Relativistic Fluid Dynamic (C.I.M.E. Summer Schools, 52)

C. Cattaneo ( E d.) Relativistic Fluid Dynamics Lectures given at a Summer School of the Centro Internazionale Matemat...

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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.

Printed on acid-free paper

Springer.com

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

<

~'

.t

r

Pfaff lineairement independantes. La consideration du repere propre est fort utile. En effet l'espace vectoriel tangent T (V) a une structure d'espace-temps de Minkowski Ie repere x

propre

V~

doit Hre identifie

a un repere galileenlocal ou Ie fluide a une vi-

tesse nulle. Si on connait les composantes d 'un tenseur t relativement au repere propre, ses composantes dans un repere quelconque (eod se deduisent des premieres par des formules de transformation connues. En effet si

(A~.)

est la matrice de passage du repere (e,) au repere

- 8 -

Pham propre (Vl.') et

(It'i)

la matrice inverse, on a CI

01

II

(3. 2)

A" " u

(3.2')

Ad. = utI

A·,~ = V(~r

0' -

"

A~

"

(.') - - Vel

Si t est un tenseur d'ordre 2, ses composantes t., dans le repere (eat) se deduisent de ses composantes t,:"., dans le repere propre par les

form~

les

(3. 3)

On a en particulier pour le tenseur metrique

Ainsi pour determiner l'expression du tenseur d'impulsion-energie d'un fluide pur, on le rapporte d'abord au rep ere propre. Le fluide y est rise par sa densite propre de matiere-energie "

caract~

son tenseur des pressions

partielles ff.'j' . Son tenseur d'impulsion-energie a pour composantes dans le repere propre

T,','

=,

Rapportons maintenant 1'espace-temps ~

T·", ,J

11'"" ~J

a des coordonnees locales xGl, on a

Q/~" ArJ. dx lil et l'application des formules (3.3) donne I

- 9-

Pham 0.5)

satisfait aux identites

(3. 6)

On voit que dans Ie cas d'un fluide pur, Ie tenseur d'impulsion-energie se decompose relativement a uti. une composante tempGrelle , u. u. et une compos ante spatiale

n.,.

en

Definition - On dit que Ie fluide est parfait si la quadrique des pressions dans Ie repere propre est une sphere i. e. si

1l't'jl = p ~~, j' , pest dite

pression scalaire du fluide. '1

..

Pour un fluide parfait, on a 1I'aI. = p .; A~ A', , soit en tenant compte de (3.2) et (3.4) 1l'G1,= P (gil, - uCil u, ). Ainsi Ie tenseur d'impuision-energie d 'un fluide parfait est donne par

(3.7)

Appelons repere principal en x Ie repere orthonorme (Wa,) dont chaque vecteurs W" est vectenr propre de la matrice (R•• ) par rapport a la matrice (gil,). Les directions definies par Wl ne sont autres que Ies direc tions principales de Ricci. Or en vertu des equations d'Einstein,

W~

sont

aussi vecteurs propres de la matrice (T",) relativement ala matrice (gil,). On peut exprimer les composantes de Tel,

a partir des valeurs propres

et vecteur propres comme S

(3.8)

TU' =

s, W(o)tJ, W(o..,-

f.i

s~ W(~ ,.W(~)~

- 10 -

Pham On voit qu' en general Ie repere propre d 'un fluide thermodynamique

cha~

ge est different du repere principal, sauf dans Ie cas d 'un fluide pur OU

pour lequel maniere

a

f =

s/l et on peut faire une rotation du

amener

( fl"."~ ~J"I.

de la matrice

§I.

3-plan espace

de

v.,1 sur W1..• s.1 sont alorslesvaleurs propres

LE FLUIDE THERMODYNAMIQUE

4. Le fluide parfait et les variables thermodynamiques. lJ..e tenseur d'impulsion-energie d 'un fluide parfait non conducteur de

chaleur est

(3.1)

II est clair que

u'

est vecteur propre oriente· dans Ie temps et

valeur propre correspondante de

la

(T,,_). Tout repere propre de ce fluide

coincide avec un repere principal qui est indetermine leur propre triple

f

a

cause de la

va-

~p.

En vue de I 'etude energetique, on decompose la densite propre " en la somme d 'une densite de matiere se

r

et d 'une densite d 'energie vitesse

r E OU t est I 'energie interne specifique

(4.2)

t=

r(1+f)

- 11 -

Pham On est amene

a introduire

l'indic e

f

(4. 3)

f

du fluide defini par

E.r

1 +£ +

Dans ces formules et dans la suite, les unites physiques ont ete choisies de

que la vitesse limite

maniE~re

il faudra remplacer E , p

Ec

par

-2

c

, pc

soit egale

a

1. Autrement,

-2

Le tenseur d'impulsion-energie d 'un fluide parfait non conducteur de la chaleur prend alors la forme

(4.4)

Du point de vue thermodynamique, la temperature propre I' entropie

specifique propre

S peuvent

~tre

e et

definis comme en

hydrodinamique classique par la relation

adS

(4. 5)

ou t t prenant

~

d£ + pdt'

est Ie volume specifique. En tenant compte de (4.3), et en

r

f, s, p

comme variables thermodynamiques (non independantes)

on peut ecrire (4.5) sous la forme equivalente

(4.6)

rdf

r9dS

dp

Ces relations expriment qu'il existe Iparmi les variables

r, 9, f, S, p,

seulement deux variables independantes pour lesquelles on choisira souvent

f

et S

ou

S

et

p.

- 12 -

Pham Si on prend donne

p

f

et S

en fonction de

I'

comme variables independantes et si l'on se f

et S, la relation (4.6) entraine

= ~p " 1f

r9

La premiere relation definit l'equation d'etat du fluide sous la forme

r

r (f, S)

et la seconde relation definit la temperature. Des conditions de conservation appliquees au tenseur d'impulsion-energie (4.4) on tire l'equation de continuite et Ie systeme differentiel aux lignes de courant

(4.7)

V. (rfu')

-

u"1,p

0

(4.8)

Tenant compte de (4. 6) on peut ecrire l'equation de continuite sous la forme

(4. 7))

d' ou l'on deduit Theoreme - Pour un fluide parfait, il est equivalent de dire qu'il y a conservation de la matiere ou que I' entropie lignes de courant.

est constante Ie long des

- 13 -

Pham

un fluide tel que De

m~me

ud,S = 0

est dit adiabatique.

tenant compte de I 'equation thermodynamique, Ie systeme

differentiel aux lignes de courant s 'ecrit avec les variables

f, S

(4.8') On en deduit TMoreme - Si Ie mouvement du fluide est isentropique Ie systeme differentiel aux lignes de courant se reduit

(S

const. )

a

(4. 9)

NOliS

montrerons que pour un tel fluide isentropique, il existe

un

principe extremal pour les lignes de courant

5. Le fluide visqueux. Pour caracteriser la deformation locale du fluide, nous introduisons Ia derivee de Lie du tenseur metrique nitaire

et nous posons

(5. 1)

g

suivant Ie vecteur vitesse u

~

14 -

Pham ou

rli~

est Ie projecteu[' d 'espace.

Les lois contraintes-deformations sont supposees lineaires, si Ie milieu est isotrope, les phenomenes de viscosite sont decrits par Ie tenseur (5.2)

011

(5.3)

(5.3')

Le tenseur d'impulsion-energie du fluide visqueux homogene est alors donne par

(5.4)

Cette expression est utilisee par C. Eckart, G. Pichon. A. Lichnerowicz proposirlt une autre expression de de Ia viscosite Ie vecteur devient alors

C"

~III'

011 intervient dans Ia definition

= fu" . Le tenseur d'impulsion-energie

- 1S -

Pham

avec

V designant

la derivee covariante dans la metrique

g=

f2 g,

6, Le fluide conducteur de chaleur, On tient compte des echanges thermiques par conduction, Celle-ci est definie par un vecteur pression de

qG(' sa

qO( orthogonal au vecteur

presence

uOl,

C'est l'ex-

dans Ie tenseur d'impulsion-energie

qui caracterise les points de vue, Eckart a choisi Ie tenseur:

(9,1)

Les equations qui regissent I'evolution du fluide sont donnees par les conditions de conservation du tenseur d'impulsion-energie, la conserva tion du courant de matiere, l'equation de definition de

'if«

(ruG)

(9,2)

et une equation thermodynamique,

0

qC(:

- 16 -

Pham L'auteur a propose en 1954 Ie tenseur d'impulsion-energie

(9. 3)

ou I 'on a neglige la viscosite

~

q«

est dMini par

(9.4)

Les equations du mouvement sont constituees par les conditions de c0E. servation

Vc( Till'

"0

et I 'equation thermodynamique est remplacee par

I 'equation de conduction qui generalise celle de Fourier

(9. 5)

C

est la chaleur specifique

a volum constant et ( la chaleur de dila-

tation du fluide. P.ichon a repris ce modele en ajoutant Ie terme de viscosite

811/, .

Pour Landau et Lifchitz,

Ie tenseur d'impulsion-energie d'un flui-

de conducteur de chaleur est identique vecteur conrant de chaleur

a celui d 'un fluide parfait,

qlll apporte sa contribution

tion de conservation d 'un certain vecteur

(9. 6)

(9.7)

POI . 11 pose

a travers

Ie 1'equ~

- 17 -

Pham Les equations du mouvement sont donnees par

VIII pOi

0

auxquelles ont ajoute l'equation de definition de

q. =

(9. 8)

01'

G

-ore

2

•

qlll:

•

l+G

(g" - u,.u )I,( -9- )

est la tiDnction de Gibbs definie par

G =E+

(9. 10)

.£. - 9 s. r

Ces modeles se justitlent par des considerations physiques et cinetiques et ont tops Ie merite de se reduire

a la

limite

a la

description

classique non relativiste. L'etude du probleme de Cauchy montre que les les systemes d 'equations auxquels ils donnent lieu, sont mixtes et com portent une partie parabolique provenant soit de la viscosite, soit de la definition du vecteur courant de chaleur

qO(. Ce qui conduit

a une vi-

tesse de propagation infinie. Pour

lever

la difficulte provenant de

qllL' Cattaneo et Vernotte

ont suggere de modifier I 'hypothese de Fourier par un terme de tion. Kranys

(9. 11)

a fait la traduction de cette hypothese en

Relax~

relativi8i~

- 18 -

Pham C e vecteur

qat n I est alors plus orthogonal

de vue de Landau- Llfchitz Mahjoreb

a

u at. Adoptant Ie point

et celui de Cattaneo- Vernotte-Kranys,

a propose une nouvelle theorie cette annee.

- HJ -

1'ham

~).

LE CHAMP ELECTROl\TAGNETIOUE

10. Representation du champ electromagnetique. S'il existe un champ electromagnetique, Ie fluide est soumis inductions electromagnetiques qu 'on peut decrire

a I' aide

des

de deux

formes: la 2-forme champ electrique-induction magnetique 2-forme induction electrique-champ magnetique

a

et

la

In,

IG

H

G. On note

leurs formes duales au sens de l'element de volume riemannien

~

de

l'espace- champ. On a en composantes

(10. 1)

(10.2)

On appelle vecteurs champs et inductions eleclriques et magnetiques les vecteurs definis par les

(10.3)

ou

i

u

e

=

i H

u

1-formes

h

diG u

i (. G)

u

b

i (lH)

u

est Ie produit interieur par Ie vecteur vitesse unitaire

u. En

composantes, on a

(10.4)

e

Ces vecteurs sont orthogonaux

a

Inversement, H, G, *II, *G

u"'. s'expriment en fonction de e,d,h,b

- 20 -

Pham par les formules

(10. 5)

H

uAe-.(uAb)

(10.6)

•H

u" b + ~u Ae)

G

Dans les deux dernieres relations Ie signe une

vari~t~

riemannienne I 'op~rateur

(uAd) - '(UAh)

+ provient du fait que sur

* satisfait

a larelation

.. 2

= Eg (-l)P (n-p) oil n = dim V ,p = degre de la forme et £g

Ie

signe du det. g. On en cteduit les relations suivantes qui donnent lascorn composantes

En theorie

electromagn~tique

de Maxwell, les inductions

d~pendent

lineairement des cilamps. Dans Ie cas isotrope oil Ie fluide a une per-

mitivit~ dielectrique (1 0.7)

et une permeabilite magn~tique ", on a

1

d =1 e

b =}'-h

Les deux relations (10.5) donnent alors

(10.8)

G

1

'"

~,,-1

,,-

H + - - uAi

u

H

- 21 -

Pham soit en composantes

(10.8')

qu 'on peut mettre sous Ia forme

(10.9)

ou

L'induction electromagnetique

H, G

satisfait aux equations de

Maxwell

(10.10)

dH

a

(10. 11)

~G

J

ou ~ est Ia codifferentielle,

June

I-forme dont Ie vecteur asso-

cie dHinit Ie courant electrique. L'equation (10.10) signifie que Ia 2-forme I-forme

H

+

est Iocalement exacte i. e. qu'il existe Iocalement une

telle que

H = d

+. +

s 'appelle Ie potentiel vecteur eIec-

tromagnetique, (10.11) donne en remarquant que

(10.12)

~J

a

~2 = a

- 22

~

Pham equation qui exprime la conservation du courant electrique. En composantes, les equations

(10.10), (10.11), (10.12) s'ecri-

vent

On decompose Ie courant electrique tion colineaire

J

en un courant de convec-

a u et un courant de conduction

r orthogonal a

r peut Nre ctefini par l'hypotMse d 'Ohm r =, e ou

u.

