FermiWalker transport to null curves. This result is discussed in Bini et al. (2006), generalizing previous results by Castagnino (1965). 15. Let u be the timelike unit vector tangent to a (timelike) world line γ parameterized by the proper time τ and the spatial vectors eaˆ (a = 1, 2, 3) mutually orthogonal and orthogonal to u, so that eαˆ = {eˆ0 ≡ u, eaˆ } is an orthonormal tetrad. Show that
Exercises
271
D(fw,u) ˆ b eaˆ = ∇u eaˆ − a(u)aˆ u ≡ C(fw) a ˆ eˆ b, dτ ˆ
b where the coeﬃcients C(fw) a ˆ are the Fermi rotation coeﬃcients. 16. If T is a 03 tensor, show that its fully symmetric and antisymmetric parts are given by
1 (Tμνλ + Tνλμ + Tλμν + Tνμλ + Tμλν + Tλνμ ) , 3! 1 = (Tμνλ + Tνλμ + Tλμν − Tνμλ − Tμλν − Tλνμ ) . 3!
T(μνλ) = T[μνλ] 17. If T is a
0 4
tensor, evaluate its fully symmetric and antisymmetric parts.
18. Show that for a diﬀerential form S, the Lie derivative can be expressed in terms of the exterior derivative and the contraction operation: £X S = X
dS + d(X
S).
19. Verify the following identity for a 1form X: δX η = ∗δX = ∗∗d∗X = −d∗X, which can also be written in the form d(X
η) = [div X] η.
20. Show that, in the Kerr metric, the Papapetrou ﬁeld associated with the timelike Killing vector ξ = ∂t , namely Fαβ = ∇α ξβ , has principal null directions which are aligned with those of the Weyl tensor. 21. Show that a constant curvature spacetime is conformally ﬂat, i.e. C αβ γδ = 0. 22. Show that ∗
[u ∧ S] = ∗(u) S,
and therefore that ∗ ∗(u)
[
S] = (−1)p u ∧ S.
23. Consider the splitting of a 2form F and that of its spacetime dual ∗ F with respect to an observer u. Show that F = u ∧ E(u) + ∗(u) B(u),
∗
F = u ∧ B(u) − ∗(u) E(u),
where E(u) = −u F , B(u) = ∗(u) [P (u)F ]. 24. Let X be nonspatial with respect to an observer u, so that X = X  u + X ⊥ .
272
Exercises Show that its spatial curl as deﬁned in Eq. (3.37) is given by curl X = 2X  ω(u)α + [curlu X ⊥ ]α .
25. Show that the symmetric spatial Riemann tensor R(sym) satisﬁes the algebraic conditions characterizing a Riemann tensor: R(sym)(ab)cd = 0,
R(sym)ab(cd) = 0,
R(sym)abcd − R(sym)cdab = 0, R(sym)a[bcd] = 0. 26. Show that the Lie derivative along u of the projection tensor P (u) is given by £u P (u) = 2θ(u), £u P (u) = −2θ(u) , £u P (u) = u ⊗ a(u) . 27. Show that £u η(u)αβγ = Θ(u)ηαβγ . Hint. From the deﬁnition of the Lie derivative and that of the unit spatial volume 3form η(u), one gets £u η(u)βγδ = η(u)μγδ θ(u)μ β − η(u)μβδ θ(u)μ γ + η(u)μβγ θ(u)μ δ . For β = 1, γ = 2, and δ = 3 (the only possible choice of indices apart from permutations), we have £u η(u)123 = Θ(u)η123 . 28. Consider an observer u with his adapted orthonormal spatial frame e(u)aˆ , with a = 1, 2, 3. If another observer U is related to u by a boost in the generic direction νˆ(U, u), i.e. U = γ(U, u)[u + ν(U, u)ˆ ν (U, u)] ≡ γ[u + ν νˆ(U, u)], show that the boost of the triad e(u)aˆ onto LRSU gives the following orthonormal triad: ν (U, u) · e(u)aˆ ] [γνu + (γ − 1)ˆ ν (U, u)]. e(U )aˆ = e(u)aˆ + [ˆ 29. Show that if X(U ) ∈ LRSU and Y (u) ∈ LRSu then X(U ) ×U Y (u) = γ {[P (u, U )X(U ) ×u Y (u)] − (ν · P (u, U )X(U ))(ν ×u Y (u)) − u[P (u, U )X(U ) · (ν ×u Y (u))]} .
Exercises
273
Hint. From the deﬁnition of cross product in LRSU we have η(U )αβγ X(U )β Y (u)γ = η μαβγ Uμ X(U )β Y (u)γ = [−uμ η(u)αβγ + uα η(u)μβγ − uβ η(u)γμα + uγ η(u)βμα ]Uμ X(U )β Y (u)γ = γ[−uμ η(u)αβγ + uα η(u)μβγ − uβ η(u)γμα ] × [uμ + ν(U, u)μ ]X(U )β Y (u)γ . 30. Deduce the algebra of the mixed projection operators P (u, U, u) ≡ P (u, U )P (U, u),
P (u, U, u)−1 .
Show that these maps have the following expressions: P (u, U, u) = P (u) + γ 2 ν(U, u) ⊗ ν(U, u) , P (u, U, u)−1 = P (U, u)−1 P (u, U )−1 = P (u) − ν(U, u) ⊗ ν(U, u) . 31. Deduce the algebra of the mixed projection operators P (u, U, u ) ≡ P (u, U )P (U, u ),
P (u, U, u )−1 .
Show that P (u, U, u ) = P (u, u ) + γ(U, u)γ(U, u )ν(U, u) ⊗ ν(U, u ) P (u, U, u )−1 = P (u , u) + γ(u, u )[(ν(u, u ) − ν(U, u )) ⊗ ν(U, u) + ν(U, u ) ⊗ ν(u , u)]. Hint. For the various observers u, U , and u , we have U = γ(U, u)[u + ν(U, u)], U = γ(U, u )[u + ν(U, u )], u = γ(u , u)[u + ν(u , u)], u = γ(u, u )[u + ν(u, u )]. 32. Prove the following relations: P (U, u)−1 P (U, u ) = P (u, u ) + γ(u, u )ν(U, u) ⊗ ν(u, u ), P (u , u)P (U, u)−1 P (U, u ) = P (u ) + δ(U, u, u )ν(U, u ) ⊗ ν(u, u ),
274
Exercises P (u , u)P (u , U, u)−1 = P (u ) + δ(U, u, u )ν(U, u ) ⊗ [ν(u, u ) − ν(U, u )], δ(U, u, u ) ν(U, u ) ⊗ ν(U, u) γ(u, u ) + γ(u, u )ν(u, u ) ⊗ ν(U, u),
P (u , u)P (u, U, u)−1 = P (u , u) −
P (u , u)P (u, U, u)−1 P (u, u ) = P (u ) + δ(U, u, u )[ν(U, u ) ⊗ ν(u, u ) + ν(u, u ) ⊗ ν(U, u )] − δ(U, u, u )2 ν(U, u ) ⊗ ν(U, u )/γ(u, u )2 , where δ(U, u, u ) =
γ(U, u )γ(u , u) , γ(U, u)
δ(U, u, u )−1 = δ(u, U, u ).
33. Consider the Schwarzschild solution in standard coordinates (t, r, θ, φ), and an observer at rest at a ﬁxed position on the equatorial plane. Show that the Schwarzschild metric can be cast in the following form (see (5.41)): ds2 = −(1 − 2AX)dT 2 + dX 2 + dY 2 + dZ 2 + O(2), where (T, X, Y, Z) are Fermi coordinates associated with an observer at rest at r = r0 , θ = θ0 = π/2, φ = φ0 (Leaute and Linet, 1983; see also Bini, Geralico, and Jantzen, 2005 for the generalization to the case of a Kerr spacetime and a uniformly rotating observer) and related to Schwarzschild coordinates (t, r, θ, φ) by the transformation −1/2 2M t t0 + 1 − T, r0 1/2 2M 2M 1 M 2 1 2 2 1− (Y + Z ) , X+ X + r r0 + 1 − r0 2 r02 r0 r0 1/2 π Y 2M 1 θ + − 2 1− XY, 2 r0 r0 r0 1/2 2M Z 1 φ φ0 + − 2 1− XZ. r0 r0 r0 All the above relations are valid up to O(3) and with the uniform acceleration A in Eq. (5.41) given by −1/2 2M M . A= 2 1− r0 r0 34. Find the transformation laws for the relative thermal stresses (q(u)) and the relative energy density (ρ(u)) passing to another observer U .
