C. Cattaneo ( E d.)
Relativistic Fluid Dynamics Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 7-16, 1970
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11097-9 e-ISBN: 978-3-642-11099-3 DOI:10.1007/978-3-642-11099-3 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1971 With kind permission of C.I.M.E.
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CENTRO INTERNAZIONALE MATEMATICO ESTlVO (C. I. M. E ) 1 Ciclo - Bressanone
dal 7 al 16 Giugno
1970
"RELATIVISTIC FLUID DYNAMIC'S" Coordinatore:
Pro,
C
CATTANEO
PHAM MAU QUAN
Problems mathematiques en hydrodynamique relativiste. Pag.
A. LICHNEROWICZ
Ondes des choc, ondes infinitesimales et rayons en hydrodynamique et magnetohydrodynamique relati vi stes.
A. H. TAUB
J. EHLERS K
B. MARATI-IE
G. BOILLAT
II
87
Variational principles in general relativity.
"
205
General relativistic kinetic theory of gases
"
30 I
Abstract minkowski spaces as fibre bundles.
"
389
Sur la propagation de la chaleur en I'elativite
II
405
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I ME.)
PROBLEMS MATHEMATIQUES
EN HYDRODYNAMIQUE RELATIVISTE
PHAM MAU QUAN
Corso tenuto a
Bressanone dal
7 al
16 Giugno
1970
Chapitre 1 LES SCHEMAS FLUIDES EN HYDRODYNAMIQUE RELATIVISTE
;1. GENERALITES SUR LA DYNAMIQUE RELATIVISTE DES FLUIDES
1. Le cadre geometrique. La mecanique relativiste des fluides a pour cadre geometrique l'espacetemps qui est une variete differentiable V de dimension 4, de classe C"', sur larquelle est donnee une structure pseudo-riemannienne g de signature +~--.
La geometrie de l'espace-temps (V, g) est celle de la connexion rie-
mannienne canoniquement associee
a
g.
La metrique definie par g est dite de type hyperbolique normal. Elle induit sur l'espace vectoriel tangent
Tx (V)
en chaque point x de V une
structure d'espace-temps plat de Minkowski. En coordon~es locales (xd.) on a
(1. 1)
Le tenseur
g,,~
( «, ~ = 0,
1, 2, 3).
dit tenseur fondamental de gravitation est assujetti
a veri
fier un systeme d'equations aux derivees partielles du second ordre qui
gen~
ralise Ies equations de Laplace.-Poisson et qui donne naissance aux condi tions de conservation. Ces equations sont Ies dix equations d'Einstein
(1.2)
ou
St(~
ne depend que de la structure riemannienne g de I'espace-temps,
TC(, est de signification purement mecanique et X un facteur constant.
-4Pham Le tenseur TIf dit tenseur d'impulsion-energie du fluide doit decrire au mieux la distribution energetique dans l'espace-temps. Le tenseur
S.,
est astreint aux deux conditions suivantes: 1.
S.,
ne dependent que de
g.~,
de leurs derivees des deux pre -
miers ordres, sont lineaires par rapport aux cterivees du second ordre 2.
SClit est conservatif, c 'est-a.-dire tel que
(1.3) On demontre (~) qu 'on a necessairement
Oll
Rca, est la courbure de Ricci, R la courbure scalaire de (V, gl, h et k
deux constantes arbitraires. k est la constante cosmologique, ne joue pas de rOle dans la description des fluides. On peut supposer k = O. En supprimant d'autre part Ie facteur surabondant h,
on prendra pour premier membre des
equations d 'Einstein
(1. 4)
S.,
sera dit tenseur d 'Einstein. Le tenseur d'Einstein Sd, etant conservatif, il en est de
m~me
du tenseur
d'impulsion-energie Til, . Les equations
(4)
E. CARTAN. - J. Math. pures et appliquees, 1. p. 141-203. (1922)
- 5-
Pharn
expriment alors la conservation de l'impulsion-energie et definiront l'evolution du fluide. 2. Le tenseur d'impulsion-energie. Dans toute theorie relativiste des fluides, Ie premier pas consiste choisir l'expression du tenseur d'impulsion-energie sion de
T.,~
T.~
a
. Chaque expres-
definit un schema de fluide. Til, doit @tre symetrique si lIon
veut satisfaire aux equations d'Einstein. Mais pour que T.c, puisse decrire un fluide physique, il faut qu'il existe un champ de vecteurs unitaires
u·
orhmtes dans Ie temps
(2.1)
pour lequelle scalaire
T., if'u'
soit positif. u· est dit vecteur vitesse un.!.
taire du fluide et ses trajectoires definissent les lignes de courant. En fait les fluides reels sont doues de proprietes diverses. Les forces de liaisons internes qui jouent un rOle fondamental dans I 'etude dynamique se traduisent par Ie tenseur des pressions propres. Les phenomenes calorifiques introduisent un scalaire
9 dit champ de temperature propre. Les
proprietes electromagnetiques sont susceptihles d'@tre representees par deux champs de tenseurs antisymetriques
H., ,G.,
comme on Ie verra.
n convient d'autre part d'etudier l'evolution thermodynamique du fluide. Ces diverses proprietes peuvent @tre envisagees dans une decomposition geometrique du tenseur d'impulsion-energie. On est aussi conduit it mettre T.c,
(2.2)
sous la forme
- 6-
Pham ou f
est un scalaire positif representant la densite propre de matiere-
energie ponderable,
1(.~
les pressions propres,
Q.~
les echllIlges
the~
miques par conduction et f ll, Ie tenseur d'energie electromagnetique. Si lIon neglige certaines proprietes, les termes correspondants ne figurent pas dans la decomposition. De m(!me on peut introduire de nouveaux termes pour etudier de nouvelles proprietes. A chaque expression de
ToI~'
correspond alors un schema fluide.
Dans chaque cas, l'evolution du fluide sera defini par les equations de conservation (1. 5) qui, tenant compte du caractere unitaire de uGl, con duisent aux equations sliivantes
(2. 3)
(2.4) (2.3) est dite l'equationaie continuite et (2.4) constitue Ie systeme differentiel aux lignes de courant. A ces equations on adjoindra eventuellement d'autres equations telles que les equations thermodynamiques, les equations du champ electromagnetique. On obtiendra de cette maniere Ie systeme fondamental des equations du schema considere. Ainsi Ie schema fluide pur a fait l'objet de nombreuses etudes divenues c1assiques, en particulier celles de L. P. Eisenhart et de A. Lichnerowicz. Le schema fluide thermodynamique a He Hudiee par C. Eckart et par l'auteur dans sa these en 1954. Le schema fluide champ electromagnetique a fait l'objet des travaux de A. Lichnerowicz, de ceux de l'auteur datant de 1955 qui ont suscite depuis de nombreux travaux, notamment ceux de G. Pichon..
n a conduit dans un cas particulier a la magnetohydrodynamique r~
- 7-
Pham lativiste dont l'etude a fait l'objet de tres beaux travaux de Y. ChoquetBruhat, A. Lichnerowicz. C 'est 1'etude mathematique de quelques uns de ces schemas qui consti tue Ie sujet de ces conferences.
3. Repere propre. On appelle repere propre en un point x de I 'espace-temps (V 4' g) un pere orthonorme
(V~ )
r~
dont Ie premier vecteur V.' coincide avec Ie vecteur
vitesse unitaire u et dont les trois autres vecteurs Vi! definissent I 'espa-
a la direction de temps
ce associe
u.
On peut rapporter I 'espace-temps dans Ie voisinage de tout point
a
un
champ de reperes propres qu 'on supposera differentiable (mais non necessairement integrable). La metrique d 'univers prend alors la forme canonique r !' _(I) sea)
(3.1)
les u>
~I
sont les I-formes duales des champs de vecteurs VJ: i. e. tel,: ~ les que W , V"., >= $ /,-', \,.', etant Ie symbole de Kronecker egal a I , 'It' si =p.' et 0 si 4 p.'. Les lA.I constituent donc quatre formes de Oll
~
4
qui donne Ia variation de I'integrale
(3. 6)
pour des extremites non fixes. Si Ies variations sont a extremites fixes,
bxo
=
bx~
0, il vient
P
0, c 'est-a-di
(3 7\
Pour que
S
est extremum, il faut et il suffit que
re
equat ions formellement identiques a (1. 1). D 'ou Theoreme. - Dans tout mouvement d 'un fluide parfait isentropique, Ies lignes de courant sont Iocalement des Iignes orientees dans Ie temps extremales de I'integrale
(3.6)
pour des variations a extremi-
tes fixes. Introduisons la met rique conforme
(3.8)
g
f2 g . On a
~
60 -
Pham Vis
a vis
de cette metrique, l'arc de courbe est defini par
ds = fds,
de sorte que les lignes de courant sont definies comme extl'e nales de
(3.9)
Ces extre::nales sont des geodesiques de
(V 4'
g ).
En posant
(3.10)
on voit que
ell
= fUll et
-II -1 fJl C = f u , de sorte que
~,
-11-' = 1
C C
et ces geodesiques ont pour equations -II
(3.11)
C
-
VA C~
= 0
Corollaire. - Lesllignes de courant du fluide parfait isentropique sont geodesiques orientees dans Ie temps de
(V, g).
4. L'invariant integral de l'hydrodynamique. Considerons un mouvement du fluide, defini par example par un bleme de Cauchy. Soit "
ro
un tube de courant s'appuyant sur un cycle
de dimension trace sur l'hypersurface initiale
lignes de courant) et soit
G
un cycle trace sur
Chaque ligne de courant de ~ est limite en Nous pouvons appliquer la formule courant
,
crit Ie cycle
(4.2)
(3.5)
P = 0 , la variation totale de
r. ' il vient
pr~
Xo
~
1:
(non tangente aux
~, homotope
ro
a chacune
et
x4 ~
r.
a roo
de ces lignes de
S est nulle quand
Xo
de-
- 61 -
Pham La 1-forme
(/J
a pour expression:
(4.2)
la propriete
(4. I)
se traduit par l'enonce suivant qui generalise un
theoreme classique sur la conservation de la circulation. Theoreme. - Etant donne un cycle
r
a une dimension non tangent
aux lignes de courant, la circulation du vecteur courant de
r
reste invariante quand
defini par Si
r
Ccc
Ie long
r se deforme sur Ie tube de courant
D est une variete differentiable
a
2 dimensions,
a bord
DD,
la formule de Stokes donne
(4.3)
L'integrale de la forme
n sur la sous-variete
dW
D se conserve quand elle se deforme de manie-
re que chaque point reste sur la
m~me
Dans Ie langage de H. Poincare,
12
ligne de courant. definit un invariant integral
pour Ie systeme differentiel aux lignes de courant Gl
dx dS -
u·
et fU definit un invariant integral relatif. La 2-forme
.n
joue un rOle
fondamental dans la description du mouvement. Elle admet l'expression
- 62 -
Pham locale
(4.4)
TMoreme. - La 2-forme nest une forme invariante pour Ie systeme differentiel aux lignes de courant
~cll
(4. 5)
£c
i. e.
= 0
est la derivee de Lie suivant
C.
En effet on a en utilisant l'identite du calcul des variations
comme
n
dW,
dn
0, il reste
die.n
or
C"n., En introduisant la connexion riemannienne associee forme
Comme
g,
a la metrique con-
on obtient
C(I.
-111- -
est unitaire, C V~ CC(
champ geodesiique d'apres La 2-forme
12
= 0 et d' autre part
(3. 11) , on a bien
est un
i/l. = o.
est une forme invariante par Ie systeme differentiel
aux lignes de courant. Nous allons rechercher tous les systemes diffe
- r3 -
Pham rentiels qui la laissent invariante. Il nous faut determiner tms les
X
champs de vecteur
L' existence de
X
tels que
depend du rang du systeme precedent, comme
!l.~ est antisymetrique, il vient :
1. - si
n
est de rang 2 , en chaqu'? point
teristiques forment un 2-plan
x
les vecteurs carac
'It'll:' Le champ de 2-plan 1T
admet des varietes integrales de dimension
2
engendries
par les lignes de courant. 2. - si
11
est de rang
O,.n
=
O. Comme.n
I-forme ferme, il existe une fonction Par suite
Car =
orthogonales
a
~,
f
=
da>,
telle que
(J)
est une
W = d
+
: les lignes de courant sont trajectoires
la famille d 'hypersurfaces
~ = cont.
Ces resultats sont importants pour I 'etude des mouvements rotation nels et irrotationnels du fluide.
tt,l\IOUVEMENTS ROTATIONNELSET IRROTATIONNEI,s
5. Tenseur tourbillon et equations de Helmholtz. Definition. - On appelle tenseur tourbillon Ie tenseur a llisymetrique d'ordre 2 defini par la 2-forme invariante [l Il constitue la veritable extension relativiste du rotationnel des vitesses introduit en mecanique classique. Si I 'on se rappelle I 'expres-2 -1-2 sion de f = 1 + te + pr ,e ou e est la vitesse de la lu -
- 64 Pham miere, on voit que
CCII
.0•• = 0« C~ -,,~ Cal
= fUg( differe de o
.a«~ = ~OI
differe de
-2
en
uCl(
par des termes en
u. -U,
UO(
e-2
par des termes
C
TMoreme. - Le tenseur tombillon satisfait aux equations de Helmholtz
(5. 1)
E;n effet un caIcul simple montre que ces equations sont une consequence de I 'equation
1/2
=
0
qui exprime que
invariante. On fait Ie caIcul en metrique initiale
.n
n.
est une forme
(V, g).
Definition. - On dit qu 'un mouvement du fluide est rotationnel
f
0 et irrJtationnel si
si
!l = o.
Theoreme. - Pour qu 'un mouvement du fluide parfait isentropique soit irrotation:aH, il faut et il suffit que les lignes de courant soient orthogonales
a une
m~me
hypersurface (locale).
Z une hypersurface orientee dans I 'espace telle que I . On peut choisir des coordonnees locales telles que
En effet soit
ntC~ = 0 sur
!.
x~
soit representee par
et que Ies lignes de courant par
= const.. (coordonnees de Gauss). Les equations de Helmholtz mon
trent alors que de
x O= 0
10.0 11 , = O.
11 en resulte que
l1Gl , = 0
au voisinage
1. .
6. Vecteur tourbillon. On suppose que Ie mouvement est irrotationnel. Au point x 6 tudions Ie 2-plan tels que
(6. 1)
ltlt
ferme des vecteurs
x
oC
caracteristiques
'Ill
e-
i. e.