~ est la conduc

tivite electrique du fluide. On a alors

(10.13)

t s'appelle la

densite de charge.

11. Le tenseur d 'energie electromagnetique. A partir de electromagnetique

HGI~' 'ru~

Gill,

on construit Ie tenseur d'impulsion-energie

dont la divergence donne la densite de force e-

lectromagnetique agissant sur Ie fluide. En generalisant un resultat con nu dans Ie cas non inductif ), Minkowski

=". = 1,

on obtient Ie tenseur donne

par

- 23 -

Pham (11. 1)

Pour interpreter ce tenseur, on va 1'exprimer teurs

a 1'aide des vec-

e, d, h, b:

On obtient

ou (11. 3)

Pet.

est Ie vecteur de Poynting et

Q" = ~~PCI . Sur (11. 2) on voit Ia

signification de chaque groupe de termes. 'r~~

n 'est pas symetrique. On peut prendre 1'expression proposee

par Abraham

(11. 4)

't'cl~

On peut penser sont obscures.

a Ie

symetriser [

1,

mais Ies raisons physiques

- 24 -

Pham Nous conservons l'expression

(11.1). En prenant la divergence

de ce tenseur, nous avons

(11. 5)

Or les equations de Maxwell du {r groupe s 'ecrivent encore

Par multiplication contractee avec

En portant dans de

Clll' , il vient

(11. 4), on a alors en tenant compte de la definition

J

(11. 5')

On peut transformer la parenthese en utilisant les equations de liaison

ce qui donne finalement

(11. 6)

011

- 25 -

Pham La signification du groupe taire (~f-1)V~u' Pf si 1".~

J'H"

est claire. Le tenseur supple'men-

).~= 1 oft P_ = 0, c'est-a.-dire

est nul si

ess symltrique. II est encore nul si

uCl

est un champ a.

d'riv'e covariante nulle. Le terme suppltmentaire

(e~ et'),l+

}-

+ h.h' ~~ .... ) correspond aux phenomenes de magnetostriction et d 'electrostriction. En fait k,rdependent des variables d'etat.

12. Cas ou 'rill, defini par

Le tenseur se est

(1", -1)

-r.,

(UGI

(11.1)

I

est symetrique.

n'est pas symetrique en general. Son antisymetri-

p~ + u, POI ). Comme

naux, I 'ant)'symetrise est nul

i. e. que

Uti

'r_,

~~= 1,

cas non inductif

2. -

p. =

ce qui est verifi' soit que

,

I

Dans Ie cas non inductif, on prend F«~

Pel

sont orthogo -

est symetrique si:

1. -

0,

et

1. =JA-=

e.

= 0 soit que hot = O.

1 , alors

HC(~ = GcI.~ =

. Le tenseur d'energie electromagnetique s'ecrit

(12.1)

et

(12.2)

Le cas

e..=O correspond a. celui de la magnetohydrodynamique ou

- 26 -

Pham fluides de conductivite rt = ()(). Comme Ie courant electrique doit Nre borne ~ e< IJO, on a necessairement

e

= 0 . Le tenseur d'impulsion-

energie electromagnetique s'ecrit alors

Le cas

h. = 0

conduit a

11 convient pour la description d 'un electron considere comme une boule continue. Dans chacun de ces cas, Ie tenseur d'impulsion-energie du fluide s'obtient a ajoutant

't'0(~

a l'expression deja connue. On obtient un

tenseur d'impulsion-energie du fluide qui est symetrique, par suite on peut ecrir

les equations d 'Einstein.

13. Le fluide parfait charge sans inductions. On a dans ce cas Ie tenseur d'impulsion-energie total

(13.1)

ou

Les equations du mouvement sont donnees par les conditions de

- 27-

Pham conservation

(13. 2)

I 'equation thermodynamique

(13. 3)

r

9d

S

rdf - dp

et les equations de Maxwell

o

(13.4)

(13. 5)

Ou supposons la conductivite i = 0, de sorte que

J'

=

1u ~

et

(13. 6)

En prenant

f, S

tions de conservation

comme variables thermodynamiques, les condi(13. 2)

donnent I 'equation de continuite et Ie

systeme differentiel aux lignes de courant

(13.7)

(13.8)

- 28 -

Pham On en deduit que si Ie mouvement est adiabatique

(uCl 0« S

= 0),

il y

a conservation de matiere

(13.9)

De

(13.6)

et

(13.9)

diOli par difference

on tire

ultOct log

1.r

= O. Le rapport

.lr

est dOIlC cons

tant Ie long des lignes de courant, on pose

(13.10)

K =

1. r

Le systeme differentiel· aux lignes de courant se met alors sous la forme

(13.11)

- 29 -

Chapitre 2 LE PROBLEME DE CAUCHY Pham Pour chaque modele de fluide, on obtient un systeme fondamental d 'equ~ tions aux derivees partielles pour etudier l'evolution du fluide. Un probleme essentiel est de voir dans quelles mesures ces equations

deter~

nent les fonctions qui representent les grandeurs physiques envisagees. Comme c 'est un probleme d 'evolution, Ie probleme mathematique pose es. Ie probleme de Cauchy. Les donnees initiales portees par une hypersurface

E

de 1 'espace-temps determinent - elles les grandeurs

dans Ie voisinage de

z..

Le seul theoreme general c1assiquement connu en reponse a cette question est Ie theoreme d'existence et d'unicite de Cauchy-Kowaleski valable dans Ie cas analytique pour un systeme de derivees partielles

a

N

equations aux A

•

N fonctions inconnus dont Ie polynome caracter2,

stique n 'est pas identiquement nul. L'hypothese d'analy1idte restreint considerablement la portee de

ce

theoreme en Physique. On peut maintenant se passer de I 'hypothese d'analyti.dte pour des systemes quasi-lineaires hyperboliques stricts. Pour de tels systemes, Leray a demontre un theoreme d'existence et d'unicite pour Ie probleme de Cauchy non analytique. Toute solution de ce probleme possede un domaine d'influence, c'est-a-dire que la valeur en un point ne depend que d'une partie des donnees initiales, celles se trouvant

a l'interieur d'un certain cOnoide de sommet ce point. C'est

cette notion d'hyperbolicite stricte et son critere que nous allons exposer en vue de I 'appliquer aux differents systemes d'equations trouvees dans Ie chapitre

1.

- 30 -

Pham

t LTHEOREME POUR LES SYSTEMES

I

D'EXISTENCE ET D'UNICITE STRICTEMENT

HYPERBOLIQUES

1. Systemes strictement hyperboliques. Soit

V

n

une variete differentiable de classe

grand) et de dimension Soit

a(x, D)

un operateur differentiel d'ordre m agissant sur les a

depend des coordonnees locales

polynome reel en ~ de degre tie principale de a (x, ~). Soit

x"

et

JeT' (V ) , a (x, ~) est un x n m. On designera par h (x, ~) la paE.

de( . Ponr

des derivees partielles

I 'equation

(k suffisamment

n.

fonctions. Localement

de

Ck

a (x, ~ ) c'est-a-dire la partie homogene de degre V (h) x

h (x, ~ ) =

Ie ctme projectif defini dans

o.

Definition 1. - L'operateur differentiel

a (x, D)

T*(V) x n

m

par

est dit strictement

si I 'hypothese suivante est verifiee: n (H) Il existe dans T* (V ) des points ~ tels que toute droite isx n sue de ~ et ne passant pas par Ie sommet du cOne V (h) Ie coupe x en m points reels distincts.

hyperbolique au point

xEV

S'il en est ainsi, l'ensemble des points ~ demi-cOnes convexes bords appartiennent a

~ appos~s

forme l'interieur de deux

. r+ (a)

non vldes

et

x

A(x, D)

x

dont

1es

V (h). x

Considerons maintenant un operateur differentiel gonal

r- (a)

suffisamment differentiable en

x

matriciel

dia-

- 31 -

Pham

A(x, D)

ou les

a.(x, D)

sont des operateurs differentiels d'ordre

1

m(i).

Definition 2. - On dit que l'operateur differentiel diagonal est strictement hyperbolique en un point 1) les

a.(x, D)

x

si

sont strictement hyperboliques en

1

A (x, D)

x

2) les deux demi-cOnes convexes apposes

r

+(A) x

=

n 1

r+(a.) x 1

r-(A) X

n 1

r-(a.) X

1

ont un interieur non vide. Pour definir l'hyperbolicite stricte dans un domaine

n.

C+(A)

(ouvert con-

r +x (A).

V, introduisons Ie cOne dual du cOne C+(A) n x x est ferme de l'ensemble des vecteurs XET (V) tels que <: f ' X > " 0

nexe) de

pour tout

4E r+(A). x

x

Le cOne

C- (A) x

dual de

n

r x-(A)

est defini. de ma-

niere analogue. SoH

Cx (A)

(1. 3)

Un chemin

r:

[0,1] .... Vn

temps relativement de

r

est dans

a

A

+

C (A) x

U Cx- (A)

differentiable est dit oriente dans Ie

si la demi-tangente positive en chaque points

C:(A). Une hypersurface differentiable

r;

est dite orien

- 32 -

Pham tee dans I 'espace relativement gent

en chaq~e point

T (E)

x

a

A

si Ie sous expace vectoriel tan

x

de

1:

est exterieur

Definition 3. - On dit que I 'operateur hyperbolique dans tin domaine

n C Vn

A(x, D)

a

CIA).

est strictement

si les deux conditions suivan

tes sont satisfaites: 1) A(x, D)

est strictement hyperbolique en tout point

x

611 x o ' xi

2) I 'ensemble des chemins temporels joignant deux points

quelconques de

.n

est compact

ou vide pour la topologie de la

convergence uniforme de l'ensemble =

Si

{~:

[0, 1] ~V n ' r(O) = Xo

xl} •

A(x, D)

est differentiable en

x

et si

A(x, D)

est strictement

hyperbolique en un point

x.' on peut montrer qu'il existe un

ge ouvert connexe

Xo hOm€l)Illorphe a une boule de Rn

lequel

,~(~)=

A(x, D)

..n

de

voisin~ge

dans

est strictement hyperbolique. Un tel ouvert sera

dit

simple.

2. Systemes de Leray. Considerons un systeme d'equations aux derivees partielles de equations

N

a N inconnues (u i ) et n variables (xIII) qqe l'on ecrit

symboliquement

A(x, u, D) u + B(x, u)

(2.1)

ou

A(x, u, D)

et

B(x, u)

est une matrice diagonale d'elements

une matrice colonne d'elements

des operateurs differentiels d 'ordre Associons

0

a.(x, u, D), i 1

b. (x, u). Les 1

= 1, ... , N

a.(x, u, D) sont 1

m(i).

a chaque inconnu u i un entier s(i} )., 1 et a chaque equ~

- 33 -

Pham un entier

tion de rang

t(j)

m(i)

(2. 2)

les entiers

s(i) , t(j)

~

1 tels que:

s(i} - t(i) + 1

ne sont definis qu 'a une constante additive

pres. Definition. - On dit que Ie systeme diagonal neaire au sens de Leray si pour tout a. (x, u, D)

les relations

(2.2)

sont verifiees et si les a., b.

suffisamment regulieres de

ts de

i, l'operateur differentiel

est lineaire par rapport aux derivees d 'ordre

1

Ss(i) - t(j) , si

d'ordre

est quasi-li-

(2.1)

.

x·,

de

1

m{i}, si

sont des fonctions

1

.

uJ et des derivees des

s(i} - t(j) < 0

a.

et

1

b.

1

sont

uJ

independa~

uj .

Ceci etant, Ie probleme de Cauchy pour Ie systeme se de la maniere suivante. Soit

n

un domaine simple de

donnees de Cauchy sur une hypersurface tent en les valeurs des fonctions

E

traces sur

~

plongee dans

V. n

12

Les

consiste

wi

admettant des deri -

s(i} + 1 de casses localement integrables dont

E sont

p~

u ~, de leurs derivees d 'ordre

< m(i) . II existe (toujours) des fonctions

vees d 'ordre

(2.1) se

les

les donnees de Cauchy envisagers.