Exercises
275
35. Deduce relations (7.40) and (7.41) and then combine them to obtain (7.46). 36. Deduce Eq. (7.50). 37. Show that by taking the covariant derivative of both sides of (7.59) one gets the relative acceleration equation. Hint. The covariant derivative of both sides of the equation ∇U Y = ∇Y U gives ∇U U Y = ∇U ∇Y U = [∇U , ∇Y ]U + ∇Y ∇U U = [∇U , ∇Y ]U + ∇Y a(U ) = R(U, Y )U + ∇Y a(U ), since [U, Y ] = 0. 38. Show that the coordinate transformation 2 M r = r¯ 1 + 2¯ r leads to the isotropic form of the Schwarzschild metric, 2 −2 M M 1+ ds2 = − 1 − dt2 2¯ r 2¯ r 4 M + 1+ (dx2 + dy 2 + dz 2 ). 2¯ r 39. In Schwarzschild spacetime let U be the unit timelike vector tangent to a √ circular orbit, U = Γ(∂t +ζ∂φ ). Consider the vector ﬁelds m = (1/ −gtt )∂t √ (congruence of static observers) and eφˆ = (1/ gφφ )∂φ (congruence of spatial φloops). Show that ΓM erˆ, r2 1/2 2M 1− sin θ erˆ + cos θ eθˆ . ∇U eφˆ = −Γζ r ∇U m =
40. With the Kerr metric written in Painlev´eGullstrand coordinates X α (see (8.79)), the locally nonrotating observers have 4velocity N = −dT . (a) Show that they form a geodesic (a(N ) = 0) and vorticityfree (ω(N ) = 0) congruence, with nonvanishing expansion tensor given by θ(N ) = θ(N )αβ ∂X α ⊗ ∂X β ,
276
Exercises with θ(N )RR θ(N )ΘΘ θ(N )ΦΦ θ(N )RΘ θ(N )RΦ
rΔ 2Mr(r2 + a2 ) (r2 + a2 )2 − 4Mr3 M , = − Σ3 Σ2 2r(r2 + a2 )
r 2Mr(r2 + a2 ) =− , Σ3 2Mr r =− , r2 + a2 (r2 + a2 )Σ sin2 θ a2 sin θ cos θ
=− 2Mr(r2 + a2 ), Σ3 2aMr2 =− 2 , (r + a2 )Σ2
being the only nonzero components. (b) Show that the trace is also nonzero, and given by M a2 + 3r2 . Θ(N ) = − Σ 2r(r2 + a2 ) 41. Consider a set of particles orbiting on spatially circular trajectories in Kerr spacetime, and let U = Γ(∂t + ζ∂φ ) be the unit timelike tangent vector. (a) Show that
2Mr (1 − aζ sin2 θ)2 − (r2 + a2 )ζ 2 sin2 θ Γ= 1− Σ
−1/2 .
Consider the following adapted frame: Etˆ = Γ(∂t + ζ∂φ ), Erˆ = (Δ/Σ)1/2 ∂r , Eθˆ = Σ−1/2 ∂θ , ¯ φ ). ¯ t + ζ∂ E ˆ = Γ(∂ φ
¯ ¯ and ζ. (b) Determine the expressions for Γ (c) Show that the frame components of the electric part of the Riemann tensor associated with U are given by Γ2 Mr 2 (r − 3a2 cos2 θ)[(3Δ + 2Mr)J Σ4 + Σ(2Δ sin2 θζ 2 + 1)], Γ2 Mr 2 E(U )θˆθˆ = − (r − 3a2 cos2 θ)[(3Δ + 4Mr)J Σ4 + Σ(Δ sin2 θζ 2 + 2)], E(U )rˆrˆ =
Exercises
277
√ 3a sin θ cos θMΓ2 Δ E(U )rˆθˆ = [a − ζ(r2 + a2 )] Σ4 × (1 − a sin2 θζ)(3r2 − a2 cos2 θ), Mr E(U )φˆφˆ = 3 (r2 − 3a2 cos2 θ), Σ where J = −(1 − aζ)2 sin2 θ − ζ 2 r2 sin2 θ − cos2 θ. (d) Show that the Fermi rotation of the above frame has components Γ2 cos θ Δ [2Mra(1 − aζ sin2 θ)2 − ζΣ2 ], ζ(fw) rˆ = − Σ2 Σ Γ2 sin θ M(r2 − a2 cos2 θ)[(r2 + a2 )ζ − a]2 ζ(fw) θˆ = − aΣ5/2 + ζ[M(a2 − r2 )ζ + Σ2 a(r − M)] . 42. Deduce the 4vector U(g) as in Eq. (8.196). Hint. Use coordinates (u, v, y, z) and the Killing vectors. ∗
∗
∗
43. Prove that (i) y − < yc− and (ii) yc− ≤ y + < yc+ , where y is given by (9.51) and yc± are given by (9.48). Hint. To prove point (i) let us ﬁrst notice that yc− > −1/a,
lim yc− = 0− ,
r→∞
yc− ≥ 0 for r ≤ 2M, yc− < 0 for r > 2M. ∗
∗
On the other hand, we have from (9.51) that y − < 0 and ∂ y − /∂r > 0 for 0 < r < ∞, and 1 ∗ ∗ lim y − = −∞, lim y − = − . r→∞ r→0 a ∗
Then it is always the case that y − ≤ −1/a < yc− . ∗ ∗ The function y + is always positive (y + > 0) and satisﬁes the condition ∗ 2 y + ≤ a/r for Δ ≥ 0, so, because yc+ ≥ a/r2 whenever Δ ≥ 0, we also have ∗ y + ≤ yc+ . ∗ To prove point (ii) we need to show that y + > yc− in the range r+ < r ≤ 2M, where yc− > 0. At r = r+ (where Δ = 0) we have, as stated, ∗ yc− = y + , but these two functions leave the point r+ with diﬀerent slopes; in fact ∂yc− /∂rr+ = −∞, while −1/2 ∗ " ∂ y + "" 8Ma2 6Ma 1 + = − 3 ∂r "r+ r4 r+ is ﬁnite and negative. Hence for r = r+ + ( > 0 and suﬃciently small) we ∗ certainly have y + > yc− . However, since yc− goes to zero monotonically as ∗ r → 2M, the function y + could intersect yc− (it should do it at least twice!) ∗ only if the plot of y + is convex somewhere in the range r+ < r < 2M. But
278
Exercises ∗
this is never so because the second derivative of y + is always positive, being equal to ∗
∂ 2 y+ 24Ma 1 + 5Ma2 /r3 = . ∂r2 r5 (1 + 8Ma2 /r3 )3/2 44. Show that the Riemann tensor Rα βγδ has in general no speciﬁc symmetries with respect to the indices β and γ. 45. In Schwarzschild spacetime, consider a particle moving in a circular orbit in the equatorial plane θ = π/2 with 4velocity U α = Γ δtα + ζδφα , where Γ is the normalization factor given by −1/2 2M 2 2 −ζ r . Γ= 1− r If of the particle, namely a(U ) =
the magnitude of the 4acceleration gαβ a(U )α a(U )β , where a(U )α = U β ∇β U α , measures the eﬀective speciﬁc weight of the particle with respect to a comoving observer, calculate the value of ζ at which the particle’s weight is maximum. 46. Given the FriedmannRobertsonWalker solution describing a spatially homogeneous and isotropic universe, dr2 2 2 2 2 2 ds2 = −dt2 + R2 (t) , + r dθ + r sin θdφ 1 − κr2 where R(t) is a diﬀerentiable function of time and κ = ±1 or 0 according to the type of spatial section, i.e. a pseudosphere (κ = −1), a ﬂat space (κ = 0), or a sphere (κ = 1): (a) Show that the cosmic observer with 4velocity uα = δtα is a geodesic for any κ. (b) Given two cosmic observers very close to each other, their relative acceleration in the radial direction depends only on the component of the Riemann tensor Rr trt . Calculate this component. 47. Given a null vector ﬁeld with components α , so that α α = 0, if it satisﬁes the condition ∇[α β] = 0, show that it is tangent to a family of aﬃne geodesics.