-65Pham On aurait deja dans TI'jt Ie vecteur passant par
x
ull tangent a la ligne de courant
. Pour achever de determiner
liz ,
de rechercher un second vecteur non colineaire a
9
sissons un tel vecteur
il nous suffit
uc(. Nous choi -
orthogonal au premier. Ce vecteur est
defini par les equations
o
(6. 2)
Le vecteur
9'
n'est defini qu'a un facteur pres, on a par un cal-
cuI algebrique
(6. 3)
ou ~Il'r"
est la forme element
On remarque que
e
a=
0
Definition. - Au vecteur
de volume riemannien de
nIl' = o.
entraine
Gel
(V, g).
defini par
(6.3)
on donne Ie nom de
vecteur tourbillon, a ses trajectoires Ie nom de lignes de tourbillon. D 'apres la definition les lignes de tourbillon sont orthogonales aux lignes de courant. D I autre part Ie systeme differentiel aux lignes de tourbillon dx Cl = dt admet la
2-forme.n.
e'
comme forme invariante. On en deduit imme-
diatement les proprietes suivantes: Theoreme. - Etant donne un cycle
r
a une dimension non tangent
aux lignes de tourbillon, la circulation du vecteur tourbillon Ie long de
r reste invariante quand on deforme r sur Ie tube de tourbillons
-66-
Pham
r,
defini par Soit topes soit
"
un tube de lignes de courant,
sur "
e et
r
et
r
,
deux cycles
hom~
' Chacun de ces cycles definit un tube de tourbillons,
6)',
Soit
r4
un cycle sur
Ie tube de courant passant par suivant un cycle
r j'
homol 'Ope
e homoiope
a
r , Alors,
~ coupe Ie tube de tourbillons
a
r' ,
Comme OJ
®
est un invariant
integral relatif pour les lignes de courant et aussi pour les lignes de tourbillons, on
3.
Cette propriete constitue la generalisation relativiste d 'un theoreme de Helmholtz en dynamique classique, Enfin Ie champ de 2-plans racteristique de la forme
x-+ 1l':Jt
defini par Ie systeme ca-
fl
est un champ completement integrable, Aux varietes integrales dimensions
WI
a deux
on donne Ie nom de varietes caracteristiques de.£l ,
Ces varietes peuvent Nre engendrees par dE'S lignes de courant et par des lignes de tourbillon qui sont orthogonales sur
Wt
' Si donc on
mene les lignes de courant passant par les points d 'un ligne de tourbillon, les trajectoires orthogonales de ces lignes de courant sur
W,
sont lignes de tOllrbillon, Cela veut dire que si une ligne fluide est de I)
'rbillon
a un instant, elle reste de tourbillon a tout instant,
- 67 -
Pham • ~ • MOUVEMENTS PERMANENTS
7, :Espaee-temps stationnaire, On dit qu 'un espace-temps groupe connexe
d'isometri~s
variant aucun point de
V4'
(V 4' g) globales
est stationnaire s'il existe un
a un
a trajectoires
parametre ne laissant inorientees dans Ie temps
et tel que 1} chaque trajectoire ~
est homeJmorphe
2} il existe une variete differentiable un diffeomorphisme la droite facteur V4
V4 .... V3 It R
V3
a R a trois
dimensions et
appliquant les trajectoires % sur
R
apparait comme une variete fibree triviale de base
R'
bre type
V3
de fi
Les fibres sont des trajectoires d'isometries, On
appelle les lignes de temps, On appelle espace la variete de base Celle-ci est diffeomorphe
a la
variete quotient de
V4
les V3'
par la relation
d 'equivalence definie par Ie groupe d'isometries,
t
Si
est Ie vecteur generateur infinitesimal du groupe d'isome -
tries il satisfal.t aux equations de Killing
(7. 1)
II resulte de la definition qu'il existe des systemes de coordonnees locales
(x', )} tel que les
cales sur
I'
V3
et qqe
x~
soient un systeme de coordonnees 10
xO definisse les points sur les trajectoires de
de sorte que les sections d'espace
definies et Eiiffeomorphes
a
xO = const sont globalement
V3' On dira que ces coordonnees locales
- f8 -
Pham (x', )) sont localement adaptees au groupe d'isometries si Ie gene-
rateur infinitesimal
g;.~
y
admet les composantes contra rariantes
.
~o
(7. 2)
Si
~
,~ " 0
sont les composantes du tenseur metrique dans ces systemes
de coordonnees, les composantes covariantes de
Les equations de Killing
(7. 1)
t
sont
se traduisent comme
spit
Ainsi dans les coordonnees adaptees les
gr,
sont independantes de
xO En ctecomposant la forme metrique suivant la variabl€ directrice
x'
on a
(7.3)
oli
(7.4)
g
- 69 -
Pham definit une metrique definie negative sur les sections
d'espace. EI-
Ie est invariante par tout changement de systeme de coordonnees a daptee de la forme
On munira
V3
de cette metrique
A
g
8. Mouvement permanent. On dit que Ie mouvement du fluide parfait isentropique est perma nent si I' espace -temps stationnaire et si Ie groupe d'isometries laisse invariants
l'indice
f
et Ie vecteur vitesse unitaire
i. e.
o
(8. 1)
Si les coordonnees sont adaptees ces conditions
(8. 1)
se traduisent
par
a, f
(8. 2)
0
~D
Ull(
0
Theoreme. - Pour que Ie mouvement du fluide parfait isentropique soit permanent, il faut et il suffit que 1'espace-temps soit stationnai-
reo Choisissons des coordonnees locales adaptees I 'hyper surface d 'equation
xO =
const , on a
et
soit
x 0 = 0 . Zest orientee dans Ie temps. 11
resulte du probleme de Cauchy que sur voisines
(xo, x")
1. et
sur les hypersurfaces
(chp II, 5, (5.9) )
L
- 70 -
Pham
En
coordonn~es adaptees, ~. S~ = 0 , ~o gGl~ = 0 . Il en resulte qu 'en
derivant par rapport modynamique
a x 0 on obtient compte tenu de I 'equation ther-
dp = rdf -
9d
(d S = 0)
S
soit en tenant compte du Ch. II, 5, (5.7)
Ou en deduit
"a.f
=
0
sur
1.. ,
puis ~o u~
O. Le mouvement
est donc permanent. Theoreme. - Dans tout mouvement permanent du fluide, la fonction scalaire
(8. 3)
conserve une valeur constante Ie long de chaque ligne de courant. II nous suffit de montrer que
i c ( Lt
(c) ) =
0
ou
Or on a en utilisant I 'identit~ du ca1cul des variations
W =
Cel dx el
- 71 Pham
Or il est manifeste que
J,'" = 0
ce qui veut dir que Ie systeme di.!
ferentiel aux lignes de courant admet la transformee infinitesimale on en cteduit egalement
Jt
II)
J'
= 0 . II vient
Remarque.- Le systeme differentiel aux lignes de courant admet la forme invariante
t,6
=
n.
et la transformee infinitesimale
I
Comme
0 , on voit que Ie systeme differentiel aux lignes de tourbil-
Ion possede la m~me propriete. On en cteduit que
H = Col til(
galement constant Ie long des lignes de tourbillons. H tant sur chaque variete caracteristique
W,
est e-
est donc cons
de II .
On a
(8.4)
formule qui rend les resultats precedents evidents .
• 9. Le theoreme de Bernouil1i;.. Introduisons la grandeur d 'espace du vecteur direction de temps
t .
SoH
- v
2
uel
relativement
a la
- 72 -
Pham En vertu du caract ere unitaire de
u, on a
d'ou
(9. 1)
(uo)
L'integrale premiere
Co =
2
2 g,o(1+v)
H a pour valeur en coordonnees adaptees
fuo . On en deduit
En posant
U = goo
' i1 vient
Theoreme. - Le mouvement permanent d'un flidde parfait isentropique satisfaits Ie long de chaque ligne de courant
(9.2)
ou
U
f
2
a
2
U (1 + v ) = const .
est Ie potentiel principal de gravitation.
Ce theoreme generalise Ie tMoreme de Bernouilli,. En effet de l'e quation thermodynamique, i1 vient
On en deduit aux termes en +
.1
C
pres v2 +
fP Po
dp r
const.
- 13 -
Pham
H. PROJECTIONS
DANS L'ESPACE
la, Un probleme du calcul des variations,
On se propose d 'etudier Ie mouvement permanent dans l'espace
V 3'
Pour cela il nous faut etudier les projections des geodesiques de (\'
" sur I' espace quotient (V3 g). 4 Un tel probleme a ete resolu dans Ie cas plus general d 'une varie(V"+ l
,t)
Mfinie par une variete differentiable
munie d 'une fonction
~ (x,
X)
te fuislerienne
sur Ie fibre des direetions
VM~
positivement homogene de degre
D(V""+4). On supposera que
(V"~4
' :
admet un groupe connexe d'isometries globales definies par un champ de vecteurs ~ tel que
On rapportera adaptees
a
(VIM I
,£) a des
coordonnees locales
son groupe d'isometries et on designera par
te quotient, L"e systeme differentiel aux ext) emales de
.t
(x ~ '. xo) V
n
la varie-
admet l'invariant inte-
gral relatif
w
(10,1)
, £ ne premiere provenant
(10,2)
depend pas de
xO
,
'ot
de l'equation d 'Euler en
,.[ o
h
=
x·
a,
on a l'integrale
- 74 Pham de sorte que
60£dxo = Ie dx O
famille
des extremales correspondant
(E k )
constitue un invariant integral pour la
a la valeur h. Il en re
sulte que
(10.3)
est un invariant integral relatif pour la famille Si
'oo!
fa,
on peu t
,
(10.2)
xo= ~ (x',. xJ ,
(10.4)
ou
resoU.dre
par rapport
Par suite la forme des variables
x~,
*
=
xj ,
£ on
x~ 'a~£
a
xII,
soit
h)
est une fonction homogEme de degre 1 en
vertu de I 'homogeneite de
(E h ).
.
xJ .
D'autre part en
a
peut s 'exprimer par ime fonction
k , soit
(10.5)
et lion a
On a demontre Ie theoreme. TMoreme. - Les projections sur une valeur
h
Vn
des extremales
donne sont les extl'emales de la function
(E h ) pour
L.
Elles
L
- 75 -
Pham sont dHinies pour un systeme differentiel qui admet l'invariant integral relatif
11. Cas d 'une metrique riemannienne. Considerons Ie cas OU la fonction
t4,~
= 0,1, ... , n. On suppose que Le procede de descente
go.
conduit
!
est definie par
r o. a
former I 'equation
(11. 1)
et
a eliminer
~, entre cette equation et
(11. 2)
L
l - h~'
L 'elimination donne
(11. 3)
o,
gIl
on a
+ g ..
(11. 4)
On supposera
g . x~ + h ....!L-
L
~J
go~
.r 0 .
x~
Le procede de descente
conduit
a
eli-
- 76 -
Pham miner
x' entre
(11. 2)
et les relations
L =
£- h x
G
L 'elimination donne
(11. 5)
L
Application aux mouvements permanents. - II suffit de remplacer 2 dans les formules precedents gClf, par f glt, et on obtient la fonction
L
dont les extremales donnent Ie mouvement dans l'espace. II
y a un seul cas car
(11. 6)
goo
f
O. On obtient ainsi: f 2g..
L
~J
x' ~ x';
II serait interessant de developper les calculs.
12. Projection des geodesiques de longueur nulle.
On les considere comme limites des geodesiques orientees dans Ie temps. Dans notre probleme
.t.· t par rappor t en d"nvan
d>.
x' • , on a
h ...r
."
go. x
, ce qui montre
- 77 -
Pham que
h ....
lorsque
oIJ
extremales de
L
.r -
0 , h
garde Ie signe de
coincident avec celles de
gOl!
xII. .
Les
L/k..
Par suite les extremales cherchees qui definissent les projections des geodesiques isotropes de
(V 4' g)
sont les extremales de la fonc
tion
(12.1)
lim h+c»
1er cas
(12.2)
ou
goo
f
0
A II
Le passage
=H - I
a la limite donne
1"· ~ .j g ij x x
goo
~I est Ie signe de gOIl
x·
et t Ie signe de
goo' puis
(12. 3)
2e cas
o . Le passage a la limite donne
(12.4)
L
(12.5)
x'= -
Nous appliquons ces resultats
a I 'etude du principe de Fermat.
13. Le principe de Fermat. On sait que les rayons lumineux [16] sont geodesiques isotropes de
- 78 -
Pham (V 4' g)
la variete riemannienne
definie par l'espace-temps
V4
mu
ni de la metrique.
(13.1)
Supposons que Ie mouvement est permanent. Si Ie groupe d'isometries de
(V 4' g)
1
et}k
sont constants,
induit un groupe d'isometries sur
(V4 ,g)
On choisira des coordonnees adaptees. Mais alors que les trajectoires d'isometries de
(V 4' g)
sont orien-
tees dans Ie temps, les trajectoires d'isometries induites sur peuvent effet si
~tre
(V4' g)
orientees dans Ie temps, dans l'.espace or isotropes . En
t est Ie generateur infinitesimal du groupe d'isometries de
(V4' g) on a pour les composantes contravariantes
o
1
Ie carre de ce vecteur a pour valeur
(13.2)
gllO
2
ou w
1
-;;;:
2 - (1 - w )
u, u,
est Ie carre de la vitesse de propagation de la lumiere
dans Ie fluide. Si nous introduisons la grandeur d'espace du vecteur vitesse unitaire ulll relativement
a la
direction de temps
~,
vu que
(u.)
2
t:
2 g •• (1 + v )
.,1 = -
g~j u~ u j
, on a
- 79 -
Pham
b, il
En portant cette valeur dans
goo
(13. 3)
-
goo
(13. 2
=
2 2
goo (v w
vient
+
2
W
-
2
v )
peut changer de signe. En appliquant les formules du paragraphe precedent, on obtient Ie
theoreme suivant qui donne la loi de propagation de la lumiere dans I 'espace. Theoreme. - Si Ie mouvement du fluide est permanent et tel que
to,
g.,
les rayons lumineux dans I' espace sont les extremales de
l'integrale
x~
f'[E£1
jXI
(13.4)
" du x,
}o
goo
Xo
glj x~ ~j
go~ x· - -..--g,.
dx~
]
du
pour des variations a extremites fixes dans V 3 . Le du ' temps mis par un rayon pour aller du point Xo au point x 4 est don ou
ne par
(13.5)
II est extremum . Dans Ie cas
(13. 6)
g.o
=
0 , on a
JXIA
du
=
Xo
Xo
(13.7)
f4
f' f' dt
x.
=
Xo
gO
. ~ •j
x x
*'
2g.~
-.