Une solution du probleme de Cauchy pose est alors une solution (u~)

de

(2.1)

dont les derivees d'ordre 's(i) sont de carres locale

ment integrables et coincident su

E avec

celles de

i

w. Pour ce

bleme, J. Leray a demontre un theoreme d'existence et d'unicite

pr~

que

nous allons enoncer sans demonstration. Theoreme. - Si les donnees de Cauchy sur des fonctions

i

w

verifiant les hypotheses:

1:

sont definies

par

- 34-

Pham 1)

dans

n

l'operateur differentiel

L

et l'hypersurface

A(x, w, D)

est hyperbolique strict

orientee dans l'espace relativement

a

A(x, w, D) 2)

les

a.(x, w, D)w + b.(x, w) 1

derivees jusqu 'a l'ordre Alors pour tout

s'annulent sur

1

1:

t(O - 1.

x E1:, Ie probleme de Cauchy pour

au moins une solution dans un voisinage de sont deux solutions et si

ainsi que leurs

iii

et

u

i

x. Si

(2.1)

(n i )

et

admet (u i )

ont des derivees d'ordre

~ s(i)+l

de carres localement integrables, alors elles coincident. Definition. - Un systeme quasi-lineaire qui verifie les hypotheses precedentes sera dit un systeme quasi-lineaire strictement hyperbolique, ou systeme de Leray. Les systemes quasi-lineaires qu 'on rencontre en Physique ne sont pas toujours des systemes diagonaux. Pour appliquer Ie theoreme de Leray, il faudra les ramener

a la forme diagonale. On sait qu 'on Ie

peut toujours. Mais Ie caractere strictement hyperbolique doit

~tre

de

montre. Nous allons exposer la methode dans un cas .

• ~. APPLICATION AUX EQUATIONS DE L'HYDRODYNAMIQUE DES FLUIDES PARFAITS

3. Les coordonnees harmoniques. Dans l'etude du probleme de Cauchy relatif au systeme des equations d 'Einstein, les coordonnees harmoniques sont un outil precieux. Definition. - Un systeme de coordonnees locales harmonique si chaque fonction coordonnee

(x')

est dit

x fest une solution de l'e

• 35 •

Pham quaiion de Laplace

(3. 1)

ij~

etant les coefficients de la connexion

riemannienne.

On remarque que les varietes caracteristiques de tangentes en chaque point au cOne elementaire C Si

(x')

x

(3. 1)

sont

de l'espace.temps.

est un systeme de coordonnees locales, on pose

(3.2)

F'

dependent de

g.~

et de leurs derivees premieres. Si

F'=

0,

Ie systeme de coordonnees locales est harmonique. On associe

a

F'

les quantites

LCII~

definies par

(3.3)

Lemme. - Dans un systeme de coordonnees locales arbitraires, les composantes du tenseur de Ricci peuvent se mettre sous la forme

(3.4)

ou

les

F.~

etant des fonctions regulieres.

- 36 -

Pham II suffit pour demontrer Ie Iemme de chercher dence Ies derivees du second ordre de On a modulo

soit

D'autre part

soit

On en deduit

Ies termes en

g1,. et

gl}!l

dans

'6r g1....

a mettre LG!~

en e.vi-

et

Rcr~'

- 37 -

Pham puis Ie lemme. Corollaire. - En coordonnees locales quelconques, on a

(3. 6)

ou S (h)

(3. 7)

OI~

R (h) ~~

1 R (h)

'2

gct~

(3. 8)

avec

R

(h)

g

harmoniques

4. Application

~.. R(h)

~I"

K.~ =

et

L = g1J"

L1-,..,.

Si les coordonnees sont

O.

a I 'etude

des solutions des equations d 'Einstein.

Theoreme. - Toute solution du systeme des equations d'Einstein

(4.1)

est une solution en coordonnees harmoniques du systeme

(4.2)

(4. 3)

- 38 -

Pham Inversement toute solution du systeme une hypersurface

1:

(4.2), (4.3) satisfaisant sur

orientee dans l'espace aux conditions

(4.4)

(4.5)

est une solution du probleme de Cauchy correspondant pour Ie me

(4.1)

des equations d'Einstein.

En effet toute solution de niques verifie

(4. 2)

car

(4.2)

L«~ =

ecrite en coordonnees harmo-

0 et comme solution de

Ie doit verifier les conditions de conservation

,,,S II~ -_

n'

Reciproquement considerons une solution de pondant

a des

donnees de Cauchy satisfaisant sur 't'

(4.4) et (4.5). Sur"" on a

et (4.5)

0

SCI!

=

S

(h).

cI

+

• K«

0

(4. 1), eli. e.

()

4. 3 .

(4.2), (4.3) corres-

E

aux conditions

et en vertu de (4.2)

on a

"-

K' - 0

En explicitant l'expression de

sur

I

K~ , on a

soit en tenant compte de la condition (4.4) qui entraine sur

=0

syst~

I , '~F'

- 39 -

Pham K' = G(

Co:nme

'()o

F'

l:

2..2 g"t

gOO '1 F~ = 0

II,

est oriente dans 1'espace , g"

fa,

on a necessairement

= O.

Ainsi la solution consideree de

(4.2), (4.3) satisfait sur

(4.6)

Pour cette solution on a encore

6)n en deduit en vertu de

(4. 7)

Or

donne par derivation

(4. 3)

V~K~,lt'

= 0

r.

a

- 40 -

Pham II en resulte que l'equation

(4.7)

peut s'ecrire

(4.8)

ou

,

A1,-

Ainsi

sont des fonctions regulieres.

F'

satisfait

a un

systeme lineaire hyperbolique qui ad -

met un tMoreme d'existence et d'unicite. Si

l:

pace, la seule solution de (4.8) qui satisfait sur

'ao

F'

=

0

est oriente dans l'es

t a

F'

=

0

et

est la solution nulle. 11 en resulte que la solution consi-

cteree est une solution ecrite en coordonnees harmoniques des equa tions d'Einstein

(4. D) •

5. Analyse formelle du systeme fondamental de I 'hydrodynamique. Le systeme fondamental des

~quations

de I 'hydrodynamique des

fluides parfaits adiabatiques est forme des equations

(5.1)

(5.2)

(5.3)

(5.4)

dp

rdf - redS

Pham On prendra f r(f, S)

et

et

S

p=p(f, S)

comme variables thermodynamiques sont alors des fonctions connues de

C 'est un systeme de fonctions inconnues

16

r = f

et

S.

equations aux derivees partielles aux 16

g«" f, S, uct. On va d 'abord faire une analyse for-

melle du probleme de Cauchy. Pour cela on se donne sur une hypersurface

L d'equation

et on cherche posons que

I.

x lS = 0, les valeurs des

locale

a determiner

la solution au voisinage de

1:.

n'est pas tangente aux cOnes elementaires

(5. 5)

et que sur

g•• , ~,gOl' ,S

gH

Z on

f

Nous suE.

i. e.

°

a

(5. 6)

Vne etude classique

montre que si

gO'

f

0, les quantites

S'III sont connues en fonction des donnees de Cauchy Les donnees de Cauchy

(g."

gO!, ,

1. gill,

'a o gill, ' S) doivent alors verifier les

conditions de compatibilite

S'CII =

X (rfu'11a! - pg'jI()

Supposons provisoirement connus les valeurs sur tions precedentes s 'ecrivenil

(5.7)

1: de

f. Les equa-

- 42 -

Pham En tenant compte du caractere unitaire

(5.7)

donne alors la valeur de

(5.3)

de

uel,

on tire

u'

(5. 8)

pui celle de

(5. 9)

L'equation de

(5.9)

definit implicitement la ou les valeurs possibles

f. On va 1'ecrire

En derivant par rapport

a f,

il vient en tenant compte de

soit d'apres (5.7)

On voit que

fest connue sur

1'.

si

F'f

f

0 i. e.

(5.4)

- 43 -

Pham

g Dne fois

f

(JD

-

(1 _

connue sur

fr ,.; ) u 'u·.,tr l:,

0

on en deduit

ul

a I'aide

de

(5.7)

si

Ceci etant en vertu des raisonnements faits au paragraphe

4

cedent, on peut substituer au systeme (5. I), (5.2), (5.3), (5.4) Ie

pr~

s~

steme

(5.10)

(5.11)

(5.12)

(5.13) oli (5.12), (5.13) proviennent des conditions de conservation Vo(TIII~= 0 et (5.3). On remarque que (5.13) entraine

- 44 -

Pham

ce qui montre que si

u~ est imitaire sur

Z,

i1 Ie reste au voisina

Z.

ge de

8upposons que les donnees de Cauchy sont donnees en termes de series

formelles par rapport aux coordonnees locales et cher

chons les solutions formelles

du systeme (5.10), (5.11),

(5.12), (5.13).

O"glll,' 1,8, ~.f, l,u'

En mettant en evidence les derivees dans ces equations, il vient

(d. c. )

(5.14)

u'1,8

(5.15)

(d. c.)

(5.16)

(5.17)

oil. les seconds membres sont connus en fonction des donnees de Cauchy. (5.14) uO

f

donne

0, (5.16)

et

~ gil'

si

~

=

i

donnent

1. u'

si

g" u' f

0; (5.15)

donne

00 8

si

~= 0 determinentalorsO,u',?,f

pour

(5.17)

si Ie determinant du systeme pour

gel;.

o.

(1 -

1...Q) u' u' r

f

0; enfin (5. 17)

- 45 -

Pham Ainsi sous la condition que:

uOf

0

100 glll~' ~o S, 'a, f

on peut calculer

et

'aou·

. Les

m~mes

conclusions

s 'etendent aux derivees d 'ordre superieure qu 'on obtient en derivant par rapport

a x O les differentes equations, de sorte que les series

formelles cherchees sont uniquement determinees. Si Ie probleme de Cauchy est analytique, il resulte du tMoreme de Cauchy-Kowaleski Ie resultat suivant. TMoreme. - Dans Ie cas analytique; si les donnees de Cauchy

'ao g"P'

S satisfont aux conditions

get~

1)

la forme

2)

I 'hypersurface lement par

3)

sur

gCIIP'

Z on

a

XO

XCIX' est hyperbolique normale

1:

F' =

portant les donnees de Cauchy et dMinie loca0, est orientee dans I 'espace 0

et

So.

=

t T'"

alors Ie probleme de Cauchy pour Ie systeme des equations de I 'hydr~ dynamique admet une solution analytique et une seule au voisinage de tout point

x E Z.

Varietes caracteristiques. - L'etude precedente montre que les tes caracteristiques du probleme de Cauchy sont dMinies par les tions suivantes

vari~

equ~

- 46 -

Pham

Les premieres definissent les ondes gravitationnelles , les secondes les ondes materielles ou entropiques et les troisiemes les ondes hydrodynamiques. Un calcul classique montre que ces trois sortes des se propagent avec les vitesses

v =

1, 0, v

d'o~

respectivement avec

~ ;fr t r~

.L:.f ~ 1 pour que v 4- 1 r ce qui entraine que les varietes caracteristiques definissant les on Sous I 'exigence relativiste on doit avoir

des hydrodynamiques doivent Hre orientees dans Ie temps.

6. TMoreme d'existence et d'unicite. Reprenons Ie systeme etudie au paragraphe precedent

(6. 1)

(6.2)

(6.3)

(6.4)

- 47 -

Pham Il ne se presente pas sous forme diagonale sauf pour Nous Ie transformons de maniere Gardons pour Ie moment Pour avoir 1'equation de

(6.4)

en

(6.1)

et (6.2).

a obtenir un systeme diagonal.

(6. 1) et (6.2)

tels quels.

en f, prenons Ia derivee contractee

V,

il vient

f, 1 en uol

ou Ia notation au

2e memerre signifie que Ies termes non explicites

contiennent des derivees d'ordre inconnue . Pour avoir

maximum

indique pour chaque

uC(V~ V. u~ , considerons I'equa.tion (6.3) qui

se developpe comme

en Ia derivant suivant Ies lignes de courant -et utilisant 1'identite de Ricci

(VatV~ -V.~)

u'

= - RoI ,

En portant dans 1'equation en

u',

f, on obtient

(6. 5)

1 en

S, 1 en

on a

f, 1 en

ull )

- 48 Pham

Appliquons l'operateur

2 en

S, 2 en

r a cette

urV

f, 2 en

equation

ud }

Or l'identite de Ricci donne

(6.6)

si on tient compte de

et l'equation erl

(6.7)

f

on a

s 'ecrit finalement

appliquons l'operateur

(1 -

F (3 en gCII~' 2 en

S,

u~ considerons Ie systeme (6.4) auquel nous

[g't~

-

(1 -

lrfi ) uCll u'1 V. V.

:i )u" u'l u V~ V,), - [

f

=

f, 2 en u GI )

Pour les inconnues

[g"':

= 0,

t r! }uclu.] u1 ~OlPr f [ g'" - (1 - r;t

2 en

f

ud. ~el S

rvef,

gd.' - (1 -

lrl"t

}u· u'] (l~ u' ul ).