Exercises
279
48. Given an electromagnetic ﬁeld with energymomentum tensor 1 1 ρσ σ Fασ Fβ − gαβ Fρσ F , Tαβ = 4π 4 show that the curvature it generates satisﬁes the condition R = 0, R being the curvature scalar. 49. In Schwarzschild spacetime, consider two static observers with unitary 4velocities −1/2 2M u1 = 1 − ∂t R1 and u2 =
−1/2 2M 1− ∂t , R2
with R2 > R1 . Determine which of these observers ages faster. 50. Consider Kerr spacetime written in standard BoyerLindquist coordinates. Discuss the parallel transport of a vector X = X α ∂α along a circular orbit on the equatorial plane with unit tangent vector U = Γ(∂t + ζ∂φ ). 51. Considering the setting of the previous problem, discuss the parallel transport of the vector X = X α ∂α along a φloop with parametric equations t = t0 ,
r = r0 ,
θ = π/2,
φ = φ(λ),
and unit tangent vector eφˆ = [r/(r3 + a2 r + 2Ma2 )]1/2 ∂φ . (a) Discuss the conditions for holonomy invariance after a φloop. (b) Compare with the limiting case ζ → ∞. 52. Under what condition does the relation (£X Y ) = (£X Y ) hold for any pair of vector ﬁelds X and Y ? 53. Consider Minkowski spacetime in cylindrical coordinates {t, r, φ, z}: ds2 = −dt2 + dr2 + r2 dφ2 + dz 2 . Let U be an observer in circular motion on the hyperplane z = 0, i.e. with 4velocity U = γζ [∂t + ζ∂φ ],
γζ = [1 − ζ 2 r2 ]−1/2 .
280
Exercises (a) Show that a(U ) = −γζ2 ζ 2 r∂r . (b) Show that the triad eˆ1 = ∂r ,
eˆ2 = γζ
1 ζr∂t + ∂φ , r
eˆ3 = ∂z ,
with eˆ0 = U , is identical to a FrenetSerret frame along U . (c) Show that the associated curvature and torsions are given by κ = −γζ2 ζ 2 r,
τ1 = γζ2 ζ,
τ2 = 0.
54. Consider the setting of the above problem. Show that the triad Eˆ1 = cos(γζ ζt)eˆ1 − sin(γζ ζt)eˆ2 , Eˆ2 = sin(γζ ζt)eˆ1 + cos(γζ ζt)eˆ2 , Eˆ3 = ∂z with eˆ0 = U , is identical to a FermiWalker frame along U . 55. Consider Minkowski spacetime in cylindrical coordinates {t, r, φ, z}: ds2 = −dt2 + dr2 + r2 dφ2 + dz 2 . Let u = ∂t be a static observer. Show that this observer is inertial, that is a(u) = 0,
ω(u) = 0,
θ(u) = 0.
Consider then an observer in circular motion, i.e. with 4velocity U = γζ [∂t + ζ∂φ ],
γζ = [1 − ζ 2 r2 ]−1/2 .
Show that (G)
∇U eφˆ = −γζ ζ∂r .
F(fw,U,u) = 0, Use this result to show that (C)
a(fw,U,u) = −[ sign ζ] rζ 2 ∂r . 56. Consider the setting of the above problem, but with two rotating observers: U = γζ [∂t + ζ∂φ ],
γζ = [1 − ζ 2 r2 ]−1/2 ,
u = γω [∂t + ω∂φ ],
γω = [1 − ζ 2 r2 ]−1/2 .
Show that (G)
F(fw,U,u) = γζ γω ωζr∂r ,
(G)
F(lie,U,u) = γζ γω ωr[(ζ − ω)γω2 + ζ]∂r .
Hint. For a detailed discussion, see Bini and Jantzen (2004).
Exercises
281
57. Show that in a threedimensional Riemannian manifold, conceived as the LRS of an observer u, one has the following identity for the Scurl operation: Scurlu ∇X = −X × Ricci, where X is a vector ﬁeld. Hint. Start from the Ricci identities, 2∇[b ∇c] X a = Ra dbc X d , and take the symmetrized dual of each side: η bc(a ∇[b ∇c] X e) = −X d η (e dg Ra)g . 58. Consider the setting of the above problem. Show that if the vector ﬁeld X is a gradient, Xa = ∇a Φ, then the following identity holds: Scurl [X ⊗ X] = −X × ∇X. 59. Consider Kerr spacetime in standard BoyerLindquist coordinates, and let m denote the 4velocity ﬁeld of the static family of observers. (a) Show that the projected metric onto LRSm is given by γ =
ΔΣ sin2 θ Σ dr ⊗ dr + Σdθ ⊗ dθ − dφ ⊗ dφ. Δ Σ − 2Mr
(b) Evaluate the Ricci tensor Rab associated with the above 3metric. (c) Show that the only nonvanishing components of the corresponding Cotton tensor are the following: [y rφ , y rθ ] =
6a2 M2 [Δ cos θ, −(r − M) sin θ], [Σ(Σ − 2Mr)]5/2
where y ab = −[Scurlm R]ab . (d) What conclusions can be drawn in the limiting case of Schwarzschild spacetime? 60. Consider Kerr spacetime in standard BoyerLindquist coordinates and let m denote the 4velocity ﬁeld of the static family of observers. (a) Evaluate the acceleration vector a(m) and the vorticity vector ω(m) associated with m. (b) Evaluate the electric (E(m)) and magnetic (H(m)) parts of the Weyl tensor as measured by the m observers. (c) Deﬁne the complex ﬁelds Z(m) = E(m) − iH(m) and z(m) = −a(m) − iω(m) and show that S(m) = z(m) × Z(m) = 0.
282
Exercises Note that S(m) is proportional to the socalled Simon tensor (Simon, 1984), whose vanishing is a peculiar property of Kerr spacetime. (d) Show that if in general a × A = 0 (with a a spatial vector and A a spatial symmetric 2tensor) then necessarily A ∝ [a ⊗ a](TF) .
61. Let u be a congruence of test observers in a general spacetime. Show that the vorticity and expansion ﬁelds have the representations ω(u) =
1 d(u)u , 2
θ(u) =
1 £(u)u g . 2
62. Consider Minkowski spacetime in standard Cartesian coordinates: ds2 = −dt2 + dx2 + dy 2 + dz 2 . Let t=
1 sinh Aτ, A
x = 0,
y = 0,
z=−
1 cosh Aτ, A
the parametric equations of the world line u of an observer uniformly accelerated with acceleration A along the negative zaxis. (a) Show that the frame eˆ0 = cosh Aτ ∂t − sinh Aτ ∂z , eˆ1 = ∂x , eˆ2 = ∂y , eˆ3 = −sinh Aτ ∂t + cosh Aτ ∂z is FermiWalker transported along u. (b) Use this result to set up a system of Fermi coordinates {T, X, Y, Z} on u. Hint. The map between Cartesian and Fermi coordinates is given by 1 − Z sinh AT, t= A x = X, y = Y, 1 − Z cosh AT, z=− A and the Minkowski metric in coordinates {T, X, Y, Z} is given by ds2 = −(1 − AZ)2 dT 2 + dX 2 + dY 2 + dZ 2 . 63. Consider the Kasner vacuum metric (2.134), ds2 = −dt2 + t2p1 dx2 + t2p2 dy 2 + t2p3 dz 2 , with p1 + p2 + p3 = 1 = p21 + p22 + p23 .
Exercises
283
(a) Deﬁne the complex tensor C˜αβγδ = Cαβγδ − i∗ Cαβγδ and form the two complex curvature invariants I=
1 ˜ 1 ˜ Cαβγδ C˜ αβγδ , J = Cαβγδ C˜ γδ μν C˜μναβ . 32 384
Show that the ratio S=
27 27J 2 = − p1 p 2 p 3 , I3 4
i.e. that it is a constant. Hint. The quantity S is also known as the spectral index or speciality index of the spacetime and plays a role in the Petrov classiﬁcation of spacetimes as well as in perturbation theory. (b) Consider the static family of observers u = ∂t with adapted orthonormal frame eˆ1 = t−p1 ∂x ,
eˆ2 = t−p2 ∂y ,
eˆ3 = t−p3 ∂z .