~
xj
x - gil 2g,:, xl
du
du
- 80 -
Pham II est clair que les resultats ne dependent pas de la variable auxi liaire
u. D'autre part si l'espace-temps est statique orthogonal et
si les lignes de courant coincident avec les lignes de temps, on a la metrique d 'univers
et la metrique associee
g
ou
n2
~f.
On peut alors mettre
(13.5) sous la forme
dr est I 'element lineaire de cas d 'un espac e temps plat
U"
(V 3' g)
.
Dans
Ie
1 , Ie tMoreme precedent se tra -
duit par
o C 'est I 'enonce du principe de Fermat en optique classique. Ce tM£ reme que nous avons demontre constitue I 'enonce du principe de Fermat en relativite generale , dans Ie cas OU Ie fluide est en mouve ment. Par ce theoreme se trouve egalement demontree I 'equivalence entre Ie principe d'action et Ie principe du moindre temps.
- 81 -
Pham 14. Application: loi relativiste de la composition des vitesses. Pla T et
VT
dMinies sur
tend vers zero par valeurs negatives (resp. positives),
convergent uniformement vers des fonctions
L
a valeurs tensorielles
etnotees To' (VT)o (resp. T 1 ,(V'T)l L
Nous introduisons les tenseurs-discontinuites sur
L
:
- 97 -
Lichnerowicz
[TJ Dans
=T - T
1
n , se trouvent
0
definis de maniere naturelle Ies tenseurs distribu-
y\7 T.
tions yOT, Y°V'T, ylT, par Ie tenseur T
Si TD est Ie tenseur-distribu.tion defini
defini presque partout dans
n ,
on a en termes de
distributions:
Le tenseur-distribution
V T D, cterivee au sens des distributions de T D,
s 'ecrit:
avec: o
-
V(Y T)=- Ih®T +y
o
0
V
T
On en deduit, compte tenu de (3. 1):
(3.2)
ou (V T)D est Ie tenseur distribution defini par Ie tenseer ordinaire defini presque partout
" T, derivee covariante usuelle du tenseur T.
b) Etudions Ia cterivee du tenseur-distribution coordonnees (y~), on a:
J[T]
de
n . En
- 98 -
Lichnerowicz
OU
V.1 ~
=L • 1
JI =0.
V.1 [T] =['V.TJ 1
D 'apres les_ hypotheses de convergence uniforme
. Ainsi l'on a
b ["Y.TJ = \7.( 1 1 ~ T,
existe un p-tenseur distribution
~ [TJ). Il en resulte qIT'il
a support sur 1.. , tel que l'on
ait la formule:
(3. 3)
c) Nous considerons maintenant des tenseurs satisfaisant toujours aux hypotheses AI' A2 mais qui sont supposes continus dans seurs definissent d 'une maniere naturelle une algebre
tA
n.
Ces ten-
de tenseurs.
La formule (3.3) devient alors:
(3.4)
Considerons I' application:
J : TE~ OU
~T est un tenseur-distribution
que l'application T, ue ~
6
--+ ~T
a support sur
~ . On deduit de (3.4)
est une derivation: si a et b sont deux reels et si
sont deux p-tenseurs, il resulte de (3.4):
b(aT+bU) = a bT+b ~U Si T, U ~CA
sont respectivement un p-tenseur et un q-tenseur, on a:
- 99 -
Lichnerowicz
b(T ® U) '" bT0 U + T
J Test
@
bU
appele la discontinuite infinitesimale de T et
~ l'operateur de
discontinuite infinitesimale.
4. Formule concernant les derivees secondes. Pla,
il vient, compte-tenu due caractere unitaire du
vecteur-vitesse:
(5.7)
soit:
D'apres (5.3) cette relation peut s'ecrire:
x
Ainsi (5.4) entraine l'equation dite de t10t adiabatique:
(5.8)
En reportant (5.7) dans (5.6) on obtient Ie systeme differentiel aux lignes de courant:
(5.9)
Le systeme (5.4), (5.5) est equivalent au systeme forme par (5.4), (5.8) et Ie systeme differentiel (5.9) aux lignes de courant.
6. Vitesse d 'une hypersurface par rapport au t1uide et ondes soniques. a) Soit tion
'f =0
L..
une hypersurface reguliere dans un domaine de V4' d'equa-
(avec I = d 0
il vient:
- 114 -
Lichnerowicz Nous n'etudierons pas, pour
elles-m~mes,
les ondes de choc de
l'hydrodynamique relati viste, mais nous considererons une telle etude comme un cas particulier de l'etude complete des ondes de choc de la magnetohydrodynamique,
a Ihaquelle nous procederons.
III. LES EQUATIONS DE LA lVIAGl\TETOHYDRODYNAMIQUE RELATIVISTE.
9. Le tenseur d'energie de Ia magnetohydrodynamique. a) Supposons Ie fluide envisage soumis Mcrit par l'ensemble de deux
a un
champ electromagnetique
2-tenseurs antisymetriques H et G; H est
ici Ie tenseur champ electrique-induction magnetique et verifie Ie premier groupe des equations de Maxwell dH=Q (ou d designe la differentia-
*'
tion exterieurel. Si
est I'operateur d'adjonction sur les tenseurs
antisymetriques, les vecteurs orthogonaux
e~ =l!l
01..
a u, donc spatiaux
Ho{~
sont respectivement Ie vecteur champ electrique et Ie vecteur induction magnetique relatifs
a la direction temporelle u. Soit
nee, la permeabilite magnetique que h est suppose relie
1"
constante don-
du fluide. Le vecteur champ magneti-
a l'induction
magnetique b par la relation:
Le courant electrique Jest sensiblement la somme de deux termes:
- 115 -
Lichnerowicz
J~ = -VUP-:> ou
+
()e~
vest la densite propre de charge electrique du fluide et () sa
conductivite. b) La magnetohydrodynamique est ici I 'etude des proprietes d'un fluide ideal relativiste de conductivite infinie () = 00; J etant essentiellement fini, il en est de
m~me
pour
rr
,e, et I 'on a necessairement
e=O. Par rapport
a la direction temporelle definie par Ie vecterr-vitesse u
du fluide, Ie champ electromagnerjque est reduit au champ magnetique h. D'apres des resultats c1assiques, ce champ admet Ie tenseur d'energie:
ou
IhI2=_h~ h f
est strictement positif pour h
f
fO. Le tenseur d'ener-
gie total fluide-champs s'en deduit:
(9. 1)
ou l'on a pose:
c) Le systeme differentiel fondamental de la magnetohydrodynamique relativiste est constitue par les equations suivantes: I 'equation de conservation de la densite de matiere: (9.2)
les equations de Maxwell (dH=O) qui peuvent s'ecrire ici:
- 11 G -
Lichnerowicz
"""~ Vo( (u n -no{(O u )=0
(9.3)
et les equations de la dynamique relativiste:
\J0( To(~ =0
(9.4)
ou
T.ol~
est donne par (q-l).
10. Consequences du systeme fondamental. Nous utiliserons dans la suite un certain nombre de relations, conSeql!enCeS
du systeme (9. 2), (9. 3), (9.4).
a) Partons des equations de Maxwell explicitees sous la forme:
(10. 1)
et projetons-les successivement sur les directions definies par les deux vecteurs orthogonaux u et h. Par produit scalaire par u il vient, compte-tenu du caractere unitaire de u:
(10. 2)
Par produit scalaire par h, on obtient:
soit, compte-tenu de l'orthogonalite de u et h:
(10. 3)
- 117 -
Lichnerowicz b) En explici tant (9. 4), on a:
Par produit par u ~, on en deduit:
c 'est-a.-dire:
qui, compye-tenu de (10.3), peut s'ecrire \7
2
eX.
Yo p+c rn~ U h
2 c rf 0
On obtient par deve1oppement:
-(U(\/l~l~ _(U~1~)2) O(u~ 10{) hOi 1 C/
- )J.hd.. 1o( 19 1
0
2 c rf
- 123 -
Lichnerowicz Pour que Ie systeme considere admette des solutions autres quella solution nulle, il faut et il suffit qhe H=O. S\~l en est ainsi, les relations (11. 1}, (11. 5) et (11. 6) fournissent fonction de
L
de u ~
et
h~
)
1~)1~=0.
~ p et les composantes normales
. Decomposons ces vecteurs selon leurs compo-
santes tangentielles et normales
ou
en
~ p.
c) L'etude precedente concernait
a
l~ ~ u~ , l~ ~ h ~ , ~ , h \2
a L . 11 vient:
Compte-tenu du b, les formules (11.3) et (11.4)
fournissent relations de la forme:
(11. 9) (11.10)
(hO\>:)
(c 2rf+p: IhJ2)(uo{
ou Ie symbole
a ~ p. Le ~ v~, ~t~
(V
~ v~ -(u'\,()~ t~ ~O
~ ~ v~
.:.}l(ht> ¢h +1~ ~ h +1",-1(0 h
- 137 -
Lichnerowicz c) Des relations (11. 1) et (11. 2), on deduit comme precedemment:
(14.9)
Des relations (10. 2), (10. 3) consequences des equations de iVraxwell,
~ 7:
on deduit par derivation, en raisonnant comme au
et
En multipliant la premiere de ces relations par
u~,
h~
, la seconde par
et retranchant, il vient:
De (10.8) il vient de
m~me:
(14.11)
l{>e (14.10) on tire, compte-tenu de (14.11) et (14.8):
- 13B -
Lichnerowicz d) Prenons enfin la derivee covariante contractee de (10.7). On obtient, compte-tenu de (14 ..5):
(c2rf+)llhI2)l~[V'~ Vo\~ _(go/~_u«u~ +}J-uo/
u~~[Y'c{ V~
Ihl 2J+t l h\2l
)JlV'... ~p] -
trb[~ V~lhI5+
~[V~~o(uo/J -2Jh~ ~ [V~ Vo( ho(] ~ 0
Ce qui peut s 'ecrire: 2 ~- [ 01.1 c rf u b ~Vo( u
_(go(l~P.
j?l
_ur:i.. u )
-
b [Vo( ~pJ
-
1 - ~ 2"JL 6 LV~'i~Ihl 2
J+
I u~ ~[~ ~ u1 +uol.u~b[V~~ Ih\2] -2h~ ~ [~~h1]~0
+t{2 h\2
ou la seconde ligne disparait en vertu de (14.10). En utilisant (14.8) il vient:
(14.13)
Introduisons l'expression:
On a:
et (14.12) peut s'ecrire:
- 139 -
Lichnerowicz
u u -h h
Apres produit par
c2rf(u~ l~
)2, la formule (14.13) prend la forme:
soit en ordonnant:
(14.15)
tc2rf((O-l)uol.
u~ +i~ )(u~lf)2+}A-lhl\ Ifl?Uol.U~_
-y-191?holh~}X[Y't¥\7~pJ +f"ll~rf(u~l?) {(uVl~)l~ -l~ u~l V~ ~lhI2~ Substituons
a 1['Yo( V~
pJ sa valeur tiree de (l4.4). Le coefficient de
pest:
soit:
Ainsi Ie coefficient de pest nu1.
n
vient ainsi
a partir
de (14.15):
0
- 140 -
Lichnerowicz 2 { c 2rf ("t _l)(uol. 10( )3u~ +c 2rf(u« 10( )2l +~I hl 21
(14.16)
-
l~ If (uo\ le()}
-
}'-l~ If (hoi ho( )he}v~ b p + r:c 2rfu ~ l~ {(U9 l~ )l ~ -l~ l~ u~l V~ ~ Ih \2 ~ 0
D'apres Ie calcul precedent concernant Ie coefficient de p, Ie vecteur
m~me
L. II en est de 'V~ J I h ~2. La relation
~6P dans (14.16) est tangent a
coefficient de
manifestement pour Ie coefficient de
precedente peut donc s 'ecrire apres division par 2:
t
c 2rf (O _l)(uo( 1ell/V ~ -
+,fI-1 hl 21
11> If (UC)(lo(, )v~ -f If If (ha!
trvc2rfl91~(uolltJ()V~~-cS lhl2~
1
lo()t~ 'l~ p
0
soit d' apres la relation (14. 1):
{2C 2r
f(r
_l)(uO< 10()3 vf.> +(c 2rf+}-Lj h/21 )1 f l~ (ue( lo()v
-}A-l~l~(hQ(lol)t~l'7~~p~
13_
0
c'est-a-dire d'apres (14.3):
Nous obtenons ainsi : Theoreme. Sous les hypotheses du males
~ 4, les discontinuites infinitesi-
~p, ~u').., Jh" relatives a une onde magnetosonique L
se
propagent Ie lang des rayons associes selon les systemes differentiels:
- 141 -
Lichnerowicz
N~ 'V, J h'A ~
15.
des rayons associes aux ondes d' Alfven.
Propri~e
a) Soit
0
=
'f'
une soluzion de I 'equation aux ondes d 'Alfen par example
d'espece A. On a: AoI. "d =A01.. I =( ~ uo( +hO()l
(15. 1)
0(
0(\
r
eX
=0
Nous SuppoBons que pour les ondes envisagees p(l)fo. Par suite:
~p =0
I
De plus (14.15) se reduitt
01..
~ uol. = 0
a P(I)p=O; par suite 1>=0 et l'on a:
(15.2)
De (14.8);, (14.11)(14. 12) il resulte:
b) Pour les ondes d'Alfven envisagees ici, seules les discontinuites
~vA, ~t').
peuven1
uollo(fO et ou
~tre
non nulles. Des relations (11. 9), (11.10) ou
~t'A.
les seconds membres sont nuls, on deduit que
est proportionnel
a ~ v OX.
Le symbole
~
dulo des termes lineaires par rapport aux
signifie dans ce
bv~
(ou aux
§
mo-
~ t'l- ).
- 142 -
Lichnerowicz Compte-tenu de (15.2), (15.3), les equations de Maywell est Ie systeme aux lignes de courant donnent par derivation:
et:
Multipliant la 1ere relation par
h~
, la seconde par u ~
il vient
soit en explicitant:
Le coefficient de U?.
soit d'apres (15.1):
Nous obtenons:
est nul et il reste:
et retranshant,
- 14:3 -
Lichnerowicz
Theoreme. Les distributions.{)(vA, V(t A d 'Alfven
L
a supports
sur une on de
d 'espece A se propagent Ie long des rayons associes
selon les systemes differentiels:
ou ~ signifie modulo des termes lineaires par rapport aux (resp.
~ vJA-
6t}l.l ).
Des resultats symetriques sont valables pour une onde d'Alfven d'espece B. c) Reprenons une onde d'espece A et etudions l'action de la derivation
b
sur Ie vecteur
D 'apres l'etude du a, on a:
1
c\.