- 49 -

Pham

=

q (3 en gOl~' 2 en

en tenant compte de cette relation l'equation en

=

Or

ur~,s = 0

r(3 en gee"

2 en

S, 2

en

f, 2 en uCl)

u 1. devient

S, 2 en f, 2 en

ucc )

entraine (6.6) de sorte que on obtient finalement

Nous avons ainsi trans forme Ie systeme Ie systeme diagonal suivant.

(6.1), (6.

2~,

(6.3), (6.4) en

- 50 -

Pham

(6.11)

2 en

f, 2

u at )

en

(6.11)

2

ou

en

f, 2

s 'obtient

(!6.9)

en

u«

a partir de (6. 1) gr~ce a l'expression de R (h~~.

Ordonnons l'ensemble {g",~, S, f, ulll }

en Ie numerotant de 1

a

16.

Nous avons avec les notations evidentes

m(~~)

2

0,1,2,3

m(11) = 1

m(12)

AS80cions aux inconnus a(N)

=

~IIlP -(1

-

et aux

!;f )ullll]

Associons aux inconnus

~quations

l?pr

3

les indices suivants m(N)

= 3

N = 13, 14, 15, 16

et aux equations les indices suivants

· 51 -

Pham s(d~)

4

s(11)

3

s(12)

t( ~ ~)

3

t(11)

3

t(12 )

3

s(N)

3

t(N)

et dressons Ie tableau de l'ordre maximum des derivations et celui des differences

~

0

s(i) - t(j):

0

0 ~

l

~

l

t

,

0

l

0

0

0

0

4

0

0

0

1

)

$

!

t

t

3

~

t t

ordre maximum des derivations

matrice (

s(i) - t(j) )

Nous voyons que l'ordre maximum des derivations ainsi que l'ordre des operateurs differentiels

a(i)

est compatible avec Ie choix des indices.

Nous avons ainsi montre que Ie systeme

(6.9), (6.10), (6.11), (6.12)

est un systeme quasi-lineaire au sens de Leray. Montrons que c 'est un systeme strictement hyperbolique. Les donnees de Cauchy sur calculees au • 5 mes, pour

n

1:

sont par example les series formelles

(il suffit de prendre la somme des

n

premiers ter-

convenablel. Leurs derivees d'ordre 's(i)+J

sont mani

festement de carres localement integrables. Nous supposons de plus que les donnees d~ Cauchy 1)

(gtA~, 0, g •• ' S) sur

la forme quadratique

gel~

x·

X.

1.

satisfont aux conditions:

est hyperbolique nor-male

- 52 -

Pham 2)

sur

Z

on a

F' = 0 ,

S'"

nit une valeur admissible de

= XT·. f

; la

seconde relation defi

11 ~ 1.

telle que

r

. r Avec ces donnees on peut rechercher les deml-cOnes A l'operateur Ie demi-cOne

r x ( ~)

A I 'operateur ne

r

+

a(~~) correspond Ie cOne I(

a( 11)

dMini

f

-

+ x

l' uftolt.tr f

(ai). 0

1

et

par

correspond Ie cOne

uti.

5«

o

et Ie demi-cO

+ (11) defini par x

Aux operateurs

a(12)

et

a(N)

correspond Ie cOne

dMini par

(6.12)

Sous I 'hypothese

r

11"1 , nous voyons que l'intersections de ces + (N) n'est autre trois demi-cOnes r + (cl~), r+ (11), + (12) = x x x x que Ie demi-cOne (6.12) qui a un interieur non vide.

r

r

Nous avons donc prouve que l'operateur differentiel hyperbolique strict en chaque point en

x

A

=

(a.) 1

est

x. Comme il est differentiable

, il est strictement hyperbolique dans un voisinage ouvert con-

- 53 -

Ph am

U de

nexe

x. L 'hypersurface initiale

dans I' espace relativement sur

t

a

A, il reste

des derivees d'indices

0

de

!

a ete choisie orientee

a preciser

~"

les valeurs

S, f, uGC

d'ordre

~

s(i) -1,

soit:

,1,s,

S

?.,S

f, 'a,f, ?'of

ot

U

g",., ~gll"

t5

~.I\ ,110 U,

'00 Ud

S ont ete donnes dans Ie probleme de Cauchy formel du

"g.. ,S , \S, f, ~f,

de ce probleme, pour avoir de deriver les equations

u"',

~ou· sont obtenus par la resolution ~,g."

100 S,

'blOf,

~"l', il suffit

(6.1), (6.2), (6.3) et (6.4).

Les hypotheses du theoreme de Leray sont donc satisfaites. On en deduit que Ie probleme de Cauchy pose pour Ie systeme non analytique (6.9), (6.10), (6.11), (6.12) au voisillage du point II reste

a montrer

x

admet une solution unique

(gGl,' S, f, uel)

,1: .

que cette solution est solution du systeme initial

(6.1), (6.2), (6.3), (G.4) . Or si les donnees sont analytiques, la solu tion

(&i~, f, S, uGl)

est analytique et est necessairement la solution

- 54 -

Ph am analytique du premier probleme. Si les donnees ne sont pas analytiques, on approche Ie systeme par des systemes analytiques et dans chaque cas, la solution du se cond probleme de Cauchy est solution du premier probleme de Cauchy. Par passage

a la limite, la solution du probleme de Cauchy de

Leray est encore solution du systeme initial. Et nous avons demontre Ie theor!me d'existence et d'unicite pour Ie probleme de Cauchy non analytique relatif au sisteme fondamental des equations de 1'hydr~ dynamique relativiste des fluides parfaits.

- 55 -

Pham

Chapitre 3 L'HYDRODYNAMIQUE RELATIVISTE DES FLUlDES PARFAITS ISENTROPIQUES

tL

LA FORME DIFFERENTIELLE INVARIANTE

1. Systeme diffhentiel aux lignes de courant. Le fluide parfait est dit isentropique si I' entropie Dans ce cas

(cf. Ch. 1,

.4)

S est constante.

Ie systeme differentiel aux lignes de cou

rant s 'ecrit

(1. 1)

ulA

uotv.cl u~ - (gIA" ~ -u u, )

(1. 2)

ou

s

~f - 0 -f- -

est l'absciae cuririligne sur les lignes de courant; fest l'indi-

ce du fluide. On a vu que

Nous nous proposons de mettre en evidence les proprietes geometri ques du mouvement du fluide.

2. Variation d 'une integrale. Soit

V

n

une variete differentiable de dimension

Ie fibre des vecteurs tangents en tous les points de

n ,'/'(: T(V ) ..... V n

V

n

et

n

D(V )-+V n

n

- 56 -

Pham Ie fibre des directions tangentes. D(V)

de dimension

n

T(V) n

2n - 1. Comme

est de dimension

T(V) n

et

D(V) n

2n

et

sont loca-

lement triviaux et qu 'on peut choisir comme cartes locales des cartes locales induites par celles de point de

T(V)

point de I' ouvert

U

Soit gine

(x~, X"') ou (xlt) est un

de coordonnees locales de

d'un vecteur tangent en

coordonnees (xCII) ; un point de (x", u·) ou

n

est defini par I 'ensemble

n

composantes

V, en coordonnees locales un

D(V) n

V

n

et

(XCi()

les

(xU) E U relativement aces est defini par I 'ensemble

uC( sonLa, parametres directeurs de la direction.

C: [to, ta -

xo= X(tD)

Vn

une courbe differentiable dans

et d'extremite

x~

V n d'ori-

= x(t.). En coordonnees locales,

elle est definie par la representation patarnetrique

(2.1)

On posera

(2.2)

Cette courbe se releve dans T(V) n

et si

x-(t)

Soit

n'est pas nulle pour tout

(2.3)

t, en la courbe

r dans

n

n

F(x·,

L: t_(xOl(t),x"(t) D(V). n

F: W(V ) -+ Rune fonction a valeurs scalaires donnee sur

T(V ) positivement homogene de degre tout

en la courbe

x

fixe, F(x,

x')

lX)

=

1 F(x, X).

est une fonction de

t

1 par rapport a A toute courbe

X i. e.

C: [to, t.] -

et on peut calculer 1'integrale

pour Vn'

- 57 Pham Cette integrale est en fait intrinsequement attachee

a

F

et

C

et ne

depend pas de la representation parametrique. Calculons la variation de cette integrale pour une variation quelcon-

a extremites

que

non fixes de

C. En supposant que

C

est dans un

ouvert de coordonnees locales, on a

dt

soit d'apres un raisonnement classique du calcul des variations

Sx > dt

(2.4)

ou

est la

fA)

1 furme definie sur

T(V) par n

(2.5)

et

P

un covecteur defini en composantes par d

(2.6)

les

ill Pc(

C

~

?F

xii - 'fX"

ne sont autres que les premiers membres des equations d'Eu

ler du calcul des variations. Si

~F

ix

est un vecteur.

n'est pas dans un ouvert de coordonnees locales, on la re -

couvre par un nombre fini de cartes locales et on peut etudier la varia tion de l'integrale

qui est la

m~me.

- 58 -

Pham 3. Principe d'extremum pour les lignes de courant. Appliquons les resultats precedents au cas OU

Vest I 'espace-

temps et

(3. 1)

On a

a calculer

les variations de l'integrale

(3.2)

supposee evaluee Ie long d 'un courbe

C

orientee dans Ie temps.

On a

fg", x.

~

~F

~g ').,. x~~,.,.

to ~U d

fgd~ x~

dt

Vg)." ~x'"

PCI(

~F

, --

~fg~r x·~·r 1 0.ilg~r x~ x + Zf

x"'

Vg,.~ x~ x'"

'hOi

~dfg~~ ~p ~r

+ -} f ~g~r ~, ~r

Vg,.,.. x" x,..

t

quelconque, prenons l'arc S de courbe com .J dx~ me parametre. Le vecteur x'" = est alors Ie vecteur vitesse unitai ds re, de sorte que l'on ait par un calcul facile

AUf lieu d'un parametre

(3.3)

(3.4)

Pat

=

111." II. (I f [ u 'd. u~ - (g, - u u~

'atl/

]

- 59 -

Pham On obtient ainsi Ia formule

~S

(3. 5)

=(W,

~x>~

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<;ons

a

nous dans I' espace temps de Minkowski rapporte

coordonnees orthonormales. rivers clont les

uel

est Ie vecteur vitesse unitaire d'u-

composantes sont determinees classiquement

tir de la vitesse d'espace

•

~

des

si

,

a par-

est prise comme unite. Un

calcul facile donne la metriql'le associee que nous ecrivons sous la forme

(14. 1)

1 - V2

+ --2 1 - ~

(~;dx~)

2

Cette metrique est du type hyperbolique normal. II est changement de l'ordre dans la signature au passage de Ie met en evidence en choisissant l'axe

}

parallele

V

a noter un 2

=

112

I'

.

On

a .., (vitesse

du fluide), On a ainsi

(14.2)

que l'on peut mettre sous la forme canonique par une decomposition en carres . Si

V2

f ~2

, on obtient

J'

2 2 2 1-~ 2 [ V-~ D (1-V)~ I - 22 --2- dx + 2 dx V 1-' 1_ ~

-f

-

2 2 (1-~)V V 2 _ ~2

- 82

-

Pham

2 et I 'on voit que pour V ~, rR2 on a la signature + - - - et pour 2 2 V < ~ la signature - + - - . Pour V2= rII. 2., on obtient

qui reste de signature

+ - - -

A partir de la metrique associee

Ie theoreme en prenant Harc

x·

remplacer dans (13.5) 2 + (dx') . On en tire ainsi: dt

d

Vi -

(14. 1)

cherchons

a exprimer

du rayon comme parametre. On a a ~I d~ ov d.1 = (dx) 4 2 par ~~ AI = dx + (dx 2):2 + ~

\

O_V2) (

I

H

,d)

v2.. ~a

V2 _,,2./. ' d onne r T 0 ,cette re IatlOn

S1·

/1. 2 W 2 1-"12 -(1-r)

Si on interprete

.. V

2

(1- V ) (1- W

~ 2 fi A)

comme vitesse absolue et

~

W

0

comme vitesse reJafuB

de propagation de la lumiere , on a manifestement

(14. 3)

-2 ....P + 2W·. .;. ...P + (W.... . -~ )- - W11'1] P

W +

2.

....

On verifie par un calcul direct que cette relation reste valable dans Ie cas

V 2= ~ 2 . C 'est la formule relativiste de la composition des vites

ses. Il est aise

de verifier qu 'on peut la mettre sous la forme

- 83 -

Pham

Nous obtenons ainsi

a partir du principe de Fermat une demonstra -

tion de la loi relativiste de composition des vitesses.