Show that they see a purely electric Weyl tensor, i.e. E(u) =
p1 p3 p 1 p2 p 2 p3 eˆ ⊗ eˆ1 + 2 eˆ2 ⊗ eˆ2 + 2 eˆ3 ⊗ eˆ3 , t2 1 t t
H(u) = 0.
64. Consider Minkowski spacetime with the metric written in cylindrical coordinates, i.e. ds2 = −dt2 + dr2 + r2 dφ2 + dz 2 . (a) Show that under the change of coordinates t = t,
r = r,
φ = φ − Ωt,
z = z
the metric becomes ds2 = −γ −2 dt2 + 2r2 Ωdt dφ + r2 dφ2 + dr2 + dz 2 , where γ −2 = 1 − Ω2 r2 . Minkowski spacetime endowed with the latter form of the metric will be referred to as rotating Minkowski. For convenience the spacetime coordinates will be relabeled t, r, φ, z hereafter. (b) In a rotating Minkowski spacetime, consider the static family of observers, m = γ∂t . Show that they are accelerated inward with acceleration a(m) = −γ 2 Ω2 r∂r ; that the vorticity vector is aligned with the zaxis, ω(m) = γ 2 Ω∂z ; and that the expansion vanishes identically, θ(m) = 0.
284
Exercises (c) In a rotating Minkowski spacetime, consider the ZAMO family of observers n = −dt or n = ∂t + Ω∂φ . Show that they are geodesic (a(n) = 0), vorticityfree (ω(n) = 0, by deﬁnition), and expansionfree (θ(n) = 0). (d) In a rotating Minkowski spacetime, consider the family of circularly rotating orbits U = Γ(∂t + ζ∂φ ). 1. Show that Γ−2 = 1 − r2 (ζ − Ω)2 . 2. Show that for null circular orbits ζ = ζ± = −Ω ± 1/r.
65. Consider a rotating Minkowski spacetime, ds2 = −γ −2 dt2 + 2r2 Ωdtdφ + r2 dφ2 + dr2 + dz 2 , where γ −2 = 1 − Ω2 r2 , and the family of static observers m = γ∂t . (a) Show that the projected metric onto LRSm is given by P (m) = dr ⊗ dr + γ 2 r2 dφ ⊗ dφ + dz ⊗ dz. (b) Show that the Ricci scalar associated with this 3metric is given by R = −6Ω2 γ 4 . (c) Consider the r − φ part of this 3metric, P (m)z=const. = dr ⊗ dr + γ 2 r2 dφ ⊗ dφ. 1. Show that the Ricci scalar associated with this 2metric is again given by R = −6Ω2 γ 4 . 2. Show that the embedding of this 2metric is completely Minkowskian (hRR = γ −6 < 1) and that the surface Z = Z(R) can be explicitly obtained in terms of elliptic functions (Bini, Carini, and Jantzen, 1997a; 1997b). 66. Using the purely imaginary 4form Eα1 ...α4 = iηα1 ...α4 , E α1 ...α4 = −iη α1 ...α4 instead of η for the duality operation on 2form index pairs deﬁnes the “hook” duality operation, with symbol ˘. For example, for a 2form F this leads to ˘ F = i∗F and hence to ˘ ˘ S = S. One can then ﬁnd eigen2forms of the operation ˘ with eigenvalues ±1, called selfdual (+) and antiselfdual (−), respectively.
Exercises
285
(a) Show that the complex quantity C+ αβ γδ = C αβ γδ + i∗ C αβ γδ , where C = Cαβγδ is the Weyl tensor, is antiselfdual, i.e. ˘ C+ αβ γδ = −C+ αβ γδ . (b) Show that the complex quantity C− αβ γδ = C αβ γδ − i∗ C αβ γδ is selfdual, i.e. ˘ C− αβ γδ = C− αβ γδ . Hint. Recall the property
∗∗
C = −C of the Weyl tensor.
67. Suppose that S is a tensorvalued pform, i.e. it has p antisymmetric indices (indices of a pform) in addition to other tensorial indices: S = S α... β...α1 ...αp = S α... β...[α1 ...αp ] . DS, the covariant exterior derivative of S, is deﬁned so that it acts as the ordinary covariant derivative on the tensorial indices and as the exterior derivative on the pform indices, that is DS α... β...α1 ...αp+1 = (p + 1)∇[α1 S α βα2 ...αp+1 ] . Show that the covariant exterior derivative of the curvature 2form Rα β =
1 α R βγδ ω γ ∧ ω δ 2
vanishes identically, i.e. DR = 0 or, in components, [DR]α βγδμ = ∇[μ Rα βγδ] = 0. Hint. The righthand side of the above equation is identically zero due to the Bianchi identities. 68. Show that Maxwell’s equations, ∇β ∗ F αβ = 0,
∇β F αβ = 4πJ α ,
where Fαβ = 2∇[α Aβ] is the Faraday 2form, Aα is the vector potential, and J α is the 4current vector, can be written in indexfree form as follows: dF = 0, with δJ = 0.
δF = −4πJ ,
286
Exercises
69. Consider KerrNewman spacetime in standard coordinates. The associated electromagnetic ﬁeld can be written as F =
Q 2 (r − a2 cos2 θ) dr ∧ [dt − a sin2 θdφ] Σ2 2Q + 2 ar sin θ cos θ dθ ∧ [(r2 + a2 )dφ − adt]. Σ
Introduce the ZAMO family of observers, with 4velocity ΔΣ n = −N dt, N= , A and the following observeradapted frame 1 Δ ∂r , erˆ = √ ∂r = grr Σ 1 1 eθˆ = √ ∂θ = √ ∂θ , gθθ Σ 1 1 Σ ∂φ . eφˆ = √ ∂φ = gφφ sin θ A (a) Show that the electric and magnetic ﬁelds as measured by ZAMOs are given by Q √ [(r2 + a2 )(r2 − a2 cos2 θ)erˆ Σ2 A √ − 2a2 r sin θ cos θ Δeθˆ], Q B(n) = − √ [2ar cos θ(r2 + a2 )erˆ 2 Σ A √ + a sin θ(r2 − a2 cos2 θ) Δeθˆ]. E(n) =
(b) Show that electromagnetic energy density as measured by ZAMOs is given by E(n) =
1 Q2 (r2 + a2 )2 + a2 Δ sin2 θ 1 [E(n)2 + B(n)2 ] = . 8π 8π Σ2 (r2 + a2 )2 − a2 Δ sin2 θ
Compute the limit of E(n) for r → r+ . 70. In KerrNewman spacetime, consider the Carter family of observers, Δ (−dt + a sin2 θdφ). u(car) = Σ Let sin θ u ¯(car) = √ [−adt + (r2 + a2 )dφ] Σ
Exercises
287
be the unit vector orthogonal to ucar in the t − φ plane. Show that the background electromagnetic ﬁeld can be expressed as 1 Q 2 2 2 F = 3/2 − √ (r − a cos θ) dr ∧ u(car) + 2ar cos θ dθ ∧ u ¯(car) . Σ Δ Using the splitting relation F = u(car) ∧ E(u(car) ) + ∗(u(car) ) B(u(car) ), show that E(u(car) ) and B(u(car) ) are parallel. 71. Consider the setting of the above problem. Introduce the following orthonormal frame adapted to Carter’s observers: eˆ0 = u(car) ,
eˆ1 = erˆ,
eˆ2 = eθˆ,
eˆ3 = u ¯(car) .
(a) Show that the energy momentum tensor of the ﬁeld is given by T = E(u(car) )[−eˆ0 ⊗ eˆ0 − eˆ1 ⊗ eˆ1 + eˆ2 ⊗ eˆ2 + eˆ3 ⊗ eˆ3 ], where E(u(car) ) is the energy density of the background electromagnetic ﬁeld as measured by Carter observers. (b) Show that T 2 = E(u(car) )2 [−eˆ0 ⊗ eˆ0 − eˆ1 ⊗ eˆ1 + eˆ2 ⊗ eˆ2 + eˆ3 ⊗ eˆ3 ], so that the mixed form is such that [T 2 ]μ ν = E(u(car) )2 δ μ ν . 72. In KerrNewman spacetime, consider the two electromagnetic invariants I1 =
1 Fαβ F αβ , 2
I2 =
1∗ Fαβ F αβ . 2
(a) Show their expressions in terms of the electric and magnetic ﬁelds as measured by ZAMOs, E(n) and B(n). (b) Evaluate them. Hint. A straightforward calculation leads to Q2 2 2 [4r a cos2 θ − (r2 − a2 cos2 θ)2 ], Σ4 Q2 I2 = 4 [4ra cos θ(r2 − a2 cos2 θ)]. Σ I1 =
73. Show that A = aB0 [∂t +
1 ∂φ ] 2a
is a vector potential for an electromagnetic test ﬁeld on Kerr background.