6uo(
=0
10(
J ho( = 0
II en resulte:
soit d'apres (11. 9) ou Ie second membre est nul:
- 144 -
Lichnerowicz
II vient
a l'ensemble
de
et de celIe de la composante normale. Ainsi Ie scalaire: 2 2 c a e = q- - - 't 10( 1 01.
est invariant.
est equivalente
- 148 -
Liehnerowiez Considerons en partieulier Ie produit sealaire invariant au eours duo ehoe:
D'apres la definition de: qt , on obtient
Xf> Vf->
=e 2ab, OU b est Ie
sealaire invariant:
(16.7)
d)
Consi~rons
enfin Ie sealaire invariant:
K
2 2 e a
done nous allons donner deux expressions importantes: Lemme 1. L'invariant K admet l'expression:
(16.8)
OU l'on a pose:
(16.9)
Il admet aussi l'expression:
(16.10)
- 149 -
Lichnerowicz En effet de:
on deduit
Il vient aussi:
(16.11)
2 a2 0(2}LeX 2 0/ )2 K =(r _ - - ) +2 - - r2(ht:X 1 )2+u? r (11 ld II ? 22 0( r 2? _.. - l ()( 1 ()( c (' a c a
En substituant
a sa
valeur, on a:
soit.:
Or d I apres 1a definition de H:
- 150 -
Lichnerowicz
n
vient aussi:
ce qui etablit (16.8). De cette relation on deduit:
Ce qui peut s 'ecrire:
K
=
22 2 c f +)1! hI (2't-
En reintroduisant 0O
p
2 c f' = €J 8
>
0
On en deduit par derivation:
(23.2)
Les etats Zo et Zl sont relies par la relation d'Hugoniot (21. 4) qui est symetrique en 0 et 1. Au cours d 'un choc, on a bien entendu 8 0 ~ 8 1 ell chaque point de L
. Nous allons etablir les resultats
- 171 -
Lichnerowicz suivants, valables en chaque point de
L ,
Theoreme 1. Pour un choc qui cn'est ni nul, ni d'Alfven, on a sous les hypotheses de compressibilite (HI) ~2t
En effet supposons qu 'au point x de L. , on ait Sg=S 1 et p jP I' En modifiant au besoin supposer
Po
et Z , on peut o 1 PI' On a alors 't' 0> "t'1 puisque 't~ « 0, De (23,2)
0
De la relation d'Rugoniot il resulte alors:
('t' - '?:
1
0
}{p 1-p0 + 1..2/M.(k 1-k 0 }2l _6(ho/.lol)Jf Adoptons en x un repere orthonormee
{e(o()}
_ho( 10/
£J3] =0
tel que eO) soit coli-
neaire a. I et e(3) a. Ia direction n. Dans ce repere il vient:
u
3 =0 o
3 u =0 1
Le systeme differentiel (24. 3) se partage en deux systemes dont Ie premier contient exclusi vement Ies perturbations
du 3,
dh 3, soit:
(24.4)
(24.5)
Nous supposons que seuis
{3
(3
ou, oh
---~o--o
sont
to
avant Ie choc. Les
variables thermodynamiques n' ayant pas ete perturbees, il en resulte que, dans les etats respectivement ante rieur ou posterieur au choc [
, de telles perturbations correspondent a. des chocs d'Alfven in-
finitesimaux, c'est-a.-dire a. des ondes d'Alfven. C'onsiderons, dans l'etat anterieur a. type A. Le vecteur
L ,
une onde d 'Alfven de
A0( etant invariant a. la traversee de cette onde 0 .13 infinitesimale, une telle onde porte en x une perturbation (JuoA' Jh oA ) .
- 176 -
Lichnerowicz telle que:
(24. 6)
De
une onde d'Alfven de type B porte en x une perturbation 3 3 ( bu oB )' bhoB) telle que: m~me
~ ~u
(24.7)
;-0
3· 3 - Jh =0 oB oB
La superposition en x d Tune onde de type A et d Tune on de de type B
~h 03)
fournit une perturbation ( hu 3, o
c)
b Les vecteurs
A(J. et Bot' verifient en x E L : o 0
01. !\ 1 =A
a
ocXTor
(24.8)
arbitraire avec:
()/.
+l:\ I
001
o
FfXo 1eX. = Fo P, ~ r
-hoallo(
o
On en deduit: 01.
-(h 10()
o
Convenons d 'orienter choc
L .
2
a
=-
2
fA.
0(
0
1 de 1 'etat anterieur vers 1'etat posterieur au
On a alors a
d-
~0 )(~) -(q-q0 )=0 d'C''lIC.
Aussi au point Zo' on a (dS/d"t:)~ =0. En derivant (26.6), on obtient:
(26.7)
2 2f 9) (d ] d't
1f(.
+2{ d{f 9 )) (dS) +('t' -"t" )( d2q) =0 d t: 11' dt; 'II' 0 d t:; 2 'll'
et en Zo' on a aussi (d 2S/d'L2)1' =0. Aussi la courbe d'Hugoniot et l'isentropique S=S
ont un contact du second ordre en Z . En o 0 ce point, nous avons done:
(26.8)
En derivant (26.7), on obtient en Z
2f @)
II resulte ainsi de (26.8):
3 d S
o
(g't:>3~
2-
+(~) d -z;2
~
=0
- 190 -
Lichnerowicz
(
(26. 9)
11 M) 2f@ ~ z0 1; p
D 'apres Ie contact entre ~ et l'isentropique
( d-r;) = ( d't') (0 dP1b dp'-j'='tpZ
-S
(S=S ), on a en Z : o
0
o
et il vient en ce point:
On obtient ainsi en Z : o
(26. 10)
et dans Ie voisinage de Z , nous avons Ie long de 1t o
(26. 11)
ou Ie coefficient de (p-p )3 est positif. Nous enonO, S~~o.
Par une nouvelle
d~rivation,
on obtient:
Nous supposons ici que la fonction S( p ,'t:') satisfait les hypotheses (H 1) et (H 2) pour 't: '" ~ et paur des valeurs arbitrairement grandes o . A MR ~ de p. Nous nous proposons de montrer que Sl v < V V - - ML 1: A 0 0 0 (ou v < v < v.A. ) c 'est-a.-dire 0< P(l) >0 les ~quations g~n~rales o 0 0 0 0' de c)1oc admettent une solution unique non triviale avec eXo 0(1 ,. O.
qui donne
- 199 -
Lichnerowicz
N
(3
2 a2 \3> 2 Ih 12 ~ $ =2c't (0'-1) 2" arv +(c 't+fl 2 r)(1 l~)ar vr r
Compte-tenu de:
il vient:
Or d'apres (29.1) (c'est-a.-dire P(l)=O)
2 a ( 1) (2_ c 't 2 D - + c ... r 2
+r
2
Ihl )It?l -2- ~ p -
r)
fA
d 2 (h l~) l~l --0 2 a f
et l'on a:
N~ =c2~ :: (0' -1)ar v ~+r; (Ii' loi)l fIr v f3 c 'est-a.-dire en tirant
1-1/r
2
de (29.1):
- 200 -
Lichnerowicz ;? 2 2 ~ ~ Nj"-" = - _c_a_'C'_l_l-ll..."_ _ ad.r v~+u ~a (he{ la>l )l\' lIZ) VI'" 2 2 Ih \2 1~ If I , c a 't
+r-
On deduit de (16/5):
La direction de N ~ est donc celle du vecteur proportionnel:
En utilisant b=f If lei et en divisant par 1:';, on obtient Ie vecteur colineaire
a N F:
ou lIon a pose:
(29.3)
30. Action de
b
sur la direction du rayon.
a) Pour que la direction de et il suffit que:
N~ soit
invariante par
~
, il faut
- 201 -
Lichnerowicz II en est donc en particulier ainsi pour b=O, c'est-a.-dire si Ie champ magnetique est tangentiel b) Cherchons a. evaluer ~ Q. II vient:
(30.1)
'C2~Q=-c2a2~~+tI~Iff-lh\2_1~1't-'-lh\2 ~
D'autre part, d'apres I'invariance de l
par
~
, on a:
c 'est-a.-dire:
En reportant dans (30.1), on en deduit, compte-tenu de
soit, comme: 2
I
c"C p =-
.1....:..!.. 2 r
on a:
(30.2)
&8=0
- 202 -
Lichnerowicz Pour que ~ Q soit nul, il faut et il suffit que
1 c 'est-a.-dire: que:
(39.3)
Nous avons ainsi etabli que, contrairement aux resultats concernant les rayons aessocies aux ondes soniltues et waux d'&des d'Alfven (en magnetohydrodynamique), la direction des rayons associes aux ondes magnetosoniques n 'est pas en general invariante par I 'operateur
b
de discontinuite infinitesimale. II y a invariance seulement si Ie champ p magnetique est tangentiel
ou si la relation (30. 3) est satisfaite. Mais celle-ci n 'est autre que (13.1). Cette relation correspond donc au cas singulier ou Ie cOne P(I)=O
admet deux generatrices doubles. A I 'approximation classique la relation (30.3) s 'ecrit:
(30.4)
2
I \ ~h
2
=rv.
- 203 Lichnerowicz
BIBLIOGRAPHIE
[d
F. Hoffman et E. Teller, Phys. Rev. t.80, p.692, (1950).
[2]
A.H.Taub, Arch. Rat. Mech. Anal. t.3, p.312, (1959).
[31
Y. Choquet-Bruhat, Astron. Acta t. 6, p.354, (1960)
[ 4]
W. Israel, Proc. Roy. Soc. A. 259, p.129, (1960).
[ 5]
Pham-Mau-quan, Ann. Inst. If. Poincare t. 2, p.151, (1965).
[61
A. Lichnerowicz, Relativistic HydrodyIIllmics and Magnetohydrodynamics, W. A. Benjamin New York (1967).
[ 7]
A. Lichnerowicz, Ann. Inst. Poincare t. 5, p. 37, (1966).
[8J
A.Lichnerowicz, Ann. Inst. Poincare t. 7, p.271, (191i7).
[9J
A. Lichnerowicz, Comm. Math. Phys. t. 12, p.145, (1969).
UOJ
A. Lichnerowicz, Comptes rendus Acad. Sc. Paris t. 268, p.256, (1969).
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E. )
VARIATIONAL PRINCIPLES IN GENERAL RELATIVITY
A. H. TAUB
Corso tenuto a Bressanone dal 7 al
16 Giugno
1970
LECTURE I VARIATIONAL PRINCIPLES IN GENERAL RELATIVITY 1.
Introduction In these lectures we shall derive the Einstein field
and the equations of motion for uncharged and charged selfgravitating fluids from variational principles.
We shall
also see how singular hyper-surfaces (shock waves) and the equations governing their behavior may be treated by means of these principles.
In addition we shall show how the
"second variation" problem is related to the discussion of the stability of the solutions of the Einstein field equations. Before taking up these problems we shall discuss some general properties of variational principles and show how a form of the principle of equivalence may be used to formulate in general relativity a field theory described in special relativity by a variational principle involving a Lagrangian fUnction which is a scalar function in Minkowski space-time and which in turn depends on tensor fields and the first derivatives. Given a four dimensional space-time with a metric tensor Vu
and a tensor field over the space-time.
vm
Let
(written as ¢A) defined
be a scalar function formed from
the metric tensor of the tensor field
¢A
and the derivatives
of these tensor fields.
=
Jf ;:g d 4 x V
where
V is a fixed but arbitrary four volume in the space-time,
- 208 -
Taub
is a functional of the metric tensor and the tensor field
For given
g~v'
I may be evaluated in any coordinate
system in the space-time by using the assumed transformation properties of the function I .
I (x)
Thus under the
= I (x(x*»
= I *(x*)
and we may write I(g , ~A)
=
J
I(x)
r-g
d4 x
V =
J
I(x(x*»
r-g
(x(x*»
J d4 x*
V* (1.1)
=
J
I * (x*)
;:gw
d 4 x*
V* where
J
is the Jacobian of the transformation of coordinates,
that is, J
we of course assume that
= J
#
0,
and V* is the same four volume as x*~
V but now expressed in the
coordinate system. If the tensor fields
g~v (x) and ~A (x) are embedded
in families of tensor fields
g~v(x)
= g~v(x;o)
and
g~v(x;e), ~
~A(x)
A
(x;e)
such that
= ~A(x;o) the functional
- 209 -
I(g ¢A)
becomes a function of the parameter ICe)
e.
= I (g~v(xie) i ¢A (x
Taub Thus we have
e)).
An example of the embedding referred to above is given by = g~v (x) + e h ~v (x), ¢A (xi e ) = ¢A (x) + e ~A (x) Where and as
h
is an arbitrary symmetric second order tensor field
~v
~A is a tensor field with the same transformation properties ¢A . A variational principle is said to apply if (dI) de e=O
= I' (0)
= 0
for arbitrary dg de
(~)
e=O
= g'~v(O)
or (d¢A) de e=O
= ¢' A( 0)
or both.
2.
Embeddings Induced by Coordinate Transformations. Let
g'-'
~v
tensor fields
(x,~)
and
¢*A(~*)
and
¢A
be the components of fixed
in the
x*~
coordinate system and
let the equations of transformation to another coordinate system
x~
depend on the parameter
e,
that is, let (2.1 )
- 210 -
Taub
Then since
and
~A(X)
'V
lln (x) \)m
~lll \)1
a n d/ l 1m dx*Ol
foOl = ~ 1
1
'V
the
and
~
A
dxlln dX *1, ox i : On dx\)l
.1.
-----
1 dX* m dX \)m
(xje)
~A also depend on the parameter e.
Such
dependence will be said to be induced by the coordinate transformation (2.1). It may be readily verified that if
=
= + ~ll
(x)
then
= - ~ll;\) - ~\);ll
(2.2)
where the covariant derivative is taken with respect to the metric
= gll \) (x *; 0 ) ,
- 211 -
Taub
and
(2.3)
These expressions for
g~v(O)
and
¢,A(O)
when the
e
dependence is induced by a coordinate transformation will be used below. When the volume over which of
e,
I(e)
is defined is independent
that is the limits of integration in the right hand
side of the first of equations (1.1) are independent of
e,
then the limits of integration in the other integrals occurring in that equation depend on function depending on the
e.
Thus if
f(x)
is a scalar
whose dependence on
g~v(x;e)
e
is
due only to the fact that this dependence has been induced by a coordinate transformation then the expression for I, He)
= Jf
;g
d
4
x
V
depends on
e only because the integrand depends on
e.