- 84 -

Pham

BIBLlOG RAP HIE

M. ABRAHAM - R.C. Circ. Mat. Palermo 22 (1909), 27-35 N. L. BALAZS - The propagation of light rays in moving media. Jour. Optical Soc. Amer. 45 (1955)

[3]

C. CATTANRO - C. R. Ac. Sc. Paris, 247 (1958), 431-433

[4]

Y. CHOQUET-BRUHAT - Fluides relativistes de conductivite in finie, Astronautica Acta, 6 (1960), 354-365 Y. CHOQUET-BRUHAT - Etude des equations des fluides charges relativistes inductifs et conducteurs, Comm. Math. Phys. 3 (1966), 334-357

[6] .. [7] [8J

MM. KRANYS - Il Nuovo Cimento, 50B, (1967), 48-63 LANDAU -

LIFCHITZ - Fluid mechanics, Pergamon, London (1958)

J. LERA Y - Hyperbolic differential equations - lnst. for Advanced Studies, Princeton (1953) , notes mimeographines A. LICHNEROWlCZ - Theories relativistes de la gravitation et de l'electromagnetisme, Masson, Paris (1955) A. LlCHNEROWICZ - Relativistk hydrodynamiqBeand magnetohrdr) dynamics, Benjamin (1967) -

[lil

B. MAHJOUB - CR. Ac. Sc. , Paris, 247, (1968), 668-671 268A (1969), 1440-14l:2.

et

C. MARLE - Sur l'etablissement des equations de l'hydrodynamique des fluides relativistes dissipatifs. Ann. lnst. Henri Poincare vol. X,n.2, (1969), 127-194 C. ECKART - The thermodynamics of irreversible process, Phys. Rev. 58, (1940) PHAM MAU QUAN - Sur une theorie relativiste des fluides thermodynamiques, Ann. Mat.. Pura e appl. (4), 38 (1955) PHAM MAU QUAN - Etude electromagnetique et thermodynamique d'un fluide relativiste charge, J. Mech. Anal. 5 (1956), 473-538

- 85 -

Pham

[i~

.

PRAM MAU QUAN - Inductions electromagnetiques en relativite generale et principe de Fermat, Arch. Rat. Mech. Anal., 1, (1957) PRAM MAU QUAN - Thermodynamique d'un :t1w.ide relativiste, Boll. U.M.I. (1960) (3) vol 15, 105-118. PRAM MAU QUAN - C. R. Ac. Sc. , Paris, 261 (1965) 3049-3052 PRAM MAU QUAN - Magnetohydrodynamique relativiste, Ann. Inst. Renri Poincare 2 (1965), 21-85 PRAM MAU QUAN - Sur les equations des fluides charges inductifs en Relativite generale - Rendiconti di Matema tica, 1. 2; vol. 2 Serie VI (1969) G. PICRON - F,tude relativiste des fluides visqueux et charges, Ann .. lnst. Henri Poincare, 2 (1965) 21-85 AH.

.

TAUB - Relativistic hydrodynamics, Arch. Rat. Mech. Anal. 3 (1959) et in Lectures in Applied Mathematics, 8 (1967)

CENTRO INTERW\ZIONALE MATEMATICO ESTIVO (C. 1. M. E. )

ONDES DES CHOC, ONDES INFINITESIMALES ET RAYONS. EN HYDRODYNAMIQUE ET MAGNETOHYDRODYNAMIQUE

RELATIVISTES

A. LICHNEROWICZ

Corso tenuto a

Bressanone dal

7 al

16 Giugno

1970

ONDES DES CHOC, ONDES INFINITESIMALES ET RAYONS EN HYDRODYNAMIQUE ET MAGNETOHYDRODYNAMIQUE RELATIVISTES

Introduction. On sait 1'importance mathematique et physique prise recemment par la magnetohydrodynamique relativiste. Dans ce cadre, on s'est propose d'etudier en ctetailles ondes infinitesimales et les ondes de choc. L 'extension relativiste des conditions de compressibilite de Hermann Weil joue dans la theorie developpee un rOle important. On a introduit systematiquement un instrument mathematique commode (les tenseurs-distributions) par 1'analyse des differentes ondes. Cet instrument est d'abord applique en hydrodynamique,

a l'etude des ondes soniques

et des rayons correspondants. En passant au cadre de la magnetohydrodynamique, on etudie ensuite successivement Ie systeme differentiel fondamental, la structure des ondes magnetosoniques et des ondes d 'Alfven, les rayons correspondants, enfin les ondes de choc qui font 1'objet d 'une analyse detaillee. On etablit en particulier grs.ce

a 1'introduction d 'une fonction d 'Hugoniilt convenable, un important

theoreme d 'existence et d 'unicite par les solutions non triviales des equations de choc. En traitant les ondes magnetosoniques par Ie formalisme des ondes de choc, on met en evidence Ia non-invariance de la direction des rayons par I 'operateur de disci.ntinuite infinitesimale. Ces le90ns qui se suffisent

a elles-m~mes represent une synthese de

certains de mes travaux durant la periode 1966-69 - Tenseurs-distributions II

- Hydrodynamique relativiste

III

- Les equations de la magnetohydrodynamique relativiste

- 90 -

Lichnerowicz IV

- Ondes de choc en magnetohydrodynamique

V

- Fonction d 'Hugoniot et orientation des ondes de choc

VI

- Ondes de choc et ondes d 'Alfven

VII - Vitesses des ondes de choc et theoremes fondamentaux VIII - Retour aux rayons magnetosoniques.

- 91 -

Lichnerowicz

1. TENSEURS-DISTRIBUTIONS ET DISCONTINUITES

I, Tenseurs-distributions sur une variete riemannienne, a) Soit V

une variete differentiable orientee, de dimension

ch+1n(h~0);

n

et

classe nous disposons sur V d'une metrique riemannienne 2 n h ds de signature arbitraire et de classe cC (O~ k~h). Localement: 2 ds =g

Cl(j

dx

r::I.

()

dx P

(01,

Si T estU sont deux p-tenseurs, nous notons (T, U) de T et U au point x de V, n

(3 =0, 1"", n-l), Ie produit scalaire

x

En coordonnees locales:

0/1. , , CXfi' (T, U)

x

= T

()( I' ,

(x)U

a

SoH D(p, V ) l'espace des p-tenseurs n

V, n pos~r

(x)

'O(p

h

support compact de classe C

sur

Si T est un p-tenseur localement sommable arbitraire, nous pouvons pour U E D(p, V ): n

(1. 1)

ou y

(1. 2)

est l'element de volume riemannien de la variete, Localement:

1J,='VAr. jgl

0

dx ;\ .. , ;\ dx

n-1

,

Un p-tenseur-distribution T de Vest une forme lineaire continue, n

a va-

leur scalaires, sur l'espace D(p, V l. Continu est ici entendu au sens nusuel en tMorie des distributions, Si U E D(p, V ), nous designons par 1f[UJ n

- 92 -

Lichnerowicz ou (T, U> la valeur pour U du tenseur-distribution

T,

Un p-tenseur ordinaire localement sommable T de V

n

peut Nre

identifie avec un tenseur-distribution au moyen de la formule (1. 1). Le tenseur-distribution est note T D , ou parfois T par abus de notation, quand aucune confusion n 'est possible, b) Si T est un scalaire-distribution et V un p-tenseyr ordinaire,

TV

est Ie p-tenseur distribution defini naturellement par:

TV[UJ = T [(V,

U)]

Cela pose, soit.f2. Ie domaine d'un systeme de coordonnees locales (xo() et donnons-nous dans

(l, nP scalaires distributions T 0(1' ,

(1. 3)

T=T "'1" '"

0<'1

dx

®,"

'O
L'expression

~

® dx P

'0(

P

0< definit un p-tenseur-distribution: si (eo(.) est Ie repere dual du corepere dx : 0( l'

U=U

, ,0< P eo(1 ®, , .8> eo
et T[U] est donne par la somme:

Inversement tout p-tenseur distribution T dans.n. peut Nre rapresente par une telle expression: designons par T

0(1' , ,O(p

definis par:

les scalaires-distributions

Lichnerowicz

To(

I'

(f) = T

[f eO( ® .. , ® ~ 1

"o(p

P

f€D(O,fi)

]

On a: oll' ,

U

=T

,0(

P

o(l'''~

et T

admet bien l'expression (1. 3), Ainsi, dans Ie domaine d'un systeme

de coordonnees locales, tout p-tenseur-distribution peut

~tre,

comme un p-

tenseur ordinaire, rapporte aces coordonnees, les composantes etant des scalmrres-distributions, Nous supposons maintenant h, k ~2, c) Soit T un p-tenseur ordinaire, connexion riemannienne, Si

\7 T sa derivee covariante dans la

U e D (p+l, V )

n

=

!v

V0 T

V

n

J

0(1"

PO<"I' , ,c(, p "l

U ,0(

p

soit, par integration par parties:

<'VT,U)

Le premier terme due sec-ond membre est nul. On introduit ainsi I 'operateur

~ de coderivation sur les (p+1)-tenseurs, definispar:

- 94 -

Lichnerowicz Ainsi la formule precedente s 'ecrit:

.( V"T , U>

(1.4)

=

QU >

Cela pose, on definit naturellement la derivee covariente d'un p-tenseur distribution T

comme Ie (p+1)-tenseur distribution "T determine par

la relation (1. 4) ou

U E D(p, V ). Il est aise de voir que toutes les pron

prietes c1assiques de la derivee covariante dans une connexion riemannienne, ainsi que les formules correspondantes, demeurent valables pour les tenseurs· distributions.

2. Les distributions -/, Y- et

D relatives a une

shypersurface.

a) Nous considerons maintenant exc1usivement un domaine.o. de V correspondant

a des

coordonnees locales. Soit

definie par l'equation locale deux domaines no et

c:P> O.

f

=0

('P de

qui vaut

f,

Y

n 1 correspondant

respectivement

a

D- en

cp.( 0 et

ou plus brievement yO (resp Y~ ou (;) la fonction surD.

1 (resp 0) dans

finissent dans

n

une hypersurface reguliere

classe C 2) qui partage

Nous notons par 1/0 Ie gradient de

Soit

L

n

Q0

et 0 (resp 1) dans

n;:

ces fonctions de-

des distributions designees par les m~mes notations,

par l'intermediaire des formules:

(2. 1)

b) Considerons la c1asse des (n-l)-formes coverifiant la relation:

(2.2)

- 95 -

Liehnerowiez Si

et

(J

Wi- sont deux formes de eel te, classe, il existe une (n- 2)-forme

W'= c..J +dfAf. Soit

).I. telle que

de

.n

0

et

11 1 (d U o =- d

.n1).

'dno

d.o lIes

et

bords orientes sur

L

D'apres la remarque preeedente, l'in-

tegrale:

a une valeur bien cteterminee, independente db ehoix de se envisagee.

d
flans la elas-

Nous pouvons ainsi definir un sealaire-distribution

ou plus bir.evement

(2.3)

W

l , par la

relation:

f € D(O,n.)

est la mesure de Dirac relative

a q>

; son

support est porte par

L.

c) Proposons-nous d 'evaluer les tenseurs-distributions derives des sealaires-distributions

yO et yl. A eet effet, introduisons dans

steme de eoordonnees

0

0(

f;

(y ) tel que y =

I

{,;j dy~

verifie (2.2). Si UE D(1,n)' on a:

soit

... 1\ dy

ty)

L

que la (n-l)-forme

w

un sy-

dans ee systeme 1 a pour eom-

posantes 1'=1,1.=0 (i,j =1,2, ... ,n-l). Del'expression de O

Il

n-l

on cteduit

- 96 Lichnerowicz

soit encore, d I apres 1a formule de Stockes:

=

-

fo

~nl

uo,W 'V Igi dy 1A

i anI

... I\.dy n-l =..

"( 10< Uc;I.W =(10, U )-

o En raisonnant sur Y , on obtient ainsi:

(2.4)

d

On verifie de m~me qu 'il existe un sca1aire-distribution

L

I

de support

tel que:

(2. 5)

3. Tenseurs-discontinuites

a la

traversee d'une hypersurface.

a) Considerons un p-tenseur T sur

n.

satisfaisant les hypotheses

n

l' Ie tenseur T est un

suivantes: AI) Sur chacun des domaines

n

0

et

tenseur ordinaire de classe C 1. A2) Quand q> 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<;ons-nous maintenant dans les hypotheses suivantes: B l ) Le tenseur Test continu sur

n ,

n . Sur

chacun des domaines

T est un tenseur de classe C 2

n.o

et --1 B 2) (,luand c:p tend vers zero par valeur negatives (resp. positives),

VT

et

'i1 'V T convergent uniformement vers des fonctions a valeurs ten-

sorielles definies sur.L

et notees ('1 T)o ('V''V'T)o (resp. (\l T)l' ('V''V T ) /

Ainsi Ie tenseur

vrT satisfait

De (3.3) applique

a vr T, il resulte:

lui-m~me

aux hypotheses AI' A2.