288
Exercises (a) Evaluate the electric and magnetic ﬁelds as measured by ZAMOs and make a plot of the lines of force of these ﬁelds. Hint. A straightforward calculation shows that E(n)rˆ = −aσ[(r2 + a2 )Y + rΔΣ sin2 θ], √ E(n)θˆ = −aσ Δ[X − Σ(r2 + a2 )] sin θ cos θ, B(n)rˆ = σ[(r2 + a2 )X − a2 ΔΣ sin2 θ] cos θ, √ B(n)θˆ = −σ Δ[a2 Y + rΣ(r2 + a2 )] sin θ, where B0 √ , A Y = 2r cos2 θ(r2 + a2 ) − (r − M)(1 + cos2 θ)(r2 − a2 cos2 θ), σ=
Σ2
X = (r2 + a2 )(r2 − a2 cos2 θ) + 2a2 r(r − M)(1 + cos2 θ). (b) Evaluate the invariants of this ﬁeld. Note that this electromagnetic ﬁeld was found by Wald in 1974 (Wald, 1974). 74. Consider Kerr spacetime. Show that the 2form ﬁeld f=
1 fμν dxμ ∧ dxν = a cos θ dr ∧ (dt − a sin2 θ dφ) 2 + r sin θ dθ ∧ [−a dt + (r2 + a2 ) dφ]
satisﬁes the relations ∇(γ fβ)α = 0. Note that the ﬁeld f with this property is known as a KillingYano tensor. 75. Consider the metric (Euclidean TaubNUT) with coordinates {ψ, r, θ, φ}: 2 2 [dr2 + r2 (dθ2 + sin2 θdφ2 )] ds = 1 + r 4 2 (dψ + cos θdφ)2 , + 1 + 2 /r where is a parameter which can be either positive or negative. (a) Solve the equations ∇(γ fβ)α = 0. The tensor ﬁelds f are KillingYano tensors. (b) Show that the above metric admits four KillingYano tensors.
Exercises
289
Hint. The KillingYano tensors are 2 dθ ∧ dφ, f1 = X ∧ dr − 1 + r 2 dφ ∧ dr, f2 = X ∧ dθ − 1 + r 2 dr ∧ dθ, f3 = X ∧ dφ − 1 + r r sin θdθ ∧ dφ, f4 = X ∧ dr + 4r(r + ) 1 + 2 where X = 4 (dψ + cos θdϕ). (c) Show that ∇α fiβ = 0 when i = 1, 2, 3. 76. Consider the ReissnerNordstr¨ om spacetime. Show that the equilibrium condition for a massive and charged particle to be at rest at a ﬁxed point r = b, θ = 0 on the zaxis is given by 1/2 Q2 + b 1 − 2M 2 r r . m = qQ Mb − Q2 What can one deduce from this relation? What happens when the black hole becomes extreme? 77. Consider the FriedmannRobertsonWalker spacetime, with metric dr2 2 2 2 2 + r (dθ + sin θdφ ) , ds2 = −dt2 + R2 (t) 1 − κr2 where R(t) is a scale function and κ = −1, 0, 1 according to the type of spatial sections: a pseudosphere, a ﬂat space, or a sphere, respectively. (a) Compute the ratio between area and volume of the spatial sections in the three cases κ = −1, 0, 1. (b) Discuss the geodesic deviation equation and evaluate the acceleration of the relative deviation vector in the three cases κ = −1, 0, 1. 78. Show that the two metrics ds2 = −dt2 + e2H0 t (dx2 + dy 2 + dz 2 ),
H02 = Λ/3
and −2 H2 ds2 = 1 + 0 (x2 + y 2 + z 2 − t2 ) (−dt2 + dx2 + dy 2 + dz 2 ) 4 are equivalent. Note that both these metrics represent de Sitter spacetime (see Section 5.4).
290
Exercises Hint. Introduce standard polar coordinates, x = ρ sin θ cos φ,
y = ρ sin θ sin φ,
z = ρ cos θ.
The ﬁrst metric thus takes the form ds2 = −dt2 + e2H0 t [dρ2 + ρ2 (dθ2 + sin2 θdφ2 )]. Applying the coordinate transformation 1 Re−H0 τ H0 τ 2 2 t= ln e , 1 − H0 R , ρ =
H0 1 − H02 R2
θ = θ,
φ = φ,
gives ds2 = −(1 − H02 R2 )dτ 2 +
dR2 + R2 (dθ2 + sin2 θdφ2 ). 1 − H02 R2
The further transformation 2 2 1 H0 ρ¯ − (H0 t¯ − 2)2 τ= , ln 2H0 H02 ρ¯2 − (H0 t¯ + 2)2 θ = θ,
ρ¯
R= 1+
H02 ρ2 4 (¯
− t¯2 )
,
φ = φ,
ﬁnally gives −2 H 2 2 ¯2 ρ −t ) [−dt¯2 + d¯ ρ2 + ρ¯2 (dθ2 + sin2 θdφ2 )], ds2 = 1 + 0 (¯ 4 which reduces to the second metric once the Cartesian coordinates are restored by using standard relations. By combining the two transformations we get t= ρ=
1 ln χ, 2H0 ρ¯ H02 ρ2 4 (¯
1+ θ = θ, φ = φ, where
H02 ρ¯2 − (H0 t¯ − 2)2 χ= H02 ρ¯2 − (H0 t¯ + 2)2
− t¯2 )
χ−1/2 ,
1−
H02 ρ¯2 [1 +
H02 ρ2 4 (¯
− t¯2 )]2
,
which allows us to pass directly from one metric to the other. 79. Show that the metric −1 r2 r2 ds2 = − 1 − 2 dt2 + 1 − 2 dr2 + r2 (dθ2 + sin2 θdφ2 ) α α
Exercises
291
represents de Sitter spacetime. Consider spatially circular orbits U = Γ(∂t + ζ∂φ ) and show that the magnitude of the 4acceleration is given by 2 2 2 2 1 2 4 2 + ζ sin θ + ζ sin θ cos θ r2 1 − αr 2 2 α . a(U ) = 2 2 1 − αr 2 − r2 ζ 2 sin2 θ 80. Consider G¨ odel spacetime, with the metric given by (2.133). Compute the vorticity of the congruence of timelike Killing vectors. 81. Consider the metric tensor of a general spacetime. Show that the inverse metric is also a tensor. 82. Consider the MathissonPapapetrou equations for spinning test bodies (see Chapter 10, Eqs. 10.2, 10.3, 10.4), 1 DP μ = − Rμ ναβ U ν S αβ , dτ 2 DS μν μ ν = P U − P νUμ , dτ S μν Pν = 0 , where D/dτ = ∇U and with P = mu (u timelike and unitary). (a) Deduce the evolution equations for the spin vector S α = η(u)αβγ Sβγ . (b) Show that the quadratic invariant s2 =
1 Sαβ S αβ 2
is constant along U , that is ∇U s = 0 . (c) Determine the evolution equation for the mass m along U . (d) Show that ν(U, u)α =
P (u)μα [∗ R∗ ]μνρσ S ν S ρ uσ . (1 + [∗ R∗ ]μνρσ uμ S ν uρ S σ )
Hint. See Tod, de Felice, and Calvani (1976) for details. 83. Consider the FriedmannRobertsonWalker spacetime with metric dr2 2 2 2 2 2 2 2 ds = −dt + R (t) + r (dθ + sin θdφ ) , 1 − κr2 where R(t) (which we leave as unspeciﬁed) is a scale function and κ = −1, 0, 1 is a sign indicator. Let U be the 4velocity of a massive particle moving along a radial geodesic and let its velocity, relative to a static observer, be ν0 at some initial time t = t0 .