Whereas the equivalent formula He)
depends on
e
= J V*
f'~!gil
d4
x'~
only because the limits of integration which
determine the volume over which the integral is to be carried out depends on
e.
The integrand is by assumption independent
- 212 -
of
e.
Taub
We may write the last equation for I(e)
f
=
fi:
I(e)
as J- l d 4 x
(/'(x)) Vg 1:(x":(x»
V
and now the integrand depends on of the functions
x*(x)
1'(0)
on
e.
e
through the dependence
It follows then that
f (fE;°);or-g d 4 x
=
(2.4)
V
3.
The Principle of Equivalence If in the special theory of relativity a field theory can
be described by a variational principle then there exists a scalar Lagrangian function
£
which depends on the dependent
variables of the theory and their derivatives.
We shall assume
that the dependent variables are given by fields which behave as tensors under Lorentz transformations.
The case of spinor
fields can be treated in a fashion similar to that described below.
Thus we are considering the case where the function
£
is a scalar function of a tensor field and its derivatives under Lorentz transformations.
We postulate that the dependent
variables of the theory are tensors under general coordinate transformations in Minkowski space-time and write
£
a coordinate system as a function of the tensor field
in such
~A,
its
covariant derivatives and the metcic tensor evaluated in the general coordinate system. We now form the integral 1=
f(R-k£)r-gd 4 V
X.
(3.1)
- 213 -
Taub
In this integral, the g~v are no longer assumed to be the metric of a flat space-time and hence one can compute a combination of it, its first, and second derivatives which is a scalar curvature of a space-time with metric tensor
g~v'
The volume of integration entering on the right-hand side of equation (3.6) is an arbitrary one, and
K is a constant which
may be related to the Einstein graviataional constant. Next we form the function fields $A(x;e)
I(e)
by embedding the tensor
into the families
and
g
~v
(x;e) and
as discussed earlier and study the conditions for 1'(0)
= O.
We shall denote
ag I =
I' (0)
when $,A(O)
= 0
and 15$ I
= I' (0)
when g~v(O)
= O.
In general we shall have I' (0)
=
ag
I + 0$ I
Taub
- 214 -
The general relativistic formulation of the field equations determining the field tensor 6¢ I
=
¢A
will be given by
o
(3.2)
and the Einstein field equations for ir,e gravitationa.l field created by the sources dependent on the tensor field
¢A
will
be taken to be
= o
(3.3)
It is evident that the equations obtained from equations (3.2) are those that would hold in a general coordinate system in Minkowski space-time.
Hence in view of the principle of
equivalence which states that some non-galilean coordinate systems in Minkowski space-time are locally equivalent to the equivalent to the presence of gravitational fields, equations (3.2) should represent the equation determining the tensor field
¢A
in general relativity.
In the subsequent discussion we shall see that equations (3.3) lead to the Einstein field equations in the form
= -KTIJ\! and
TIJ\!
will be a symmetric tensor determined from the
tensor field
4.
(3.4)
¢A.
Notation In order to avoid an excessive use of indices, we employ
the following notation ¢A T
(4.1 ) n
- 215 -
The symbol A ¢ ;lJ
is to be read as
'\,
¢ol
...
a
= ¢ol
...
a
'\,
n '1
,mj].J
Ttl
,m,lJ
...
_ r¢ol
a
n,
1
...
Taub
"stands for.
n
Oi_lpoi+l .•• On rOo '1'''' m P].Jl
+ ~¢ol l
...
'j-1P'j+l
P ' mr T.].J
(4.2)
]
where the comma denotes the ordinary derivative and P r a,
PA 1 = g 2' (gaA ,, + gAT ,a - go, , A)'
We also write
IjJA
1jJ].J A
'\,
'\,
IjJA'
IjJ
IjJ
'1
IjJlJ where A
and
,m 01
'1
(4.3)
,m 01
a n (4.4)
a n
lJ
and IjIA¢A
'\,
IjI
'1
,m
al
a n
al ¢
...
a n '1
,m
(4.5)
(4.6)
That is, the quantity given in (4.5) is a scalar and that in (4.6) is a vector. We consider
£ as a function of three sets of variables: and define
- 216 e~V
qA
Taub
1 )1V = ~+ "2 g J:. ag)1V
=
(4.7)
aJ:. a¢A
(4.8)
aJ:. p)1 = -::7\ A a¢ ;)1
(4.9)
In each case the remaining two sets of variables are kept constant in the partial differentiations.
We also define
=
(4.10)
It follows from equations (4.3) that
=
= g PA
-21
(g'
OA;1
+ g'
A1;0
_ g'
01;A
) (4.11)
and from (4.2) that A
[(~) ] de e=o;)1
=
("A
),
'I'
;)1
~
0 " .0, IPO'+l"'O ¢ 1 11 n
1
11,
r' a,1 .. 1 m P)1
- 217 -
Taub where the variations are produced by varying the
¢A
and the
gjJ\)' Hence PjJ (,!,A )1
=
A 'I';jJ
(4.12)
= where
(4.13)
It follows from this equation that
It is a consequence of the definition of the Ricci tensor. RjJ\)
=
rOjJ\),O
rOjJo,\)
+
r PjJo rOP\)
rPjJ\) rOpo
(4.15)
and equation (4.1l) that (
dR
jJ\) """'"de e=o =
RI
jJ\)
=
rIa
jJo;\)
rIa
jJ\);O
and that jJ \) po jJ P \)0 1 ] = [ (g g - g g )gjJ\);P;O
(4.16)
- 218 -
5.
Taub
The Euler Equations The relation between the energy-momentum tensor for the
~-field
which appears in the Einstein field equations and the
£ with respect to the
variation of
was first pointed
g~v
out by Hilbert [1] as is noted in Pauli's classical discussion of the theory of relativity[2] where additional references may be found.
In this section we shall derive the Einstein field
equations and the equations that must be satisfied by the ~-field.
It may be verified that
[(G~v
f
=
+
Ke~v) g~v
+
K(qA~A'
+ pX
(~~~)')
V
(5.1)
_{(gPO
where pX
G~v
g
~v
- g
} ].r;:Qgd 4 x >IV;P ;0
p~vo)g'
g
is defined ln equations (3.4),
in (4.8) and
in equation (4.9). It follows from equation (4.12) that equation (5.1) may
be written as -1'(0) =
f
[(G~v
Ke~v) g~v
+
+
K(qA~A'
+ pX
(~A');~
V
+ KM~vO
,
g~v;o
)
-
{( pO ~v P~ VO) g' } ] ~g d 4 x g g - g g ~V;P ;0
- 219 -
Taub
On integrating by parts this in turn may be written as -1'(0)
=
J {[G~v
+
KT~vJ g~v
+ K(qA -
p~;~) ~A'} ;:g
d4 x
V ( 5•2)
+
J [KH~va
g'~v
+
KP~ ~A'
-
(gPag~v_gP~gva)g'~v;pJ;ar-g
V
where the symmetric tensor
= and
M~va
(5.3)
= is defined by equations (4.10) and (4.13).
By requiring
1'(0)
to vanish for arbitrary variations
which vanish on the boundary of the region
V we obtain the
Euler equations G~v
+
KT~v
= 0
(5.4)
and FA
-
~
qA - PA;~
(5.5)
= 0
Equations (5.4) are the Einstein field equations with a matter tensor.given by equation (5.3). from the variation of field
we call
I
Because this tensor arises
with respect to the gravitational the gravitational matter tensor.
that even in the Minkowski space-time e~v
T~v
Note
is different from
in a general coordinate system. Equations (5.5) are the equations for the field
~A.
They
may be obtained from the special relativity equations ln a galilean coordinate system by replacing every ordinary derivative
d4 x
- 220 -
by a covariant one.
Taub
They obviously reduce to the equations of
special relativity in case the tensor
g~v
is the metric tensor
of Minkowski space-time.
6
Conservation Laws In this section we shall use a technique similar to that of
E. Noether [3] to relate the tensor
T~v
occurring above with
a not-necessarily-symmetric tensor which, will be called the inertial stress energy tensor and to derive various conservaTion laws.
The inertial stress energy tensor of the field
¢A
is
defined by the equation Q'
t P
=
(6.1)
and it is evident that in general
= Further we see that as a consequence of the equations satisfied by the
¢ field, equations (5.5), we have
= = when
£ does not depend explicitly on the coordinates.
special relativity we have at most depend linearly on view of the Ricci identity. in equation (6.7).
In general
In can
and the curvature tensor in We shall evaluate this dependence
- 221 -
Taub
The computations carried out in the sequel make use of the fact that when
g'~v and
~,A are given by equations
(2.2) and 2.3») 1'(0) must be given by equation (2.4) with
f = R - Kt.
That is, we must have 1'(0)
=
-f«R
o
r-
Kt)~ )'o~-g
,
d
4
x
V
when equations (2.2) and (2.3) are substituted into equation (5.2). When the former equations are substituted into the latter one we use the identity 2G~v~
s~;v
_ ( po ~v _ p~ ov)(~ + ~) g g g g ~;v sV;~ ;po
=
equation (26.1) and the definition
to obtain 1'(0)
=
-f[(R V
+ K
where
N~vO
( 6 • 2)
is defined by equation (4.14).
- 222 -
Taub
Note that
= o when equations ( 5 .5) are satisfied. Since the tensor
Nl1v O'
is antisymmetric in (~
11
Nl1vO')
; va
= 0
Nl1va) ;V;O'
= 0
v
and
0'
we have
That is, (~
I1;V
Nl1v O' +
~
11
or
In view of this equation and equation (2.4) we may write equation (6.1) as J[(2T I1V + Ll1v)
~11;V
+
FA¢A;p~P];:g
d4 x
V
On integrating this equation by parts we obtain
(F ~A ..
V
A'" ;P
_ 2T
0'
P;O'
_
L 0' ) ~P;:gg d4 p;a
X
- 223 -
When the tensor field
¢A
Taub
is a solution of the Euler equa-
tions (5.5) the above equations become
o
=
(6.3)
and -2fT\1\) ;\) V
= o (6.4)
Both equations must hold for arbitrary volumes and arbitrary vectors
~. )J
If
is a Killing vector, that is, satisfies ~)J;\) + ~\);)J
=
o
then equation (6.2) inplies for such vectors that
= [t
o~p
P .,
+ NP\)O~ ] "p;\) ;0 =
Killing vectors need not exist in a Einstein field equations.
spa~e-time
o
(6.5)
satisfying the
As is well-known there are ten linearly
independent Killing vectors in Minkowski space-time--the generators of the inhomogeneous Lorentz group.
Equations (6.5) for theBe
Killing vectors are the conservation laws of energy momentum and angular momentum discussed by Belinfante and by Rosenfeld. the tensor
t PO
Since
satisfies these conservation laws, in special
relativity, we may consider it as the inertial energy tensor. Since equation (6.4) must hold for arbitrary vectors and arbitrary volumes, we must have
- 224 -
Taub Til \I ;\1
o
( 6 . 6)
= o
(6.7)
=
and
Equation (6.6) is a consequence of the Einstein field equations. We have now shown that it follows from the invariance properties of
£ and the
¢A
field equations, equations (5.1).
Thus
equations (6.6) and (6.&) hold in special relativity. Equations (6.7) relate two energy tensors, the gravitational and the inertial one
one
til\l.
It follows from equations (6.6) and (6.7) that
= (6.8)
=
This equation, which may be derived directly from the definition t PO
of
and equations (5.5), reduces to
= o ln the case of special relativity. It should be noted that as a consequence of equation (6.6), equation (6.3) may be written as J[(2T pO + tpO +
v
NpO~\I) ~p];O~
d 4x
= a
which in turn may be written as an integral over the hypersurface S boundary the volume V, namely
- 225 -
Taub
ov ] scP no dS f[2Tp O + tpO + Np;v
= 0
(6.9)
S
where nodS is the element of volume ~n S, and vector.
~p
is an arbitrary
Equation (6.9) may be used to relate the time rate of
change of the three-dimensional volume integrals of T~ and t~ by choosing the hypersurface surface S to consist of the hyperplanes t 7.
= constant
and t + dt = constant.
Generalizations The results obtained above may be readily generalized to
the case where there are a number
of~-fields
present.
In such
a case for each such field there will be an associ~ted T~v and a corresponding t~v.
The right-hand side of the Einstein field
equations will contain the sum of the T~v and this tensor will be related to the sum of the t~V by equations analogous to equations (6.6). In case the
~-field
is a spinor field a similar discussion
to that given above can be made.
The special relativistic
Lagrangian must first be generalized by replacing ordinary derivatives of the spinor field by covariant ones.
The variations
in the met·ric tensor may be performed by varying the generalized Dirac matrices which satisfy the relation
= 2g~V'
- 226 -
Taub
It should be pointed out that if we define the scalar
x
=
=
then equations (5.5) become 0
PA;o
ax
(7.1)
= - acpA
and A
,
cp '0
=
ax apA
(7.2)
0
These two equations are similar to the Hamiltonian equations for particles. Note, however, that
X is not
virtue of equations (7.1) we call related to
t
0 p
tp
p
nOr
4 t4 •
Thus if in
X the Hamiltonian it is
through the equations
x =
t P P
(7.3)
+ 3£
Thus the connection of the Hamiltonian with the stress energy tensor involves the Lagrangian function.
References 1.
D. Hilbert, Nachr. Ges. Wiss.
Gottingen, p. 395 (1915).
2.
W. Pauli, Theory of Relativity, Pergamon Press, London, p. 158, (958).
3.
E. Noether, Gottingen Nachtrichten, p. 235, (1918).
4.
L. Rosenfeld, Acad. Roy. Belgique,
5.
F. J. Belinfante, Physics 7, p. 887 (939).
~,
p. 6, (940).
- 227 -
Taub
LECTURE II A VARIATIONAL PRINCIPLE FOR PERFECT FLUIDS 8.
Co-Moving Coordinates The general discussion of variational principles given
above may be applied to the derivation of the equation of motion of a self-gravitating perfect fluid and the Einstein field equations for the case when such a fluid is the source of the gravitational field.
In order to make such an application
appropriate field variables function
£
¢A
must be chosen and a Lagrangian
must be specified.
We shall use as field variables
the rest density of the fluid temperature
S
p,
a variable related to the rest
of the fluid and a set of functions which
characterize a three-parameter congruence of curves, the world lines (particle paths) of elements of the fluid.
We shall
not vary these quantities arbitrarily but shall restrict the variations of
p,
the particle paths and the metric so
that the mass of the fluid is conserved under the variation. The use of co-moving coordinates will enable us to represent the congruence of particle paths and their variation in terms of the metric and its variation. field variables
¢A
Thus, some of the
are absorbed into the tensor g~v'
The
use of this special coordinate system greatly simplifies the calculations.