On en deduit d'apres (3.4):

soit:

(4. 1)

La metrique etant C 2, Ie tenseur de courbure est continue a la traversee de

r. , II en resulte d 'apres l'identite de Bianchi

- 100 -

Lichnerowicz

Comme

~ 1~

=

\7~ 10( = Vol \7~'P' on cteduit de (4.1)

soit:

On cteduit de (4.2) qu'il existe un p-tenseur-distribution T a support sur

L

tel que:

En substituant dans (4.1), on obtient une formule utile:

II. HYDRODYNAMIQUE RELATIVISTE ET HYPOTHESES DE COMPRESSIBILITE.

5. Fluide parfait thepmodynamique. a) Soit V4 un espace- temps muni d 'une metrique hyperbolique ds 2, de signature +- - -, satisfaisant aux hypotheses usuelles de differentiabilite. En coordonn6es locales ds 2 =go( p., dx()( dx f3 (0(,

f.> =0, 1, 2, 3). Dans un domaine de

- 101 -

Lichnerowicz V4' un fluide parfait est decrit par Ie tenseur d'energie:

(5. 1)

ou

~

est la den site d'energie proprl!, p la pression et

tesse unitaire du fluide, oriente vers Ie futur;

~

UO(

Ie vecteur-vi-

contient une densite de

matiere et une densite d'energie interne. NOlls posons:

2

E.

f=cr(1+2) c ou rest la den site de matiere du fluide et E. son energie interne specifiElu que. Considerons Ie scalaire 2 £ p o+p = c r (1+-+--) ) 2 2 c c r Nous posons

=

eo +.L = e.. +pV r

Vest Ie volume specifique et i I' enthalpie specifique.

(ou V=l/r)

A cette variable

nous substituons la variable thermodynamique equivalente, appelee l'indice du fluide, defini par: i f = 1 +c2

Avec ces notations, Ie tenseur d 'energie s 'ecrit:

(5.2)

(f) 2 To{ ~ = c rfud. ul?> - pg()( ~

- 102 -

Lichnerowicz b) La temperature propre

®

du fluide et son eutropie specifique S

peuvent Nre definies, comme en hydrodynamique classique, par la relation differentielle

~

dS = dE.

+~dV

= di-Vdp = c 2df-Vdp

( ®;>O)

Ainsi (5.3)

En relativite la variable thermodynamique Z =fV (volume dynamique) joue un rOle important et se substitue Ie plus souvent au volume specifique classique. II est commode d'adopter p et S comme variables thermodynamiques de bases Nous considerons

't: = t; (p, S) comme une fonction donnee defi-

nissante, par Ie fluide envisage une equation d'etat c) Le systmne differentiel de l'hydrodynamique relativiste est fourni par les considerations suivantes: nous supposons d 'abord que la densite de matiere (qui correspond au nombre specifique de particules) est conservative. Si

\l

est l'operateur de derivation covariante:

(5.4)

D'autre part les equations de la dynamique relativiste sont fournies par la conservation du tenseur d 'energie:

(5. 5)

Nous allons transformer Ie systeme (5 4), (5.5) en un systeme equivalent; (5.5) s'ecrit explicitement:

(5. 6)

- 10:3 -

Lichnerowicz Par produit par u

f.>,

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
de l'hypersurface

L..

par rap-

- 104 -

Lichnerowicz port au fluide, c'est-a-dire par rapport a la direction temporelle u, est donnee c1assiquement par la formule:

(v E )2 l. - 2 - =y c

(6.1)

On voit que, quel que soit

b) Dans un domaine

l~

n

avec

y

I..

est positif et que l'on a:

ou les variables thermodynamiques p, S et Ie

vecteur-vitesse u sont continues, supposons les derivees premieres discontinues a la traversee de L. , de fa<;on que p, S, uo( verifient les hypotheses du

§ 3,

c. D'appes ~3. 4), nous posons dans la suite:

Etudions a quelle condition l'une au moins des distributions

bp, ~S,

est non nulle. La relation (5.4) peut s'ecrire, compte-tenu de (5.8),

(6.2)

r \l. uel. +r' ucla p =0 01. p b(

De 't' =fV, an deduit en derivant en paS constant: 't" I = f' V

P

P

+ f V'

P

(ou r=r(p, S))

bu ~

- 105 -

Lichnerowicz De (5. 3) on deduit:

Nous introduisons dans la suite la quantite

y

definie,

a.

partir de l'e-

quation d'etat, par la relation:

(6.3)

l'

de telle sorte que

=c

2

fr~.

La relation (6.2) peut s'ecrire:

(6.2')

En ecrivant cette relation de part et d 'autre de apres produit par

b:

ce qui s'ecrit: (6.4)

On deduit de

m~me

de la relation (5.8):

(6.5)

Enfin Ie systeme (5.9) donne:

(6.6)

L et retranchant,

il vient,

- 106 -

Lichnerowicz Pour uc( 10(=0, 68 peut ~tre non-nulle. On obtient ainsi les hypersurfaces engendrees par des lignes de courant, dites ondes d'eutropie (ou de matiere). Leur vitesse par rapport au fluide est nulle. 8upposons uo(lOl./o; on a a

b 8=0

de (6.6) il r~sulte que si ~p=O, on

bu~ =0. Nous sommes ainsi conduits d'apres (6.4)

a multiplier'

(6.6)

par 1~. II vient

soit d'apres (6.4):

Ainsi si

J prO,

l'hypersurface

L.

verifie l'equation:

(6.7)

qui est l'equation aux ondes soniques du fluide; ondes, qui sont naturellement des

(6.7) exprime que des

caract~ristiques

du systeme differen-

tiel (5.4), (5.5) sont les hypersurfaces tangentes aux cOnes du second degre d~finis par dualite

a partir

du tenseur:

8i vest la vitesse des ondes soniques par rapport au fluide, on deduit de (6. 1) et (6.7)

- 107 -

Lichnerowicz Nous postulons dans la suite que v<:c (ou

1>1),

ce qui revient

que les ondes soniques sont orientees dans Ie temps.
l' > 1,

Ie tenseur hol (3

'r;/

defini~e

a postuler

Pour que v soit

soit..( 0 forme quadratique de type

hyperbolique normal; ho(r.> designe Ie tenseur covariant inverse de La generatrice de contact de

L.,

hol~.

onde sonique, avec Ie cOne est definie

par Ie vecteur:

(6.8)

Soit v

~

la composante tangente

Notons que, NI3

etant tangent

de la vitesse u (3

aL

a :z:.,

la direction de

N~

du fluide:

est celle de v 13 :

(6. 9)

7. Propriete fondamentale des rayons en hydrodynamique. a) Les bicaracteristiques ou rayons

associes aux ondes soniques (et qui

les engendrent) sont les trajectoires du champ sur Ce sont des geodesiques isotropes relatives male definie par

hol~

a la

des vecteurs N ~ =hClf.;1

0(:

metrique hyperbolique nor-

.

II resulte de (6.6) que, par une onde sonique

(7. 1)

L

2:

- 108 -

Lichnerowicz s 'exprime simplement en fonction de montrer que

~ p (et par suite les

bp

rayons, c 'est-a.-dire que

~ po Nous nous proposons de

JJ ufJ ) se propage Ie long des

verifie un sisteme differentiel de la

forme:

Nous postulons dans la suite les hypotheses Bl et B2 du

§

4 pour

p, S, u ~o De la formule (403) il r~sulte qu'il existe des distributions _ - -'A

p, S, u

telles que:

(702)

~['Yc(~~p]

(7 3)

~[\7o(~sJ =lo(l~S

(70 4)

~[~O/ ~ u]

0

=

=

\7oll~ ~p+1O(V~ ~P+l~'7ol ~P+lo(l~p

\10(

II!-

~

u" +lol 'V~

~ul). +1~ ~c( bu~ +10( l~ ul).

b) Partons de la relation (602')' divisons la partc 2rf et

d~ri

vons-lao II vient

En ~crivant cette relation de part et d'autre de obtient:

L

et retranchant, on

~

109 -

Lichne rowic z est une combinaison lineaire de termes en proportionnel

a

!J

p et

bu'>-

et est donc

~ p. Nous ecrivons (7.5) sous la forme

(7. 6)

ou

I'J

signifie modulo des termes proportionnels

En procedant de

m~me

a

J p.

sur la relation (5.8) relative

a 1 'entropie,

on a:

soit d 'apres (7. 3):

Ainsi S est

~

0 et il vient:

(7.7)

Enfin en se livrant au

m~me

raisonnement sur (5.9), il vient.

En tenant compte de (7. 6) dans (7. 8) on obtient:

soit d'apres (7.2):

- 110 -

Lichnerowicz

Or

L.

~tant sonique verifie P(I)=O et Ie terme en

p

disparait. II reste:

soit: (7.9)

On a Ie theoreme qui traduit la propriete fondamentale des rayons: Theoreme. Sous les hypotheses du males

Jp,

~ u')..

relatives

a une

§

4, les discontinuites infinitesi-

onde sonique

L. s1 propagent Ie

long des rayons associes selon les systemes differentiels:

c) Nous avons vu que I 'operateur trique et

L

etant supposees de classe

~ definit une derivation; la me-

C~

dans

iO... , on a:

~ gO/~ = 0 On peut mettre les equ ations fondamentales relatives aux ondes soniques sous une forme commode. En ecrivant (5.4) de part et d'autre de mediatement

f-

et retranchant, il vient im-

- 111 -

Lichnerowicz Ainsi Ie scalaire:

a= rue( 10(

(7. 10)

est invariant par Ia derivation ~ De m~me, en ecrivant (5.5) de part et d'autre de

L

et retranchant,

on obtient

et Ie vecteur,:

(7.11)

est invariant par ~

(7.12)

Cevecteur peut Nre decompose en Ia somme:

W

~

=

2 ~ 2 ueXIo( ~ c afv +(c af - - - p) 1 10\ 10(,

et ses composantes tangentielles et normale par rapport

a L.

sont

invariantes. Ainsi:

~ (fv ~)=o En particulier la direction du rayon, qui est d'apres (6.9) celIe de v

f.,.

est invariante par 1'operateur de discontinuite infinitesimale

~

.

- 112 -

Lichnerowicz 8. Hypotheses de compressibilite. a) J'ai ete conduit

a adopter par les iluides parfaits relativistes

les hypotheses de compressibilite suivantes portant sur la fonction

"t (p, S), Dans Ie domaine envisage des variables p et S, on suppose que l'on a:

'(;'p (0 et la condition de convexite:

~ ~2 L'inegalite

rr'p <0

>0

exprime que vest

ou que les ondes soni-

ques sont orientees dans Ie temps. Les hypotheses (HI) et (H 2 ) se reduisent

a l'approximation

classique aux hypotheses usuelles dites de

Hermann-Weil et nous montrerons qu 'elles jouent Ie

m~me

rMe qu 'elles,

pour la thermodynamique des ondes de chac en hydrodynamique et magnetohydrodynamique (relativistes). On verifie facilement que

7::

I

p

<.0

implique les deux autres conditions par 1es gaz polytropiques relativistes. Dans des travaux encore partiellement inedits Israel et Lucquiaud en ont donne des justifications du point de vue de la mecanique statistique b) En inversant la fonction

1:' = (;' (p, S), on obtient line fonction

p=p ("t', S) exprimant la pression en fonction des variables -"t; et S. On a identiquement en 't et S

(8. 1)

p = p {'t(p, s), s}

- 113 -

Lichenerowicz Nous nous proposons d 'evaluer les derivees partielles de derivation de (8. 1) par rapport

a p, a S constant,

p' 'U

(8. 2)

1;

/

P

p(~

S), Par

il vient:

=1

On en deduit:

p' =

(8.3)

'C

De

m~me,

't~

par derivation de (8. 1) par rapport

a S, a p

constant, on a:

p' 't;' + p' =0 't S S II en resulte:

'C"S p' = - - S t;'p

(8.4)

Les hypotheses de compressibilite (HI) se traduisent denc par les inegalites

Pz,
a p, a S constant,

On en deduit:

(8.5)

" P'\;'2

et (H 2) se traduit, modulo (HI)' par rf2> 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<. (c rfu )-u

0(

00( p

=0

relation indentique a. (5.7). En faisant intervenir

® et

S, il vient encore:

Ainsi (9.2) entraine toujours I'equation de flot adiabatique:

(10.6)

En utilisant (10.6) paur transformer .(10.4), on obtient Ie systeme differentiel aux lignes de courant:

-

1 1g

-

Lichnerowicz

En developpant, il vient:

(c 2rf+

}it hl2)uO
+ t )-l-uo(

u~~lhI2 + }lh'A u,u,V" h}l- u~ - ~ v;"h« h~ -fA.h~Y'o( !

=0

En tenant compte de (10.3), on obtient:

(10.7)

(c 2 rf+

+ fU"'-

}-LlhI2)J<\7~u~ -(g~~-uotu~)aol.p-tfg~~~lhI2+ }V'oc: Ihl 2+}L Ihl 2 ~ uo(

Par produit par h~,

U~ - J1:~ho( h~ - f

ho("V
on deduit de (10.7):

2 JL Ihi 201.~\7 C<" 1 0<\71121120( )u h Vo( u~ -h CJo( P - '2 Jih Yo{ h + p.- h \/0( h

(c rf+

soit, apres simplification, 2 2 0( ~r1 C( (c rf+ )-l~hl )u u ~ h[:J Hi ()().p-

fA' Ih.12 \7vol... h0(-

Compte-tenu de (10.2), on obtient la relation simple:

=0

+

- 119 -

Lichnerowicz 2\7

(10.8)

c rf

Vr;I.