292
Exercises √ (a) Show that U = cosh α∂t + sinh αerˆ, where erˆ = ( 1 − κr2 /R)∂r is the unit vector associated with the radial direction and sinh α = γ0 ν0 R0 /R, with γ0 = (1 − ν02 )−1/2 . (b) Show that the relative velocity as measured by another static observer at the generic value t of the coordinate time is ν ≡ tanh α =
ν0 γ0 R0 R2 + ν02 γ02 R02
.
(c) Discuss the limit ν0 → 0. 84. Consider the case κ = 0, R(t) = tn of the FriedmannRobertsonWalker metric of Exercise 83. (a) Show that Rα β =
n diag[3(n − 1), 3n − 1, 3n − 1, 3n − 1] . t2
(b) Discuss the two cases n = 1 and n = 1/3. 85. Consider the induced metric on the t = const hypersurfaces of a FriedmannRobertsonWalker spacetime which is the local rest space of the observers with 4velocity u = ∂t , namely dr ⊗ dr 2 2 2 + r dθ ⊗ dθ + r sin θdφ ⊗ dφ , γ = 1 − κr2 conformally rescaled by the factor R(t) (constant in this case). (a) Show that the Ricci tensor associated with this metric in its mixed form is Ra b = −2κδ a b . (b) Show that the threedimensional Cotton tensor y ab = −[Scurlu R]ab , where R is the Ricci tensor associated with the metric γ, for the three cases κ = −1, 0, 1 is always vanishing. (c) Show that the geodesics of this metric when θ = π/2 satisfy the following equations: 1 2 dr =± (r − L2 )(1 − κr2 ) , ds r
L dφ = 2, ds r
where s is the curvilinear abscissa parameter and L is a Killing constant. (d) Integrate the above equations. 86. Show that the equations of state p = (1/3)ρ and p = −ρ, p and ρ being the isotropic pressure and the energy density of a perfect ﬂuid, are Lorentz invariant.
Exercises
293
87. Consider the 1form X = Xα dxα ,
Xα = Xα (r, θ) ,
in the Kerr spacetime with metric written in standard BoyerLindquist −1 coordinates. Introduce the families of static observers
(m, m = M ∂t , √ M = −gtt ) and ZAMOs (n, n = −N dt, N = 1/ −g tt ). (a) Evaluate curlm X and curln X. (b) Compare these results in the special case X = Xφ (r, θ)dφ. 88. Consider the vector X in a generic spacetime. Let u = e0 be a family of test observers with associated spatial frame ea , so that X = X  u + X ⊥ ,
X⊥ · u = 0 .
(a) Evaluate the components of ∇X = (∇X)αβ ≡ (∇β Xα ) in terms of the kinematical ﬁelds associated with u. (b) Show that in the case X  = 0, i.e. X spatial with respect to u, one has ∇α X α = [∇(u)a + a(u)a ]X a . 89. Repeat Exercise 88 in the case of a 11 tensor S ≡ S α β . 90. Consider a threedimensional Riemannian space with metric ds2 = gab dxa dxb and a conformal transformation of the metric g˜ab = e2U gab . (a) Show that the unit volume 3form transforms as follows: η˜abc = e3U ηabc ,
η˜abc = e−3U η abc .
91. Vaidya’s metric for a radiating spherical source is given by 2M (u) du2 − 2dudr + r2 (dθ2 + sin2 θdφ2 ) , ds2 = − 1 − r where M (u) is a nonincreasing function of the retarded time u = t − r, often chosen in the form M 2 − M1 (1 + tanh αu) , M1 , M2 , α = constant. M (u) = M1 + 2 Consider the family of observers at rest with respect to the coordinates, i.e. with 4velocity 1 u ¯ = eˆ0 = √ ∂u −guu and associated spatial orthonormal triad √ 1 1 eˆ1 = −guu ∂r + ∂u , eˆ2 = √ ∂θ , guu gθθ
eˆ3 = √
1 ∂φ . gφφ
294
Exercises (a) Show that these observers are accelerated with acceleration a(¯ u) = −
M r2 − M (r − 2M ) eˆ1 , [r(r − 2M )]3/2
M =
dM . du
(b) Show that the congruence Cu¯ of the integral curves of u ¯ has a shear given by √ M r eˆ ⊗ eˆ1 . θ(¯ u) = (r − 2M )3/2 1 (c) Show that the vorticity of Cu¯ is identically zero, ω(¯ u) = 0. (d) Show that u)ˆ1 u ¯ − θ(¯ u)ˆ1ˆ1 eˆ1 , £u¯ eˆ1 = a(¯
£u¯ eˆ2 = 0,
£u¯ eˆ3 = 0.
√ Consider then the null vector k = 1/ 2∂r and show that Vaidya’s metric is a solution of the Einstein equations with Tμν = qkμ kν where 1 q = 2πr 2 dM/du. 92. The general relativistic motion of extended bodies with structure up to the quadrupole is described by Dixon’s model (Dixon, 1964; 1970a; 1970b; 1974): DP μ 1 1 = − Rμ ναβ U ν S αβ − J αβγδ Rαβγδ ; μ dτU 2 6 ≡ F (spin)μ + F (quad)μ , 4 DS μν = 2P [μ U ν] + J αβγ[μ Rν] γαβ dτU 3 ≡ 2P [μ U ν] + D(quad)μν , where P α is the generalized linear momentum of the body, U α is the unit tangent vector to the world line of the moment’s reduction, J αβγδ (a tensor with the same algebraic properties as the Riemann tensor, and hence with 20 independent components and support along U ) is the spacetime quadrupolar momentum tensor of the body, and S αβ (an antisymmetric tensor, also with support along U ) is the proper angular momentum tensor of the body. The completeness of the model is assured if one requires the further conditions S αβ Pβ = 0 and speciﬁes the type of body under consideration through the socalled constitutive equations. (a) Show that the following two expressions for the torque D(quad)μν , due to Dixon (1974) and EhlersRudolph (1977), are equivalent: (quad) μν
DDixon
4 = − R[μ αβγ J ν]αβγ , 3
(quad) μν
DE− R
=
4 αβγ[μ ν] J R γαβ . 3
Exercises
295
(b) Write down a 1 + 3 representation of the quadrupole momentum tensor with respect to an arbitrary observer family u, and identify the physical meaning of the various terms. 93. Starting from the deﬁnition of scalar expansion, Θ(u) = ∇α uα , show that the Raychaudhury equation, ∇u Θ(u) = ∇α a(u)α − Tr(k(u)2 ) − Rαβ uα uβ , holds identically. Consider then a full splitting of the terms ∇α a(u)α and show that it can be written as ∇α a(u)α = ∇(u)α a(u)α + a(u)α a(u)α . 94. In the Schwarzschild spacetime (with metric written in standard coordinates), consider the foliation T = t − f (r) , where f (r) is an arbitrary function of the radial variable only. (a) Show that the induced metric on each leaf is given by 1 2 (3) 2 2 2 2 2 2 2 , ds = γrr dr + r (dθ + sin θdφ ) , γrr = −N f N2 where N 2 = 1 − 2M/r. (b) Evaluate the threedimensional Cotton tensor y ab = −[Scurlu R]ab of this 3geometry, where Rab is the Ricci tensor and f is arbitrary. (c) Introduce the family N of observers whose world lines are orthogonal to the T = constant foliation and show that they are in radial motion with respect to the static observers n. Hint. Put N = −LdT and determine L from the normalization condition N · N = −1. (d) Show that the relative velocity satisﬁes the condition ν(N , n)rˆ = N 2 f . (e) Show that γrr =
1 N γ(N , n)
2 .
(f) Under which conditions is the previous 3metric is intrinsically ﬂat? Is it possible to forecast such a condition without studying the induced Riemann tensor components?
296
Exercises (g) Prove that the family N of observers with associated intrinsically ﬂat 3metric are geodesic and coincide with the Painlev´eGullstrand observers introduced in the text.