The resulting simplifications are one
implication of the fact that as a field theory relativistic hydrodynamics is special in that the equations of motion of the fluid are a consequence of the Einstein field equations and are not independent of them.
That is, in the case of a
- 228 -
Taub
perfect fluid, we are dealing with a situation where equations (5.5) are implied by equations (5.4). We shall be dealing with a one-parameter family of space-times with metrics
g~v(x;e).
In each space-time of
the family there is a congruence of curves determined by the solutions to the ordinary differential equations dx *~ as where the
= U*~ (x *;e)
(8.1)
are the labels assigned to events in the
space-time in an arbitrary coordinate system in which the metric has components
g
,',
~v(x
'/;
;e),
are the
and the
components of the velocity four vector of the fluid in this coordinate system.
They satisfy (8.2)
We may write the solutions of Eqs. (8.1) as (i=1,2,3)
( 8• 3)
where
are required to be the parametric equations of a hypersurface The four variables
~
i
,s
which we shall denote as
form a comoving coordinate system in each of the space-times. Eqs. (8.3) which may be written more generally as (8.4)
- 229 -
Taub
with xi x
= ~i,
4
= x
4
(~
i
(8.5) ,s;e)
may be regarded as the transformation between the
x
*
co-
ordinate system and a general comoving one which uses the x~
as labels for events.
Eq. (8.1) is then to be under-
stood as
*
,~
= u ~(x (x;e);e)
where In the partial differentiation the
xi
( 8• 6)
are kept con-
stant for when these variables are fixed a particular worldline is selected. In the general comoving coordinate system we have the components of the four velocity vector given by = u
as follows from Eqs. (8.6).
*0
*
dX~
(x)---wo
dX
Hence (8.7)
where
are the compoenents of the metric tensor in the
comoving coordinate system.
Eqs. (8.7) are a consequence of
Eqs. (8.5) and (8.2). We shall be using a comoving coordinate system in each space-time of the one parameter family of space-times with which we shall be concerned.
In a particular one of these
the metric tensor in the comoving coordinate system will be
- 230 -
written as
Taub
The tensor
g~v(xje).
( 8• 8)
x~
with
kept constant will measure the change in the metric
tensor evaluated in the comoving coordinate system at an event labelled by the coordinates parameter
e.
x~, produced by a change in the
Similar statements will apply to other tensor
fields which depend on Ul~
e.
In particular we shall have (8.9)
=
That is, the transformations given by Eqs. (8.4) for various values of
e,
produce comoving coordinates in each of the
space-times associated with that value of
e.
We shall use the notation
.*~(x *' je) V kept constant, where the V*~
with
(8.10)
= are the components
of a vector field in a general coordinate system using the labels
x *~ •
between
V'~
It is of interest to determine the relation and
V • To do this we define .~
(8.ll)
where
is given as a function of
tions (8.4) tion.
and
x~
x
and
e
by equa-
is kept constant under the differentia-
Since 6~ p
must hold for all values of
e,
it follows from the differentiatior
- 231 -
of this equation with respect to a
ae
ax v
(--rp)
ax
e
Taub
keeping
xll
fixed that
ax v a~*11 = - --r,;- -,-;;ax 11 ax P
(8.12)
= _a_~*v ax P
(8.13)
that a
ae
,\ v (~) ax P
From the transformation law of vectors we have *v * dxll = V (x (x;e);e)~. dX v On differentiating this equation with respect to xll
e,
keeping
fixed we find
V'
=
In virtue of Eq. (8.12) we may write this as =
(8.14)
=
(8.15)
where
*v with and is of course the Lie derivative of the vector V 1')J respect to ~ . It may be shown by similar arguments that for any tensor the operation of taking the prime derivative of the tensor compenents differs from the transform of taking the dot derivative by the appropriate Lie derivative of the tensor.
- 232 -
Taub
In particular for a scalar we have
f'(x;e)
(8.16)
=
where
f*(x*;e) = f(x(x *;e);e). For the metric tensor we have I
g aT
=
(8.17)
It follows from equation (8.9) that
~s
is to be expected.
If we apply equation (8.14) to the
four-velocity field, we find that U*v U*0 )Z" *
"'o;p
- ( g*vo - U*0U*v )U *
Z"
o;p'"
(8.18)
*p
Hence
= o
(8.19)
as a consequence of equation (8.18) and the fact that equation (8.2) holds.
9.
The Variations of the Field Variables The particle paths which are a three parameter C0nbruence
of curves, are described by equations (8.4). the
xi (i = 1,2,3)
In these equations
label a particular particle and
x
4
measures a "co-moving time" which is allowed to differ from
- 233 -
Taub
the proper time along the world line of a particle (cf. equation (8.5».
The vector field
defined in terms of the
former equations describes the variations in this congruence of curves.
That is,
if
x*p(x;o) is the description in the
starred coordinates of the position of and time when particle x
i
1+
is located at the comoving time
along an unperturbed path, then
x , as this particle moves
x*p(x;O) + e~*p(x;O) describes
the corresponding event as the particle moves along a perturbed world-line. measure a variation in the metric
The quantities tensor in the sense that
gOt(x;O) +
almost equal space-times.
eg~t(x;O)
describe two
In each of these space-times, the
particle paths, whether perturbed or not, are described by the curves
xi
= constant.
Because we are allowing general space-
time hypersurfaces to be described by the equations
x
1+
= con-
stant, a variation in
UV;
tte
gOt
produces a variation in
unit tangent vector to the particle paths. Equations (8.17) and (8.18) describe the variations produced at the event labeled by
of the metric tensor and
the four-velocity vector field when these tensors are varied in the comoving coordinate system and then evaluated in the x*p
coordinate system.
It should be noted that although f our
arbitrary functions, the
enter into equations (8.18),
the manner in which they enter is restricted so that equation (8.19) holds. We now turn to a discussion of the variations which we shall allow in the rest density
p and the rest temperature
®.
- 234 -
If
~(x;e)
Taub
is the rest density of the fluid and
U~
is its
four-velocity, then the conservation of matter is expressed by the equation
= __1__ (f:g p U~) f:g
= o
,~
(9.1)
In a comoving coordinate system when equation (817) holds we may integrate this equation to obtain 3 pf:g = vi44 Ho(x1,x2,x ,,e)
The function
Mo which is independent of x 4 measures x4 = constant between
the amount of fluid in the hypersurface the world-lines labeled by X3
and
x3 + dx 3 •
(9.2)
xl
and
x 2 and
xt + dx t ,
x 2 + dx 2 ,
The requirement MI o
=
0
is then the statement that for each of the space-times .
.
g
~v
(x;e)
and for every perturbation of the world-lines of the particles, the amount of matter in the region
de~cribed
above, is constant.
It follows from the results of the preceeding section and equation (9.2) that the requirement
MI
=0
is equivalent to (9.3)
In the
x*~
coordinate system, we have 1
'2
*0 *1 •
U
U
*
(gOl +
*
*
~O;l + ~t;o)
= o. (9.4)
- 235 -
The rest temperature
~
Taub
and the rest specific entropy
S of a fluid are defined by the equation i&:ls
where
p
= de: - ~ dp
is the pressure and
e:(p,p)
(9.5)
is the rest specific
internal energy given by the caloric equation of state of the fluid.
In this equation
ID(p,p)
is determined as an inte-
grating factor of the right-hand side of the equation and S
is then determined up to a constant of integration. It follows from equation (9.5) that if
functions of a parameter ®S'
=
p
and
pare
e, then
e:'-~P'
(9.6)
p
® as a field
Instead of considering the function variable we introduce another function
a
defined by the
equation (9•7)
In a comoving coordinate system 1
v.; lJ
we find that I'
= f[l=g(glJVgPO
va • g PlJ g)g
l:
A ]do +
lJV;P a
f lJV'glJV ;Y dl: t
l:
(15.2)
The requirement that SlJ l:
I' = 0
for arbitrary vectors
which vanish on the boundaries of
VI
and
V2 other than
requires that we consider the last term in this expression.
- 255 -
Taub
We shall evaluate it in the Gaussian coordinate system used in the previous section.
In such a coordinate system we have
as the non-vanishing Christoffel symbols
r~. 1J
=
-no1J.
=
k r.. 1J
=
{
..k },
1J
Hence we have ~l;l
= ~l ,1
~i;l
=
k
~ v, l-~kn.1
~l;i
= ~l , i
~i;j
=
~.
~
k
nk .
1
k
.
~k {.
1,J
. l-
1J
[1 n .. -
1J
The last of these equations may be written as
= to denote the covariant
when we have used the symbol
derivatjve with respect to the three-dimensional tensor
y ..
1J
We may then write
+
1
ij (~·I· 1
1
-
~ln ..
1J
» d~
The vanishing of this integral for arbitrary
~~
and
then implies that 1 1
11
li
ij n .. 1 1J ij 1 •
IJ
= 0 = 0 = 0 = 0
(15.3)
- 256 -
Taub
These are the equations which govern the behavior of the thin shell described by the tensor off the hypersurface
L.
1~V
which vanishes
We may use the results of the
preceeding section to determine the equations which determine 1~v
in terms of the geometry of the hyper surface
L.
If we
use equation (14.17) we may write equation (15.2) as I'
=
J(r ~v·g
~v
IY + [1Y(y
nij )g• .. ]}dL
ij'
g .. 1
lJ,
lJ
L If we impose equations (14.18) on the
g~v
II = 0
subject to these condi-
for arbitrary
.
. and
g~v
g~v,l
we find that
tions if equations (14.19) hold and =
(15.4
Equations (15.3) and (15.4) are the equations given by Israel [9] and Papaetrou
and Hamoui [lOJ for the theory of thin shells.
- 257 -
Taub References
1.
A. H.
2.
pp. 454, Approximate solutions of the Einstein equations for isentropic motions of plane-symmetric distributions _?Lp~d~ct fluids, f>hysical Rev. Vel Ie 7 II ~ t: I) ~r~~if-100 , Relativistic Rankine-Hugoniot equations, Physical Rev., vol. 74 (1948), pp. 328-334.
3.
, General relativistic variational principle for perfect fluids, Physical Rev., vol. 94 (1954), pp. 1468-1470.
4.
6.
Stephen O'Brien and John L. Synge, Jum~ conditions at discontinuities in general relatlvity, Communications of the Dublin Institute for Advanced Studies, Ser. A, no. 9 (1953).
7.
A. Lichnerowicz, Theories relativistes de la gravitation et de l'electromagnetisme, Masson et cie, Paris, 1955.
8.
A. H.
9.
W. Israel, Singular Hypersurfaces and Thin Shells in
10.
Relativit ,
General Relativity, II Nuovo Cimento, 44 (1966) pp. 1-14.
A Papapetrou and A. Hamoui, Couches Simples de Materie en Relativite Generale, Ann. Inst. Henri Poincare ~ (1968), pp. 179-211.
- 258 -
Taub
LECTURE IV FLUIDS OBEYING AN EQUATION OF STATE 16.
Equations of State In this lecture we shall derive a simpler variational
principle from which we may derive the Einstein field equations for a self-gravitating fluid that satisfies an equation of state of the form p
where
p
fluid.
= p(w)
is the pressure and
(16.1)
w is the energy density of the
That is, in terms of the quantities used previously
we have w =
where
p
p(c 2 + d
is the rest mass density and
(16;2)
e
is the rest specific
internal energy. We have previously made use of the fact that function e
= dp,p),
the caloric equation of state, describes the nature of the material with which we deal and serves to determine the temperature
e
and the entropy ®is
=
S by means of the equations de +
(16.3)
Hence for every fluid we may express the pressure as a function of two thermodynamic variables, say
wand the entropy
S.
- 259 -
Taub
Thus for every fluid we may write p
= pew,S).
06.4)
The assumption made above is that all thermodynamic variables are functions of one of them, say
w.
This assumption is
satisfied in case the fluid motion is isentropic, that is S
where
So
= S o
is a constant.
It is also satisfied in other
~ircumstances.
In the general situation when equation (16.4) holds the velocity of sound
ca
is determined by the equation 06.5)
= where
c
is the special relativistic velocity light and the
entropy is kept constant in the differentiation occurring on the right-hand side of this equation.
If there is a family of
flows such that the thermodynamic variables are functions of a parameter
e
as well as the coordinates,
then it follows from
equations (16.4) and (16.5) that p'
SI = a 2w' + (.~) as w
06.6)
When we restrict this family by the condition that SI
= 0,
we shall say that the family of motions is an adiabatic family (or that the perturbations which distinguish one member of the family from another member are adiabatic perturbations).
- 260 -
Taub
For such families we have (16.7)
17.
Integration of the Equations of Motion In case equation (16.1) holds we may derive from the
equations of motion of the fluid, TjJ\)
F;\I
=
(17.1)
0,
expressions for various components of the metric tensor in the comoving coordinate system in terms of the thermodynamic variables.
In this coordinate system we have
=
(17.2)
1 .rjJ -- u4
Ig 44
and
=
=
(17.3) Equations (17.1) are in general equivalent to the equations U\)
=
0
(w + p)UjJ;\)U\)
=
:f: \)
(w + P)U\)'\) + ,
W
,\)
(17.4)
and
,
(o\)
jJ
U\)U ) jJ
(17.5)
When equation (17.1) holds there exists a thermodynamic function
o(w)
defined up to a constant by the equation do o
dw
= w-+p
(17.6)
- 261 -
Taub
Hence equation (17.4) becomes (17.7)
In case equation (16.1) is equivalent to the statement that the entropy is constant we have
o
=
p
and equation (17.7) is the conservation of mass. In the comoving coordinate system equation (17.7) becomes
=
o
This equation may be integrated to give (17.8)
f(x i )
where
is an arbitrary function of the variables 4
xl, x 2 and x 3 but independent of
x •
If equation (16.1) holds ~ w + P
=
do + d(w + p) o w + P
Thus, in the comoving coordinate system equations (17.5) become (17.9)
=
These equations are identically satisfied when
~
integrability conditions may be integrated to give g'4 (~_1_)
o
~
44
,
,J
=
4.
Their
- 262 -
where the
F .•
variables
i
lJ
Jl
x ,x
2
Taub
are arbitrary functions of the three
= -F ..
and x
3
but not of
x
4
and are such that
= 0
F .. k + F' k . + Fk · . lJ, ] ,l l, ]
Hence we must have _0_, c.(x j ) + = w+p l
0
W"+P'
¢
,i (17.10)
where the
c· .l
are arbitrary functions of
be a function of
x
i
4
and
x
i
and
may
The solutions of equations
x •
(17.9) then become (17.n)
=
It is no restriction to take ¢
=
k
= 1
constant
for if these conditions are not satisfied we may make the coordinate transformation _4 x
=
-i x
=
where
The
x~
coordinate system is also a comoving one and in it
we have U. l
=
=
(17.12)
Taut.