O
+h ~p =0

h

11. Ondes magnetosoniques et ondes d'Alfven. a) Dans un domaine[L et les vecteurs quation

cr

=0

ue>("

(Cf

h cI.

ou les variables thermodynamiques p, S

sont continus,

de classe C 2; l=d

soit

f),

L

a la

une hypersurface d 'etraversee de laquelle

les derivees premieres sont discontinues, de fac;on que p, S, uC<, hO< verifient les hypotheses du

§ 3, c.

Comme precedemment nous posons:

~ [~SJ =k ~ S

rV h~J -

a quelle ~ u ~, J h~ est

Etudions

rJ..

=1

0(

~ h~

continion l'une au moins des distributions

~ p,

Js,

non nulle.

Le systeme differentiel (9.2), .(9.3), (9.4) est equivalent au systeme differentiel forme par (9.2), (9.3), (10.6), (10.7), Les relations (9.2) et (10. 6) donnent d 'abord, exactement comme dans Ie cas de l'hydrodynamique:

(11.1)

et

(11. 2)

(uc( 1 )

0,

JS =0

Des equations de Maxwell (9.3) ou (10.1), on deduit par un raisonnement

- 120 Lichnerowicz identique

a celui

§ 6:

du

(11.3)

Enfin Ie systeme differentiel aux lignes de courant donne de maniere analogue:

(11. 4)

(c 2rf+}-'-1 hl 2}(uo( lei.) + ]A- fttc{ 10( )u~

Pour ucJ.lo\ =0,

JS

~ u~ -(l~

_uc(

i

u~ )10£ ~p - }U~ ~I hl 2 +

~ Ih 12 + Ji Ihl 2 u~ Ie{ ~ uo(_ f" h~ 10( ~hO< -f(ho( lo()J Jl =0

peut t!tre nulle et on obtient de nouveau les ondes

de matiere. Nous supposons dans la suite uo/ lCl{FO et par suite dS=O. b) Des relations (10.2) et (10.3), consequences des equations de Maxwell, on deduit les relations suivantes consequences de (11. 3):

10{

~ho( -(JX lol.)U~ h h~

~(tFlo()~lhI2+ IhI21c<~uo( Par elimination de u p.,

~ h~

_(hoi.

=0

~)u~ cS h~

=0

entre ces deux relations, il vient:

Considerons maintenant la relation (10.8), consequence de (10.7). On en deduit:

(11.6)

2

(0/

crflolu h

0(

(

+olo(c)p=O

- 121 -

Lichnerowicz Notons que comme

bs=o, il resulte de (5.3)

()p=c 2r

~f.

Par suite

(11.6) peut s'ecrire:

~

Ainsi Ie scalaire b=fhO( 1 eX est invariant par la derivation Enfin multiplions (11. 4) par 1 ~. II vient:

(c2rf+J-LlhI2)(UO
~

Ih/ 2+

f' IhI2(uo('lo()l~ clu~-2 f(hcX\:()l~~ h~

=0

c'est-'a-dire: 2 rio. ,PI 0I.P.J 0( B c rf(u lo1.)l~ 0 ur -(g ,- -u ur )lo(l~Jp

-'21 fl ~ l~cl(I hi 2

+y.~ Ih l2(uoIlo( )1~ ~u~ +(uO/ 10(.)2 ~ ~h \2 _2(ho( 101.)l~ ~ h ~ }

+

:0

0

D'apres (11.5), on obtient:

Eliminons Jlhl2 entre (11.5) et (11.7). Par produit de (11.5) par

IN 1Y 1 f'

de (11. 7) par (uol 1~)2 et addition, on obtient:

c2rf(uollol)331~ ~u~ _(ud.l~/(g~~ _ueJ. u~ )10{ 1~ Jp+ n

1

+ )A-19 ly{lhI2(tP101)1 ~ cl u~ _tho( 10/)1 ~ ~ h~l

=0

- 122 -

Lichnerowicz c 'est-a.-dire:

~

{c 2rf(uO< 10/ + Jl'lh121 f 1 } (u"\.(

)1~ bu~ -(uollo()2(1 ~ 1~ -f) 1,e )2) ~ p- IJ, h0{

I

1 1P1 1

f'~

0{

S h~

=0

A cette relation, nous pouvons adjoindre (11. 1), soit:

et (11. 6) soit:

ot

(h 10{)

{

2

Le determinant de ces trois equations

bP,l~~h~

l~eaires

=0

aux inconnues

1~ du~,

s'ecrit:

tc2rf(UcX.\,/+ J.l'lhI21Y1~}(~lo() H:

(~

c> 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>
t~~

0

signifie encore modulo des termes proportionnels

determinant des equations (11. 9), (11. 10) aux inconnues s'ecrit: D(l) =(c 2rf+

f I h 12 )(u0( l~) 2-f.l,(h~ lei) 2

d) Nous avons ainsi mi-8 en evidence trois types d'ondes ou d'hypersurfaces caracteristiques du systeme fondamentale de la magnetohydrodynamique 1) Les ondes d'entropie ou hypersurfaces engendrees par des lignes

- 124 -

Lichnerowicz de courant dlequation

Ii la traversee desqueUes

¢ S peut ~tre #0.

2) Les ondes magnetosoniques, hypersurfaces telles que l=d

A la traversee d lune onde magnetosonique

Ip ~ J , ~ Ihl 2

peuvent

~tre

Jp

et par suite 1~

~ uf

fO

3) Les ondes dlAlfven, hypersurfaces verifiant:

A la traversee diune onde dlAlfven, il peut y avoir discontinuite effective des derivees des compos antes tangentielles de la vitesse et du champ magnetique:

dv~

Posons pour abreger

et

~

~ t~

peuvente

~tre

fO pour

='V(c 2rf+ JA-I hI 2)/fI'

.

~ p=O.

Llequation D(l)=O

aux ondes dlAlfven peut slecrire:

et lIon voit que les ondes dlAlfven sont engendrees par les trajectoires des champt de vecteurs:

- 125 -

Lichnerowicz

i- = ~ ual +h ol ce qui

d~finit

deux types d'ondes dites ondes A ou ondes B. On note que:

Les vecteurs A et B sont temporels et

12. Vitesses des ondes

magn~tosoniques

L

a) Une hypersurface nous avons

associ~

pr~cMente

(12.1)

et des ondes d'Alfven.

r

n

est (l-y

~quivalente

1..

Mfinie par:

h 2 v~rifie Ie lemme suivant. n

=0, avec l=dcp) ~tant donn~e,

~o

A la relation:

0( 2 r.o( )(u 10() = -y (1 10()

De maniere analogue, associons au vecteur h

vers Ie futur.

A u et 1 (voir (6.1)) Ie parametre:

y La dMinition

(d'~quation

orient~s

h

et A

1 ,

une composante

- 126 -

Lichnerowicz 2 Lemme 1°) On a toujours h 2n 61hl -2°) Pour que h 2 =lhI 2 , il faut et il suffit que I appartienne au 2-plan

n

n

defini par u et h. En effet considerons au point x de

L

un repere orthonorme

(e ,e.) (i, j=l, 2, 3) tel que e =u. On a hO=O et il vient d'apres l'inegalite

°

°

1

de Schwarz:

c'est-a-dire

On obtient

ce qui etablit Ie 10. Pour que I 'egalite ait lieu il faut et il suffit que si

nou~

decomposons I selon un vecteur colineaire a u et un vectaur

k orthogonal a u, k soit colineaire a h, ce qui ctemontre Ie 20. b) En introduisant h 2, Ie polynome P(l) peut s'ecrire: n

D'apres (12.1), P(l) peut s'exprimer en termes de

yL.

par:

- 127 -

Lichnerowicz

(12. 2)

ou

n

(y) designe le trinome du second degre:

c 'est-a-dire en developpant et ordannant en y:

(12. 3)

Sou3-.Jil.

seUl~potl:..~~e.. _ryJp <~

(soit

r> n 1),

(y) ales proprietes

suivantes

D'autre part:

-rr

1\

2

v (-:d c

Ce qui peut sreerire:

.. . f\lO~l

1\ • dnous " "eSlgnons par y :VlL et 1\ () Y a d eux zeros en t re O et lque

. d one pour.1es rmdes magnetosomques .. . y "vIR. II eXlste et par rapport au

Lichnerowicz f1 Ul'd e d eux ' vltesses v IVlL et v MR , d'f" e lmes par (v:VfL)2/,c 2 c y NfL et .. . 1 < 1"ltes: (V 1\11''1)2/Ie 2=y MR ,satlsialsant es' 'mega

(12.4)

lViL

v .

v e s t (hte vitesse des

ML

~ v~

~lDdes

v

:YIn

<" c

magnetosoniques 1entes et v

i\iIn

vitesse des ondes magnetoeoniques rapides. ?

c) En introduisant h~ n

dans D(l), on a:

(12. 5)

c

2 )-I.hn

_)

2

2

rf+yl h]

Ainsi les ondes d 'Alfven admettent par rapport au fluide une vitesse v

A

donne par:

(12.6)

Evaluons:

soit, apres semplifications,

(V A)2 -' --2c

Lichne rowicz

1\ I\

A)2

A)2

')

(J;-- ) ~

f- (11 ~ - IL I ") ('¥ . l) ~ 0

=

c

c

On etablit ainsi que:

(12. 7)

Soit

v

L

i\IL

~

~

v

A/

v

~

.Im

une onde d'alfven de vitesse vAiO (h 2 fa); elle verifie n .

D(1)=O. i\'otons que paur que P(l)=O i1 faul et il suffit que ')

c'est-a-dire que h~= 2 -plan

n

n

')

/111-.

n(~)=O, A)2 c~

D'aprcs Ie lemme

doit appartenir au

defini par u et h( (puur urX. 10{ f 0).

13. Representation des cOnes d 'ondes dans H 3. a) En un point x du fluide, soit

T~ x

l' espace des covecteurs tangents

a 1'espace-temps. Le ctlne fondamental de 1'espace-temps peut-Nre defini par Ie cOne

r

des covecteurs 1 verifiant:

r D'apres l'etude du

S 11,

go{

~

10(

l~

=0

les caracteristiqnes ou ondes de la magneto-

hydrodynamique sont definies par les hypersurfaces tangentes

a 1'un

des

trois cOnes d 'equations:

Introduisons en

XI

un repere orthonorme (e ,e.l (i' I, 2, 3) tel que l'on ail: o 1

- 130 -

Lichnerowicz

e

o

~

u

Nous posons provisoirement dans ce paragraphe:

1 =t o

1 = "K 1

1 2

1

= y

"

3

=

z

Avec ces notations, nous obtenons pour les differents cbnes les equations explicites: 222 2 t -x -y -z = 0 t = 0

2 42 222222 c rf(1' -l)t +(c rf+ flh\ (t -x -y -z )-

i')t

avec "( > 1. En introduisant ~ , on obtient pour equations de

fH'

r

~2=

A (apres avoir pose

r

222 2 t -x -y ··z

=

r,

)=>2/1 hJ2)

0

P, 2 1 4 -2 2 2 2 2 2 2 2 2 2 2 (l~ -l)('¥-d t +(~ +'i'-l)t (t -x -y -z )-z (t -x -y -z )=0

-22

~ t -z

2

=

0

Nous appelons indicatrices dans R3 les sections de ces cbnes par l'hyperplan too I _ Nous obtenons ainsi les trois indicatrices suivantes:

- 1:11 -

Lichnerowicz

2 2 2 1-x -y -z =0 -2 2 2 2 2 2 2 2 l' -( P + r -l){x +y +z }-z {l-x -y -z }=o

-2 ~

z

=t.~

11 est aise de discuter la forme de ces indicatrices qui admettent Oz

comme axe de rotation. II suffit de couper les indicatrices par Ie plan x= 0, ce qui conduit au cercle 2 2 Y +z =1

aux deux droites

z

et par SH

a la

=

±~

quartique:

Wi -( -2

-?

CH : ~

2

2

2

2

2

~ +~ -l){y +z }-z {l-y -z )=0

admettant Oy et Oz pour axes de symNrie. L'equation de C H peut s 'ecrire: 2 (z Y = -

2

-1

z2(

2

Pour z = l' .)

-'J

z-= f.> ~+

1:'

et

-1.