95. Consider Minkowski ﬂat spacetime in Cartesian coordinates. (a) Write the 4velocity of an observer U moving with constant speed ν along the positive direction of the xaxis until he reaches the distance (at the coordinate time t = t∗ ) from the origin (which he left at t = 0) and then suddenly reversing his direction to return to the origin with the same speed, during another interval t∗ of coordinate time. (b) Show that the associated 4acceleration is a(U ) = −2νγ 2 δ(t − t∗ )∂x , where δ denotes the Dirac δfunction and γ is the Lorentz factor. 96. Consider Rindler’s metric, ds2 = −(1 − gz)dt2 + dx2 + dy 2 + dz 2 , where g is a constant, and the family of observers at rest with respect to the coordinates, U = (1 − gz)−1 ∂t . (a) Show that a(U ) = −g/(1 − gz)∂z . (b) Write down the geodesic equations and ﬁnd the corresponding solution for both the timelike and null cases when the motion is conﬁned in the (t, z)plane. (c) Find a coordinate transformation mapping Rindler’s metric in Minkowski spacetime with Cartesian coordinates and show that Rindler’s geodesics are mapped into straight lines, as expected. 97. In Kerr spacetime in standard BoyerLindquist coordinates, consider the family of timelike “spherical orbits” parametrized by their coordinate angular velocities ζ = dφ/dt and η = dθ/dt, U α = Γ[δ α t + ζδ α φ + ηδ α θ ] , where Γ = dt/dτU > 0 is deﬁned by Γ−2 = −[gtt + 2ζgtφ + ζ 2 gφφ + η 2 gθθ ] because of the condition Uα U α = −1. Introduce also the family of static observers, m = (−gtt )−1/2 ∂t . (a) Evaluate the relative velocity ν(U, m). (b) Evaluate the 4acceleration a(U ) of these orbits.
Exercises
297
(c) Evaluate the FermiWalker gravitational force (G)
F(fw,U,m) = −Dm/dτU . (d) Study the evolution of ν(U, m) along U and determine the relative acceleration a(fw,U,m) . (e) Determine the expressions for ζ, η, and Γ corresponding to the U(geo) (G)
geodesic and evaluate F(fw,U(geo) ,m) . 98. The general stationary cylindrically symmetric vacuum solution (see Iyer and Vishveshwara, 1993) can be written in the form ds2 = λ00 dt2 + 2λ03 dtdφ + λ33 dφ2 + e2φ (dτ 2 + dσ 2 ), where coordinates are such that x0 = t, x1 = τ , x2 = σ, x3 = φ, and λα = Aα τ 1+b + Bα τ 1−b , α = 00, 03, 33, √ √ 2 e2φ = cτ (b −1)/2 , τ = 2ρ , σ = 2z , b and c being constants. The coeﬃcients Aα and Bα are related by the algebraic conditions A00 A33 − A203 = 0 , 2 = 0, B00 B33 − B03
1 A00 B33 + A33 B00 − 2A03 B03 = − , 2 and to the mass per unit length m and angular momentum per unit length j by 1 1 + b(A33 B00 − A00 B33 ) , 4 2 1 j = b(A03 B33 − A33 B03 ) . 2
m=
Note that τ here is a radial coordinate while σ is the z coordinate (rescaled by a factor). Consider a timelike circular orbit with 4velocity U = Γ(∂t + ζ∂φ ) , with ζ constant along U , £U ζ = 0. (a) Show that the normalization factor is given by Γ = [−(λ00 + 2λ03 ζ + λ33 ζ 2 )]−1/2 . (b) Show that these orbits are accelerated with purely radial acceleration 1 a(U ) = − e−2φ Γ2 (λ˙ 00 + 2λ˙ 03 ζ + λ˙ 33 ζ 2 )∂τ , 2 where a dot denotes diﬀerentiation with respect to τ .
298
Exercises (c) Show that timelike circular geodesics correspond to 1−b2 λ˙ 03 2 ζK± = − ± . ˙λ33 ˙λ33 (d) Show that geodesic meeting point observers have angular velocity ζ(gmp) = −
λ˙ 03 . λ˙ 33
99. In the context of Exercise 98, show that extremely accelerated observers ζ(crit)± have angular velocities which are solutions of the following equation: λ33 λ33 d λ33 2 2 d 2 2 d λ03 ζ + λ00 ζ + λ00 = 0. dτ λ03 dτ λ00 dτ λ00 Show that extremely accelerated observers have constant angular velocities. 100. In the context of Exercise 98, (a) study the kinematical properties of static observers, with ζ(stat) = 0 . (d) study the kinematical properties of zero angular momentum observers, with λ03 ζ(ZAMO) = − . λ33
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Index
Acceleration, 22, 42, 102, 104, 108, 156, 157, 170, 187, 238 centrifugal, 66 centrifugal stretch, 189 centripetal, 65, 104 centripetal FermiWalker, 56, 62 extreme, 207, 250 relative, 126 relative, equation for, 136–138, 186 spatial, 201 tangential FermiWalker, 56 Active galactic nuclei, 212 Aﬃne parameter, 22, 77, 81, 92 Ambiguity, 202, 205 Angular momentum azimuthal, 201 Antisymmetrization, 12, 15 of four indices, 37 Apparent horizon, 153 Arc length parameterization, 106 Basis, 11 BelRobinson tensor, 132 Bianchi identities, 19 Binormal of a world line, 64, 66 Black hole, 135, 186, 207, 212 Kerr, 136 rotating, 165 Schwarzschild, 136 Boost maps, 51, 66 Born rigidity, 48, 130, 138, 156, 158, 173, 200 Boundary conditions, 95 BoyerLindquist coordinates, 32, 166, 199 Carter observers, 176 Christoﬀel symbols, 18 Circular orbits, 157, 172 equatorial, 163, 192 Clock rates, 6 synchronization, 3 Clock eﬀect in Kerr spacetime, 246 in Schwarzschild spacetime, 246 Commutation brackets, 12, 153
Components of the connector, 76, 80 Concatenation of curves, 75 Congruence of curves, 1 Connecting tensor, 51 vector, 77, 137, 186 Connection, 17 coeﬃcients, 60, 76 compatible with the metric, 17 Connector, 75, 144 consistency, 75 diﬀerentiability, 75 linearity, 75 Constant of motion, 192, 218 Contraction left, 13 right, 13 Coordinate components, 226 Coordinates ingoing, 153 null, 153 outgoing, 153 Coriolis force, 196 Cosmological constant, 28 Cotton tensor, 20 Curvature invariants, 155, 167 scalar, 19 singularity, 155, 167 Curve relatively ﬂat, 65, 107 relatively straight, 65, 107, 181 de Rham Laplacian, 26, 27 Derivative absolute, 21, 38, 53, 75 covariant, 17, 18, 27, 38 exterior, 23, 24 FermiWalker, 22, 38, 53, 63, 64, 66 Lie, 23, 27, 38, 53, 75 of a connecting vector, 79 of the world function, 77 spatial absolute, 39 spatial covariant, 39 spatial exterior, 39
306 spatial FermiWalker, 39, 54 spatial Lie, 38 spatialFermiWalker temporal, 45 spatialLie temporal, 38, 45, 164 Deviation matrix, 187 Directional singularity, 167 Divergence operator, 25 Doppler velocity, 100, 143 Droste, 31 EddingtonFinkelstein coordinates, 153 Eﬀective radius, 159 Eikonal, 100 Einstein tensor, 20 Einstein’s equations, 28, 129 exact solutions, 29 in Maxwelllike form, 131 Electric and magnetic ﬁelds, 113 Electric part of a pform, 36 Electromagnetic 1form, 26 Electromagnetic power, 112 Embedding diagrams, 182 Energy density, 120 current, 117 of an electromagnetic ﬁeld, 113, 114 Energy equation, 119 Energy of a particle, 99 Energymomentum tensor, 113, 132 of a ﬂuid, 116 of a perfect ﬂuid, 117 of an electromagnetic ﬁeld, 28, 113 of dust, 118 of matter energy, 28 tracefree part, 117 with thermal stresses, 123 Ergosphere, 169 Event horizon, 153, 166 Expansion, 41, 238 tensor, 111, 137, 156, 187 Exterior product, 16 Faraday, 115 Faraday 2form, 26, 28, 33 Fermi coeﬃcients, 227, 256 coordinates, 80, 142, 274 frame, 151, 225, 226 metric, 81 rotation, 188, 201 FermiWalker angular velocity vector, 62, 160, 170 curvature of a world line, 65, 164 gravitational force, 55, 174, 233 relative ﬂatness, 107