- 263 -
=
IS.
=
°
(17.13)
w+ P
The Vorticity Vector c.(k j ),
The three functions
1
which enter into the
components of the metric tensor in the comoving coordinate system and which may be determined from the initial conditions satisfied by the motion of a self-gravitating fluid, determine and are determined by the amount of rotation in the fluid.
This
may be seen by examining the vorticity vector 1
=
;:g
ElJ\lOTU U \I 0,[
in the comoving coordinate system. v
v
k
1+
We have
(_o_)2
=
w+ k
=
(_0_)
(1S .1)
1
v"-g 2
1
W+P;:g
E
kij
c .. 1, 1 OS.2)
E
kiJ' ckc . . 1,]
It is evident from these equations that the necessary and sufficient condition for c..
I,]
that is
c.
1
vlJ = 0
c..
J,l
is that
= 0, In that case there
be the gradient of a scalar.
exists a comoving coordinate system In which.
= The world lines of the fluid particles are then orthogonal to the hypersurfaces
xl+ = constant In the comoving coordinate system.
In case the flow is isentropic, that is
S
is a constant, we
- 264 -
Taub
have o w+ P
1
=
where i
=
E:
+
.e.p
is the specific enthalpy of the fluid.
Further we have in
the comoving coordinate system =
1
These are the results obtained earlier [1].
19.
A Variational Principle In the comoving coordinate system used above, we may
use equation (17.13) to determine a thermodynamic variable such as
p
as a function of
g44' that is as a function
of the coefficients of the metric tensor.
Variations of the
metric tensor in the comoving coordinate system will then produce variations in the pressure.
Thus with the notation
we have used earlier we have
p'
=
as follows from differentiating equation (17.13) with respect to
e
in the comoving coordinate system.
This equation may
be written as (19.1)
Thus we have
p
as a function of
and an expression
- 265 -
Taub
for the variation of this function. Now consider the variational principle based on the integral (19.2) It follows from the results given earlier and equation (19.1) that I'
I" is the collision density
in M with respect to.n (in the sense m m
defined above). Note that if (x, p) for equals
a a (x (v),p (v))
is the phase orbit passing through
v = 0, then the expression d
(Tv
(40), evaluated at
(x, p, ),
fm (x(v), p(v)))v=O' a fact that is often useful.
The preceding considerations prove the following theorem. The distribution function
f
of a component of a (possibly heterogeneous) m gas satisfies Liouville's equation
L
(f)
m m
0
(41)
in a region where in
D C M if and only if there is detailed balancing every m D, i. e., if the average number of creations of particles of
that component equals everywhere in lations
( 1.)
( i )
D the average number of annihi-
.
Note that, in our terminology, even an elastic collision involves two annihilations and two creations.
- 329 -
Ehlers
Corollary
1..
( i )
If the particles of a particular species do not pa!:,
ticipate in any collisions in function satisfies, in Corollary
D, then the corresponding distribution
D, equation (41).
2. If the assumptions of the theorem hold, then
f is, in m
D, an integral of the motion defined by (18).
As an application
of the invariance (observer-independence) of
the distribution function, let us consider a radiation field ton gas with distribution function 4-velocity
f
r . Relative
as a pho-
to an observer with
u a , it is customary to define a specific intensity I~
of
the radiation field, as the limit of the radio "(energy of photons with frequency in d oJ and direction in solid angle normally through an area to
fr
dA/ (doJ
dndtdA)."
dn
passing in time dt
It is related (exercise)
by
(42)
Since
( 1)
V
2
rr )-1
J u a Pa I
'
the observer-independence of
f
r
im
For geodesic motion (e = 0), this assertion has first been stated by Walker (1936).
- 330 -
Ehlers
1-.,) / '\)~, a fact that is important, e. g., in cosmology;
plies that of
its direct, kinematical proof is somewhat cumbersome. If the photons are emitted by a source
S (galaxy, e. g.) and do
not interact with matter on their journey to the observer ville's equation
(41) for
fr
and
(42)
0, Liou-
give the important relation
Iv
s (I+Z between
Iv
5
(43)
)~
' "measured" near the source by a fictitious comoving
observer, and
I"
usual redshift of
o
' the intensity actually measured by O.
z is the
S relative to O. (43) is basic for the derivation of
observable relations in cosmology. Notice that the derivation just sketched holds in any spacetime, not only in the standard RobertsonWalker universes. If one assumes that the famous
3° K
"fireball"
radiation
emitted thermally from the recombination hyper surface (T~' in the early universe, one obtains from
(43)
was
3500 0
)
the predicted intensity
distribution in each direction in an arbitrary model universe, provided one can compute
z
from the null geodesics.
( t )
This idea was used by R. K. Sachs and A. M. Wolfe (1967) to estimate the influence of material "lumps"
on the radiation, and similar applications
have been made more recently. The same method has been employed by W. L. Ames and K. S. Thorne ( 1)
(1968) to determine the optical appearance
It is also assumed that no scattering occurred between emission and
absorption.
- 331 -
Ehlers
of a collapsing star to a distant observer. Several other applications of (41) have been made, particularly in cosmology and stellar dynamics.
6. Macroscopic fluid variables, balance equations, conservation laws. Let us rewrite (38) for a hypersurfa(;e to
X
is a hypersurface
Nm
whose projection in-
G. We obtain, using (34),
J f rJTt
[L]
f). {
G K
E
Kx
is that part of the mass shell
x In particular, the integral
(44)
m }
P
(x)
15"a m{
L
which is contained in Jfmpa 1tm }
gives
the
average total number of particles 1f the species considered whose world lines intersect
G. Here we have used the convention, to be maintained
J~.
throughout the remainder, that whole mass-shell
P
N
m
m
..
denotes an integral over the
(x). Therefore, the spacetime vector field
a (x): =
J
f pa
m
rc m
(45)
is the particle 4-current density of the respective species. It is always timelike and future-directed under our assumptions. (If we would permit to be a distribution,
m m
=
N a
m
could be lightlike in one particular case:
D, and theOre is no 4-momentum dispersion at any event).
- :1:l2 -
Ehlers
Similarly, a J :
e N m
a
is the electric 4 - current density In analogy with
(45)
T ab(x) m
(46)
of the species considered.
we define
f
a bf PPm T(m
(47)
as the kinetic stress energy momentum or matter tensor of the species. (If is possible to define a 4-momentum flux through hypersurface
G C X
and to show that
(47)
a
is the corresponding
4-momentum flux density, but this has no further use and is therefore not treated in detail here. ) We have assumed here, and will do so throughout these lectures, that
f
vanishes at infinity on P (x) so that integrals like (45), (47) m m exist. (Sufficient for this is exponential boundedness on P (x), as dem fined at the end of section 8.) Excluding the trivial case where
m
vanishes on P
m
(x) (and the
singular distribution mentioned below (45)) we infer from (47)
- 333 -
Ehlers
Lemma
5. If va is not spacelike and va
T
This lemma and
Lemma
ma
a
bVa v
b>
f
0, then
O.
(48)
theorem due to J. L. Synge
(i )
imply
6. Any kinetic stress energy momentum tensor is normal
(t)
i. e., admits a decomposition
T
ab m
jU
a b ab u +P
(49)
with
u u
a
a
u
a
- 1,
O.
(50)
can and will be chosen future-directed, and then (49) is unique. The physical meaning of
N a, T ab for a local observer in m m terms of 13-dimensional" quantities is obtained by evaluating (45) and
( 1)
See Synge
(1956), p.292.
( 2.)
See Lichnerowicz
(1955).
,
- 334 -
Ehlers
(47) in an inertial coordinate system at arbitrary observer at N 4 m -Nm '.
x:
is the number density 4 1Tf\,)
).4 ~ 30: m OIX ). ~ ';\ T rr! ~
-
Tm T
x. We obtain, for an
T
m
is the energy density,
X
=N
4
(m) is the momentum density, X ';) 4 {m\ ~xl4 = N m p ® v is the kinetic
m
4 aI
: infinity,
- 364 -
Ehlers
10) Stationary states, equilibrium, and thermostatics A gas given by
gab' Eab , fA
state in a region
D
sional local group
G
c:
X
is said to be in a stationary
if there exists, in
D, a one-dimen
of fixed-point free local isometries with
timelike orbits which leaves
F ab
and the
~a of
of the generating vector field
G
fA invariant. In terms
the last two conditions
can be expressed as
(103)
(104)
moreover , we then have Killing's equation
r?( a;b) =0
(105)
The last two equations imply
(106)
and similar statements for lows further that
Na
A'
Tab A
etc, Because of (105)
it fol
- 365 -
Ehlers
(107)
0,
i. e .• the entropy production is constant on the
Let us assume now that an a stationary state in '}
G-orbits.
adiabatically isolated gas is
111
D, and that the boundary of the world tube
of the gas is . G-invariant;
~C
D. Let
cross section of ~ and a E G. Then a(
~
E)
be a space like is again such a
E )] = = S rr]. Applying Gauss's theorem to the part of!J between E and a( L), using the adiabatic condition along the wall'dj- , and cross section, and because of the assumed stationarity S [a(
taking account of (101)
we obtain in '} Sa
;a
O.
(108)
This conclusion, combined with the expectation described at the end of the previous section, leads us to define: A gas. is in local equilibrium at x E X if, at x, Sa ~a = O. a The formula (100) for a summand of S;a shows the validity of the first part of the theoreme. If the collision functions positive almost everywhere
R'"
of a gas are all strictly
(w. r. t. the measure
a(.t:)p) 1tA 1\ . .. )
and continuous, then the gas is in local equilibrium at
x if and
- :1 (j G -
Ehlers
only if at
x AB f f ... fC...
(109)
whenever ~ p = 0, for all types of collisIons which occur; or, equivalently, if and only if for each particle species. on
P A (x)
there holds
o.
(110)
The second part of this theorem follows from the first part by means of equations (93) and The restriction
R::
point of view, since the
>0
(99). is not unsatisfactory from the physical
R - functions are usually analytic functions of
the momentum variables on the "collision fiber" Ap
=
0, and hence
they vanish only on sets of measure zero. The problem of finding the general continuous solutions ( fA' ... ) of (109)
has been solved for binary elastic collisions between Boltz-
mann particles, where
fl fl
A 13
(l09)
reduces to
whenever
PA+ PB = pIA + pIB·
(111)
- 367 -
Ehlers
In this case, the general solution is given by
(112)
and a similar formula for
Ja
fB and with
the same for both spe-
cies. (Chernikov
(1964), MarIe (1969) and, in the case where the 1 fA'S are assumed C, Bichteler (1965), Boyer (1965). The nicest
proof is that of MarIe, the shortest that of Bichteler.) If we consider elastic binary collisions between Bosons or Fermions
( or a mixture ) and assume that all factors in
(113)
fAfBf~f~
are positive on their mass-shells, we may divide by
and
__ I_ f - :±: 1 etc. the same relation as for Boltzmann parsA A ticles, so that we obtain
obtain for
_()( (x) - B (x)p e
Ja
A
Whereas it is easy to deduce from (111) provided that holds for some pair
pA
f
that
a
-+
fAfB
(114)
f
0 everywhere
PB' this does not seem so
- 368 -
Ehlers obvious in the case (113). Nevertheless I shall accept (114) as the general form of an equilibrium distribution at an event
x
for parjicles participating in some kind of .binary elastic collision. If particles in a gas undergo not only b-inary elastic collisions,
but in addition other kinds of reactions, then (114) and (109) show that the
OC A
must obey
Oi A
+ O(B + ...
for all permissible collisions With
(115)
A + B + ... ~ C +
(114) and (115) we have obtained the general local equili-
brium distributions (fA' fB, ... ). Since the fA'S have to vanish at infinity on the mass-shells, ,a(x) must be a future-directed timelike vector. We put
(116)
It is a straightforward matter to obtain from (114) the quantities a ab a a a PA' uK,u D defined in eqs. (45), (47),(97), NA , T A , SA' nA, (63), (64), (49), (52), respectively. Working in the rest frame of u a
rA'
one gets
(115)
- 369 -
Ehlers (116)
a b ab T ab = A (fA + p A) u u + p A g ,
( 117)
with the scalars (we omit temporarily the index
A) n,
f' p, s
by
( 118)
( 119)
( 120)
r
s --- 2 rc'l. m
These functions and further thermostatic relations obtained from them have been studied extensively; see, e;g., Landsberg and Du!:. ning-Davies (1965) and the references given there. The thermostatic meaning of the two parameters
oc..! is reco-
gnized thus: observe that
s = - Otn +
!r+2~1
f 00
m
log( 1+ ecJ.-/E)
f
E2_ m 2' E dE.
- 370 -
Ehlers
Transform the last term by partial integration and get, with (120),
s =Use (120)
(122)
and compute, again integrating by parts,
dp = }
(122)
Now,
and
r
where
(123)
T
give
r
d
(s, n)
(123)
dO(
=
! -1 ds + c:J., -1 dn
( 124)
is a thermostatic pot~tial, and
is the temperature
df
=
T ds + fdn,
and fA'iS the chemical potential
(per particle). Hence we conclude
T
-1
,
(125) can now be rewritten in terms of the
( 125)
fA's
and reveals
itself as the law of mass action. For the thermodynamics of mixtures
see Ehlers (1969), and
for applications of the preceding theory to cosmology see Ehlers
- 371 -
Ehlers
and Sachs
(1968).
Let us now investigate which restrictions are imposed the parameters 04.,
J
on
and on the mean velocity u a by the requi-
rement that there is global equilibrium, i. e., that there is local equilibrium at each event of a region theorem above, the functions
D C X. According to
the
(114) must then obey Lionville IS
equation; i. e. LA ( O(A + }apa ) = 0 in D.
This equation is ea-
sily evaluated (see, e. g., Ehlers (1969) and leads to the theorem. Global equilibrium requires that (a)
J
a is a conformal Killing vector and, if at least one comp~
nent of the gas consists of particles with positive rest masses, a Killing vector, and (b) 0(
the electric field strength
E: = F u a ab
b
is related to T and
by
T dot
= e E.