Index relative torsion, 65 strain tensor, 139, 188, 201 structure functions, 43, 45, 62, 157 torsion of a world line, 65 transport, 22, 63, 235 Fluids, 116 dust, 118 dynamics, 118, 120 evolution equations, 118 ordinary, 118, 120 perfect, 117, 122 viscous, 124 Force centrifugal, 106, 198, 244 gravitational, 107, 198, 235 in a ﬂuid, 118, 121 orthogonal splitting , 111 tidal, 136 Force equation, 103, 105, 106 Frame adapted to an observer, 2, 41, 61, 82 complex null (NewmanPenrose), 69 evolution of frame vectors, 42 inertial, 3, 60 null, 69 orthonormal, 59, 60, 238 phaselocked, 162 real null, 70 spatial, 2 structure functions, 12, 42 Sunlocked, 230 FrenetSerret frame, 104, 138, 157, 162, 175, 257 absolute, 63 angular velocity, 63, 65, 258 comoving, 65, 69, 105–107, 182 curvature, 163 relative, 64 torsions, 163 FrenetSerret procedure, 64 Frequency, 100 of a photon, 99 reduced, 201 shift, 80, 100, 194, 214 Geodesic, 22, 77, 192 completeness, 154 in gravitational wave metric, 185 meeting point, 178, 216 null, 219 spatially circular, 178 Geodesic deviation, 79, 135 generalized equation, 141 G¨ odel solution, 84 GPS, 88, 89, 94
Index Gravitational compass, 135 drag, 217 grip, 207 polarimeter, 255 strength, 198 Gravitational waves, 87, 184, 255 polarization, 87 Gravitoelectric ﬁeld, 115 Gravitoelectromagnetism, 116 Gravitomagnetic ﬁeld, 116 Gyroscope, 234 on a geodesic, 255 Gyroscope precession, 208 backward, 207 forward, 207 induced by gravitational waves, 255
HamiltonJacobi, 218 Harmonic motion, 192, 203 Heaviside function, 183 Hodge duality, 16, 37 Hubble constant, 82 Hydrodynamics, 123, 124
Inertial forces, 11, 55 frame, 59 Integrated spectrum, 213 Inverse boost, 51, 53 Inward direction, 194 Isometries, 27
Jacobi identities, 44
Keplerian angular velocity, 158 relative velocity, 164 Kerr metric inverse metric, 166 weak ﬁeld limit, 210 Killing equations, 27 horizon, 153 tensor, 28 vectors, 27, 80, 84, 86, 166 KillingYano tensor, 288 Kinematical tensor, 41, 53, 108 Kretschmann, 155 Kronecker delta, 13 generalized, 15 Kruskal coordinates, 154
307
Laboratory as a congruence of curves, 1 spacetime history, 1 Lapse, 169 Length parameterization, 57, 65 LenseThirring, 239 spacetime, 209 Levi Civita symbol, 14 Lie brackets, 12, 153 structure functions, 157 transport, 23 Light speed in vacuum, 3 Line of sight, 99 Linearity of coordinate transformations, 76 Local rest space, 49 Lorentz factor, 49, 88, 159, 172, 216, 230 force, 113 gauge, 27 invariants, 2 transformations, 2, 8, 9, 33, 59, 60, 113, 120 Lowering of indices, 14
Magnetic part of a pform, 36 Maps between LRSs, 49 Mashhoon, 244 Mass density, 120 Maxwell equations, 27, 115 Measurement ambiguity, 2 domain, 1, 91, 99, 126 local, 1 nonlocal, 1, 126 of angles, 95, 149 of curvature, 126 of spatial intervals, 74 of time intervals, 74 Metric tensor, 42 Minkowski, 60 Mixed projection maps, 49 Momentum, 103 4vector, 99, 104 density of a ﬂuid, 117 Motion in a pipe, 191
Normal neighborhood, 77, 91 of a world line, 64, 66
308 Notation threedimensional, 39 Null orbits, 177, 218 Obsculating circle, 202 Observer, 2 extremely accelerated, 180 inertial, 2 static, 3, 5, 156, 169 transformations, 108 zeroangularmomentum (ZAMO), 170 Orbit relatively ﬂat, 65 relatively straight, 65, 181 Orientation of a frame, 14 Oriented manifold, 14 Orthogonal decomposition, 34 Painlev´ eGullstrand coordinates, 154, 166 Papapetrou equations, 232 Papapetrou ﬁeld, 27 Parallel transport, 21, 100, 178 PenroseDebever equation, 28 Petrov classiﬁcation, 28, 33, 134, 155, 168 Power equation, 104 Poynting vector, 113, 114 Precession backwards, 198 forward, 198 in a weak gravitational ﬁeld, 237 in Kerr spacetime, 247 Pressure, 113 Principal null directions, 155, 168, 177 Projected absolute derivative, 55 Projection operators, 34 Proper mass of a ﬂuid, 119 Proper time, 2 Raising of indices, 14 Raytracing, 218 Recession relative motion of, 4 Reduced frequency, 203, 213 Relative boost, 49 Relative velocity, 48, 56 Relativity factor, 3, 6 of time, 2, 3 principle, 3 special theory, 2 Ricci identities, 19, 47, 130 Ricci rotation coeﬃcients, 60, 238 Ricci scalar, 29 Ricci tensor, 19, 29, 127
Index Riemann tensor, 18, 126, 143, 167 electric part, 126, 131, 136, 157, 188 FermiWalker spatial tensor, 47 force equation, 241 frame components, 46 magnetic part, 126, 131, 141 spatial symmetryobeying tensor, 47 symmetries, 19
Sagnac eﬀect, 215 Satellite attitude, 226 Scalar curvature, 127 Schiﬀ formula, 239, 248 Schwarzschild coordinates, 152 Scurl operation, 40, 128 Separation constant, 218 Shear, 41, 117 Shiﬀ formula, 237, 241 Shift, 169 Simultaneous event, 93, 95, 96, 143 Space distances, 91 Spaceship, 194 Spacetime, 11 Spacetime solutions antide Sitter, 30 de Sitter, 30, 82 FriedmannRobertsonWalker, 30 G¨ odel, 30, 84, 85 Gravitational plane wave, 33 Kasner, 30 Kerr, 32, 152, 165, 199, 218 KerrNewman, 32 maximally symmetric, 29 Minkowski, 30, 80, 88 ReissnerNordstr¨ om, 31 Schwarzschild, 31, 152, 154, 156, 157, 186 Spacetime splitting, 34 Spatial cross product, 39 curl, 40 curvature angular velocity, 160 distance, 93, 95 divergence, 40 divergence of spatial tensors, 40 duality operation, 37 gradient, 40 inner product, 39 momentum of a photon, 99 orientation, 194 velocity, 96 Spin vector boosted, 234 FermiWalker transport, 235 projected, 232
Index Splitting longitudinaltransverse, 104 of a 4momentum, 99 of a form, 36 of a tensor, 36 of a vector, 35 of a volume 4form, 37 of diﬀerential operators, 38 Strains, 126, 135, 139, 191 Stress in a ﬂuid, 121 transformation law, 121 Stress tensor, 117, 118 Structure functions, 61 Structure functions in terms of kinematical quantities, 44 Superenergy density, 134 SuperPoynting vector, 134 Supplementary conditions, 245 Symmetrization, 15 Szekeres, 135 Tensor ﬁeld metric, 14 nondegenerate, 13 Tensor product, 12 Tetrad, 59 phaselocked, 189 Thermoconduction, 117, 124 Thermodynamics ﬁrst law, 119 Thomas precession, 235 Threepoint tensor, 147 Thrust, 207
speciﬁc, 209 Tidal strains, 201 Time delay, 80, 215 Time interval, 91, 95 Time parameterization, 54–57, 61, 233 Torsion tensor, 18 Torsions of a world line, 63 Trace of a tensor, 13 Tracefree tensor, 13 Transport law, 61 Twist tensor, 139, 188 Twopoint function, 74, 75 Velocity composition of, 97, 98 relative, equation for, 138 spatial, 96, 144 Velocity parameter, 159 Volume 4form, 14, 34 Volume (bulk) viscosity, 117 Volumetric expansion, 41 Vorticity tensor, 41, 156, 158, 163, 188, 238 vector, 41, 109, 137, 170, 174 Weyl tensor, 20, 28, 127, 130–132, 167 principal null directions, 28 World function, 74, 75, 92, 145 derivatives, 77, 93 in de Sitter, 82 in G¨ odel, 84 of a gravitational wave, 87 Xray binaries, 212
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