(126)
For a gas containing (also) ordinary particles (m> 0), equilibrium requires a stationary spacetime. Defining in such a spacetime a scalar gravitation;]] potential
ra. ~ = T
oj
a
by
IT
-
e 2U = 't:' 2 ~
e
U
in terms of the
Killingvector
we obtain Tolman IS law T
o T
(127)
- 372 -
Ehlers
and if E = 0, then
Ol = const., so that
r....
depends on the
p~
tential like the temperature. (For the general evaluation of (126) see Ehlers
(1959).)
It is possible to characterize the global equilibrium solutions
in a given, stationary spacetime by means of a variational princiEle in which
S is maximised under certain constraints, see MarIe
(1969) pp.107. For examples of equilibrium solutions, see Chernikov (1964). By means of (42) and (114) it can be verified that Planck's distribution law results for
rr = 2,
OC
r= 0,
as it should be;
OJ.t °
results from the relations (115), since there are always some precesses which change the photon number but not the numbers of the other particles involved (ex.: e-e collisions). A gas is said to be nondegenerate if the
+ I-term
can be ne-
glected without serious error, so that (112) holds. Otherwise,
it
is called degenerate. One consequence of the last theorem is that a gas with m>
°
cannot maintain an equilibrium distribution if it expands isotropical~
in contrast to an (m = OJ-gas (photons, nel1ltrinos). A physical
son for this deviation from the nonrelativistic behaviour of a (m gas will be given in the last section.
re~
0)
- 3n -
Ehlers
Since the thermostatic functions of a relativistic gas are explicitely known (cf. eqs. (118)-(121) ) one can compute, e. g.,
the
velocity of sound in such a gas, and one can check the validity of Weyl's condition with m
for shock waves. For a Boltzmann gas
0 this has been done in detail by Synge (1957), with the
result that the sound velocity increases monotonically with the c temperature and approaches the limit as T.--. 00
{3'
(the value for a photon gas); shock speeds are always less than c. Shock waves in a gas of Fermions or Bosons have been investigate by israel (1960). 11. Irreversible processes in small deviations from equilibrium; hydrodynamics. Whereas the equilibrium solutions of the Boltzmann equation can be written down exactly, there is not much hope to find rigo-
> 0) processes-in fact ;a no such (relativistic) solution is known at present. In physics, however, rous solutions describing irreversible (Sa
one is mostly interest in non-equilibrium situations. Therefore, in order to proceed one has to resort to approximations. We shall briefly describe such approximation methods in this sectivil,
and
refer to research papers for details. Our main goal here will be to indicate how one may obtain from kinetic theory a complete system of equations for thermo-hydrodynamics
which is sufficiently
- :17 4 -
Ehlers
general to include heterogeneous systems in which transport processes and reactions take place, by applying suitable approxima tions to the Boltzmann equation. Partly our exposition will program
be a
rather than the exposition of a completed theory.
For
simplicity I shall consider here only neutral fluids, thus in the se quel
"e A = Ja = F a b = 0 " .
Also, we shall only consider proces -
ses close to equilibrium, whicl: will (for most of the sequel) mean states which are infinitesimal perturbations (first order variations) away
from local equilibrium.
Two distributions fA' ~ will describe nearly the same
ma-
crostate of a gas if their moments in p-space are everywhere near ly equal. This will be the case if fA = fA (1 + f is a. e. bounded on meter
!'vIA
and the numerical
~A
"perturbation" para-
Eo is small. With this motivation, we shall now consider a
one-parameter family
fA ( £)
of states which is, for
€
local equilibrium, i. e., is such that for form
~"') provided
=
(114), with unspecified spacetime fields
s ha 11 deno t e by
l th e varla ' t'lOns fA
cal equilibrium functions"
dfA
--~
J
a
O(A'
I ~
lJa
=T
=0.
0 the OiA'
€
=
0, in
f 's have the A
,a'
and
we
Notl'ce that the "10
are independent
of £.
For "small" E , the moments computed by ,means of the "pertur~ ed distribution functions"
fA(O) + E. fl
A
will be considered to be the
macroscopic variables describing a "state close to equilibrium".
- 375 -
Ehlers
It is clear that the perturbed macroscip variables will satisfy
the conservation laws
0,
(128)
and similar ones, if we impose additional "scalar" conservation laws like
b-conservation. Also, we shall have the "Clausius ine-
quality"
~:
( 129)
=
Again we can write the decomposition (65) for the total, perturbed ab . I I a a I ab tensor T , wIth = rIO) + ,p = p(O) + Eo p, q = € (q ) • 1C = = E ( rc ab ) I, because of (117) for £ = o.
£f
f
Similarly,
( 130)
and a
S
=~u
a
a
+'6.
( 131)
- :376 -
Ehlers
with
n A = n A (0) + f n
E(~a)"
A,
i/
from (115)
and
f. (i }\a)',
~
for
(116)
=
~O)
+ £~',
~/
=
£=0.
It is a straightforward matter to derive from
(128) the e-
nergy balance eguation
{7,
where the kinematical quantities
O"ab' Ua' Wab
are defined
by
u
a;b
=W
ab
+<S"
ab
1 +3
-& h
ab
.
-uu a b' (133)
cr}=w =3"[ab] ab (ab)
a
6"
and are interpreted as the rate of rotation ( crab)' rate of expansion flow given by
)' :
=
0
(Wab), rate of shear
(-&), and 4 -acceleration
u a (see, e. g., Ehlers
Here and henceforth
a
(u a)
of the
(1961), Synge (UJ60) ).
we write
) ;
;a
a
u .
Also, one obtains the momentum balance equation
( 134)
- 377 -
Ehlers (r+p)
ua + hba (qb + p • b +T(cb;c ) + (Co\) ab+
(h ab
has been defined in (6"6).) Let us now assume that there are
t3' ) qb+ -.! ab 3
N is the number of species
(135)
a
Q conserved scalar quan-
tities. like b. which we call cq A' where where
~q = 0
1
~
q
~
Q
~
A of particles; the
Nand c qA
are given. constant "gharges". Then the reactions in the system are restricted by (136)
We assume the Q "vectors" (c
•...• c N) to be linearly indeq pendent. and denote by (I' ••.•• 1' N)' l~p~R: = N - Q a Pt (") p basis in the orthogonal space. The vectors (1', •... I' N) of the
qt
latter can be interpreted as (chemical or nuclear. e. g.) reaction coefficients. as is seen from the equations Na
A;a
=
~v I' p pA
(137)
L
P
which express the general solution of (136) in terms of the constants
r pA
and the reaction rates vp' giving the spacetime den-
sities of reactions of type p. From (130).
(t)
(137)
I;e .• LCqA r pA A
we obtain the particle balance equations
0
for
l"q~Q.
l~p~R.
- 378 -
Ehlers
(138)
Similarly, we rewrite (129), using
~
+ :S{}' +
-'ja 'a =
,
(131), as
~~o.
(139)
To proceed further we vary the expression (97) for the entropy current density
Sa; because of
hog(.l. sf
and
± 1)-1
(114) we obtain
T \\'
(r- IF-AnA)
( 140)
A
and at T~ =
(q
a'
-
~
""
£..fA
.a' lA ),
(141)
A
where
the
fA
are the chemical potentials defined in (125).
If we combine (140) with the thermostatic Gibbs relation
( 142)
- 379 -
Ehlers
which results from
by summing over the species
(124)
A,
and which holds for the unperturbed equilibrium functions (on the manifold
f s, n
j , .••
,n
A)}
of equilibrium states), we get
the rather remarkable Lemma 10. The perturbed
thermodynamic variables
f' s, n
A
satisfy
( 143)
where
F
is the thermostatic potential of the system (as detel
mined from the exact equilibrium relations of section 10), It is, therefore,
"reasonable" to use, for near-equilibrium
processes, the ordinary Gibbs equation of state for the perturbed variables, neglecting the error term in
(143), as we shall
henceforth. Also, we rewrite (141) for the perturbed variables:
( 144)
We also recall that, from
(122), the thermostatic pressure
Po associated with the perturbed state is
Po
Ts
-f+
2:fA A; n
A
(145)
- :380 -
Ehlers
there is no reason why p re
p
o
should equal the total kinetic pressu-
in (65).
We are now ready to derive an explicit expression for the entropy production density, ~, in terms of appropriate thermodynamic and hydrodynamic quantities. Compute ~ from (142)
for
the perturbed state, which is permissible because of Lemma 10; insert
f
using
from (132),
(445), (144)
nA
from
(138), and rearrange terms,
and the definition
( 146)
for the volume viscosity 1C , to obtain the entropy inequality,
-
T~
=
1'Cab
e-ab +'Jt~-+~a_ 4fA i~]
[ (log T), a + Ua ] + ( 147)
+LiA A
a
( rA,a + fAUa)
+~Ivp2:fArAp} ~ O. P
A
This expression has the usual form known from ordinary irreversible thermodynamics (see, e. g., de Groot and Mazur (1962) ); in relativity, it has also been worked Qut on the basis of phenomenological assumptions by several authors (see, e. g. Stiickelberg and Wanders (1953), Kluitenberg, de Groot and Mazur (1953), Kluitenbelfg and de Groot (1954), (1955).
- 381 -
Ehlers
We wanted to show that
(147) and the previous formulae
follow, in the sense we have specified, from kinetic theory, just as in the non-relativistic case; this does not seem to have been pointed out before with the generality we have retained here, The crucial fact is tliat equations (143) and
(144) follow
from the kinetic expression (97) for the entropy current, The expression "fluxes"
T~
as given by
and "forces"
')tab,1(""
(147) is bilinear in
D)-gases with respect to proper-
ty a) ot the theorem in section 10: A gas of the latter type behaves irreversibly if expanding isotropic ally, because of the term in (147); a photon gas, however, behaves reversibly, since
1'(-& Ta
a
=0
implies that 1t = 0 always. For more details concerning the transition from kinetic theory to thermo-hydrodynamics within the framework of relativity we refer to Chernikov (1964), and to the papers cited above. a The roles of temperature T, entropy S (or s) and of the main theorems of thermodynamics are completely clear within the framework of relativistic kinetic theory; there is no room for assumptions. (Of course, this changes if one wants to leave the domain of appli cability which we have delineated above.) In particular, integration of (129) over a section of a world tube of streamlines, bounded by two space like cross sections Land,"" , gives with the help of (131) i
L....l
- 334 -
Ehlers
and
~a =
w
a (~(l44)
T
S[
2:
t] - S
):
[L:J ~
J
a T- 1 w = rfuol.uf'> _ pgo«('> ,
ou rest la densite pro pre de matiere, f l'index du fluide, (6)
f .. 1 + i,
i l'enthalpie specifique. La temperature 0 et l'entropie S sont introduites par l'equation differentielle, df .. Vdp + OdS,
v ..
l/r.
Nous ecrivons l'equation de conservation de la matiere,
et celIe du tenseur d'impulsion-energie,
(9) ou q~ est Ie vecteur (transverse) courant de chaleur
(4),
- 409 -
Boillat
(10)
quo(=O,
(uuo(=l).
0(
0(
(7),(8),
Compte tenu de
les equations
(9)
s'ecrivent,
(ll)
(rfu.t.. - qo - 't~\~rJ..p + ru'fV,,(q./r)\
(12)
rOu.... fd.J
s
'"
= 0,
= V q«" + ul\uo(V of'> = V qO< - q i·v ur' , 01.. ,,"" • J.. r at.
-tf' = l(f' -
u"'ul" •
On ajoutera eventuellement les equations d'Einstein,
et on supposera que lion se donne les equations d'etat, par exemple, p
= p(g,r),
s .. S(g,r).
II reste encore a definir Ie courant de chaleur et nous etudierons plusieurs modeles dans les paragraphes qui suivent mais des a present nous pouvons obtenir des equations aux perturbations en faisant dans
(8),(11),(12),
~cA.
Ie remplacement,
~ «0< ~ .
VrJ..
'
Pour abreger nous poserons, U ..
u""!(.t.' Q ..
q,(~a(
,
C ..
t"t.8r·
Alors, r C{a(~ua( + U br .. 0, (14)
(rfU -
Q)~uf' - ,(a~O) - UQ. ~h " O.
(32) et (29) donnent ensuite, (37)
rQU2~S + 2U 2(q ~h + dq) + UQ. ~h + )..Icbo - U2(r+ q)(jr/r) .. O.
Enfin, ou1tipliant (30) par qr et tenant compte - d'apres (33) et (29) - de,
(38)
(U ;. 0),
- 413 -
Boillat
on obtient,
q2 = _ q qO< 0(
Nous avons maintenant avec (35) un systeme lineaire et homo gene de quatre equations pour les quatre inconnues 1, • (dO, 'dr/r, ~q, ~h). Nous ecrivons que Ie determinant est nul, FO
rF
(PO -)))c
F-q
Fh
0
- UQ
(PO -,.»Q
rp C + U(A - Q) r 2 rp Q + q U r
- A
2 q U - qA
JJa + rOSOU 2
( r 2OSr
2U 2
(2qU + Q)U
(40)
r
-r- -)q U2
" O.
8.- HYperbolicite. Existence de la surface d'onde A a O. II est necessaire d'examiner d'un peu p~es cette question afin de corriger une erreur que nous avons faite (10) et qui est d'ailleurs sans oonsequence pour la Buite sinon que les conditions qui seront imposees a F (28) ne Ie Beront plus par un motif d'hyperbolicite. Le systeme des equations du champ comporte 9 variables independantesl r, Q, u~(3 composantes seulement Bont independantes car u u~ " I). Q - Pg = - p)g/~r • Le coefficient de r( 1> - Pg )U 2 devient,
rQ(Sr~g - SgPr)/Pg +
f =
rQf)s~r)p +
()f(dr)p ,
f ..
en raison de la relation,
(64)
= fdr
dt
+ rOdS,
qui decoule immediatement de (7) (at rf = Pour ce qui est du coefficient de r( V-
- V p!Po .. )} ~)p ..
e+ pl. Po)c,
il slecrit,
~)p~~)p .. ~)p ,
en utilisant (50). On a ainsi (63). En vertu de (46),(47), Ie determinant se developpe facilemant at il reste apres mise en facteur de A et un calcul de coefficients analogue a celui que lIon vient de faire,
(65)
P(X,Y) = a 2X2 + 2b 2XY X = - U2/C,
=
• 0,
o
y. QU/C,
a 2 = 2q2/p +
al
+ a
if ~)r'
r(,*)p~)g
2b 2 = [#)rp
'if lij)r'
-
bl
-
= 2(11)0
2~)0' -
~)rp
c2 .. -
2/p,
,
Dans Ie repere pro pre (cf. 16), X = ~2,
y..
~q
n
•
On doit avoi;,
(66 )
o }q
=
0,
CprO