2 -? ){z - \? ~}

~ \1'

-l}

?

on a y'"=0 et elI admet les asymptotes

- 132 -

Lichnerowicz -

l'

l'

2

Nous appelons cas general Ie cas ou f ~ . Par exemple pour -2 t... ~ , on a pour CH Ia forme de Ia fig. 1; H est decomposee

topologiquement

en deux parties

r

r

rH

et Hila partie exterieure 1 . . 2'... l~ H aux ~H correspond aux ondes Ientes et 1a partle lllt.::rleUre 2 ondes rapides.

'Y = -2 ~ ,

Nous appelons cas singulier Ie cas ou v

(13.1)

c

C

H

c'est-a-dire ou:

2 2

2 c rf+

J-"Ihl

admet des points singuliers sur Oz

2

et a Ia forme della fig. 2.

z

------.~--"-=-

-- -

-

/

/

\

/

-,

/

"-

\

0

'- ......

fig. 1

\

fig. 2

/

I

I

J

- 133 -

Lichnerowicz Le cCne

r

~H

2

admet deux generatdcessingulieres x=O, y=O, z =1' t

11 defini par ru et h,

contenues dans Ie 2-plan

rA

Dans tous les cas,

est l'ensemble des deux hyperplans z=± ~t;

leur intersection est Ie 2-plan l'orthocomplement du 2-plan

r

r

A sont tangents a

et dMinieS par z2=

n

I

n,

defini par z=O, k=O, c'est-a-dire Dans Ie cas general les hyperplans

H le long des generatrices contenues dans

~2

1T

t 2 , Dans Ie cas singulier, ils passent par ces

generatrices qui sont les generatrices singulieres, L'intersection de l'hyperplan

r

(t =0) soit avec

r H,

chacun des hyperplans de, rAn lest autre que Ie 2-plan b) Etudions l'intersection

I'H n rA,

Si nous posons

soit avec

TT

I,

z2=~2t2

r A il vient

dans I'equation de

-2 4 -2 2{ 2 -2 2 2} (~ -1)(~-l)t +(~ +t-l)t t (1-~ )-x -y - 2 2 J 2 - 2 2 2) - ~ t t (1- ~ )-x -y =

J

L

°

soit apres simplifications:

(i' -l)t 2(x 2+y 2) = ° L'intersection se compose donc du 2-plan

1t

r H contenues

I

(d'equations z=O, t=O)

dans Ie 2-plan IT (x=O, -2 y=O) et dMinies par z=± ~t, Dans Ie cas general f ~ ), les et des deux generatrices de

hyperplans de

r

A

sont tangents a

dans Ie cas singulier

(1t

2

r:

H

('1

Ie long de ces generatrices;

= ~2), nous avons vu que ces generatrices

sont singulieres, c) Designons par a(x, () ) (x f

n.)

un operateur differentiel homogene

- 134 -

Lichnerowicz d 'ordre m. Si 1 E: T~

x

a(x,l)

est un polynome homogEme en 1 de degre

m. Soit V (a) Ie ctlne de T* defini par l'equation a(x, 1)= 0, OU x est x

x

fixe. BNous considerons V (a) comme un ctlne projectif. L'operateur x

a est dit

strictement hyperbolique en x,

si V (a) verifie l'hypothese

x

suivante: II existe dans T* des points 1 telle que toute droite issue de 1 x

et ne passant pas P<:ir Ie sommet du ctlne coupe la surface du ctlne en m points reels et distincts. Etudions les operateurs P(l) et D(l) qui apparaissent dans Ie systeme differentiel de la magnetohydrodynamique. Pla90ns -nous dans Ie cas general: l'operateur P(l) est strictement hyperbolique, mais il n 'est est manifestement pas de

m~me

pour l'operateur D(l) qui n 'est que Ie produit

de deux operateurs strictement hyperboliques du 1e ordre correspondant aux deux hyperplans. Le systeme differentiel etudie n'est donc pas strictement hyperbolique. II en est a fortiori de P(l)

lui-m~me

m~me

dans Ie cas singulier OU l'operateur

n 'est pas strictement hyperbolique.

En me plarant dans Ie cas general, j 'ai etabli cependant ailleurs (Relativistic hydrodynamics and Magnetohydrodynamics, Benjamin, New York (1967))un tMoreme d'existence et d'unicite pour Ie probleme de Cauchy relatif au systeme de la magnetohydrodynamique, sur une classe convenable de fonctions Coo.

14. Propriete fondamentale des rayons associes aux ondes magnetosoniques. a) Soit

'f

une solution de l'equation aux ondes magnetosoniques

P(l)=O. Nous supposons que pour les ondes

L..

envisagees D(l) est

Lichnerowicz #0. Il resulte du

§ 11

(en particulier (11.9) et (11.10) que les ''tli-

sconiinuites infinitesimales 11 niere unique en termes de l' expression de

~ /hl

Ju ~,

~rf

peuvent s 'exprimcr de ma-

~ p. Nous aurons besoin en particulier de

22. D 'apres (11. 7):

On en deduit en exprimant

l~ ~ u~

a partir de (11. 1):

c'est-a-dire

(14.1)

b) Les ondes magnetosoniques sont les shypersurfaces tangentes

aux ctmes P(l)=O. La generatrice de contact avec Ie ctme est dCfinie par Ie cvecteur:

I() P(l) 2

'C)1~

soit:

(14.2)

n:'{~

=2c2rf(O-1)(U<><'lo()3u\'J+(c2rf+flhi1

- JA-

)uo(lll((lflfu~+(U~lo()l~)

0( 0 ~ 0( ~ h 100J.l) 1 ~~l +(h 10()1 )

- 136 -

Lichnerowicz Ce vecteur etant tangent a l'onde magnetosonique

L,

(14.2) peut

encore s 'ecrire en prenant les composantes tangentielles des differen)s termes:

(14.3)

N~ =2c2rf(l-l)(Ull/lo()3v~+(c2rf+flhI1 )(UOC:lo()(lfl~)VP

-f

(hoi.

Les bicaracteristiques ou rayons trajectoires sur de montrer que long des rayons,

lo()(l~ If)t f

associes aces ondes

L

sont les

L.. du champ des vecteurs Nf... Nous nous proposons cl p (et par suite les ~ u~ , ~ h~ ) se prop agent Ie c'est-a-dire que

~p verifie un systeme differentiel

de la forme:

ou Ie symbole ~ signifie modulo des termes proportionnels

a

~ p.

§ 4 pour distributionsp,S,u~,h:t

Nous postulons dans la suite les hypotheses 58 1 et B2 du p,S,uiX,hiX . D'apres (4.3), il existe des telles que:

(14.4)

~[Vd.V~

pJ = 'V'"

(14.5)

X[VD\V~

SJ = 10{

(14.6)

1['Vo\v~uaJ = Vd., 1~ J u'l. +10( ~ 6U~+1~ \70{ ~U?.+~l~

(14.7)

~['Vo( V~h 'A] = 'Yo( l~ ~

l~ b p+lD( 'V~

~ p+l~ VoZ~ p+lo( l~ P

1~ S -A-

u

~ Vc ~ ~ 'A -~ h +lc;( /?> ¢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

<5 }

=0 et Ie vecteur A(3

lui-m~me,

et sa direction en par-

ticulier, sont invariants par la derivation ~ .

IV. ONDES DE CHOC EN MAGNETOHYDRODYNAMIQUE 16. Le systeme fQndamental des ondes de choc. a) Dans un domaine ..0.. de V4' soit encore guliere d'equation ce

cp =0 (Cf de

c1asse

c 4,

L

une hypersurface re-

avec l=dtp). L'hypersurfa-

L. est une onde de choc magnetohydrodynamique si uO{, ho( ou l'une

au moins des variables thermodynamiques sont discontinus a. la traversee de

L.. ,

conformement aux hypotheses Al et A2 du

tablirons que sous les conditions de compressibilite HI'

~ 3. Nous e-

r.. est

neces-

sairement orientee dans Ie temps, done de vitesse admissible au point de vue relativiste. Soit Y un etat du systeme fluide-champ ,defini par les valeurs de

p,S,u~,h~ en un point x de

L.

Un tel etat est defini par la donnee

de 8 quantites scalaires, c 'est-a.-dire depend de 8 parametres. Nous notons

[Q]la

Yo 1'etat anterieur au choc, Y1 I 'etat posterieur au. choc et discontinuite, Q1- Qo d'une quantite a. la traversee de

L. .

Le systeme fondamental (9.2), (9.3), (9.4) est suppose satisfait au sens des distributions:

- 145 -

Lichnerowicz Ces equations impliquent:

II resulte de l'equation (3.2) que l'on a:

(16.1)

Po sons

a(Y)=ruc( 10(

et:

w,., (Y)=(c 21: jl

~

IhI 2

f3

0(

+ f -2-) a(Y)ru -ql - j4(H 1~)h r

II est clair que Ie vecteur V ~ est tangent

is

a I... Le systeme (16. 1)

exprime que:

Vi' (Y

1

)=V~ (Y

Le scalaire a et les vecteurs V i3

0

et

)

wfJ

definissent les invariants

du choc. b) Un choc est dit tangentiel

si a=O. S'il en est ainsi:

est de vitesse nulle par rapport au fluide dans les deux etats et

- 146 -

Lichnerowicz orientee dans Ie temps. De 1'invariance de V (? et

w (3

il resuIte:

(16.2)

(16.3)

~~ !.od~ u~

et

u~

etant unitaires et colineaires d'apres (16.2),

orientes vers Ie futur, on a

u~ =u

t

h~ 10( =h~ 1«.

et par suite

De

(16. 3) il re suIt e :

et par suite

[q] =0. De (16.3) resulte ainsi

scontinuite de r 0(

[if]

°

restant indeterminee si he< lex' = -0-

_

-

=0,

[PJ

=0, la di-

il resulte de (16. 2)

hI 10< -0 et les equations de choc donnent seulement ici:

les autres discontinuites restant indeterminees. c) Nous supposons desormais alO (choc non tangentiel) et introduisons Ie scalaire H, invariant par Ie choc defini par:

(16.4)

H(Y)

01

= ..!. V~ V(.l, = (h 10/) a

En substituant dans

2

wi3 a

\~

a

2

2

Ih I r

2

2

h(3 son expression en termes de

vt3

et ui3 :

- 147 -

Lichnerowicz il vient (nous supprimons la reference

W

a Y lorsqu'elle est inutile):

2 Ihl 2 i3 f3 (hO< 10d 2 {3 rhO/leX Vi3 =(c"C +}'--2 ) aru . -ql - ~ ru + f - r a: a

{3

soit:

ou l'on a introduit la variable importante:

0(

Si nous decomposons

a L,

2

= c t -fH = D(l)/a

wf3>

2

ell sa partie tangentielle et sa partie normale

on obtient

ou

(16.5)

est tangent celIe de

aL .

xP->

(16.6)

L'invariance de W f->

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 0<.,

2

II ##-)-a c2

rX c 2l I c\

(c

II

rc:

2

2 2 2 -2c 't:,l.lH+}J- H ) /

on obtient (16.10), ce qui demontre Ie lemme.

Transformons enfin l'expression de HK, ou K est donne par (16. 11). De

il resulte en substituant

a

,..H sa valeur

2

crt: - 0(:

- 151 -

Lichnerowicz 2 2 2 2(ho( 1 ) 2 2 2 HK = H(r - _a_)~ + r c{ (c'i:' +c< Hc 1:" -01) 2 2 I e( 1 0( c 2 c a soit: ?

HK= H(r~

(2 2 2 0< )2 _a_) ~ + r (h Lx (c 41: 2_ 22 1.'Y. 1eX c 2 c a

OIh

Nous obtenons ainsi:

2 HK ={iI(r2 __ a_)_ 10( 10(

soit: 2

(16.2)

HK = c 2 a

Ainsi L= stitue

?

b~

a

X0( 2

c

)CO<

2

2

est un invariant qui q, H, K etant fixes peut

~tre

sub-

b en convenant que Ie signe de (hOI 10() demeure inchange au

cours du choc. e) En ce qui concerne qui nous sera utile

'X. '

a differentes

Lemme 2. 1) On a (1 Id,) X{, c(

nous allons etablir Ie lemme suivant reprises.

a.

ta, pour que :x =a, il Henne au 2-plan IT Mfini par (u, h) 3) Si 10( lex est ';y a, ion a ll~a et par 2) Si lo{levest

En effet d 'apres les definitions de

'X

faut et il suffit que I appar-

suite 0< et de H

> a.

- 152 -

Lichnerowicz

. en mtro . d' SOlt Ulsant h 2 : n

§ 12,

ce qui, compte-tenu du lemme du L'inegalite

h2~ Ihl 2 n

demontre les 1) et 2).

peut s'ecrire:

\ hj2 10( 10( 2 a Pour

~

1 lct)O, on a donc

H~O

et

2

0( =c't:

-fH)O.

17. Analyse des equations de choc et chocs d 'Alfven.

~ 16, il resulte que les deux arariables thermo-

a) De l'etude du

dynamiques et les trois scalaires Ih1 2, ue< 10(, ho( 1 cl( verifient les cinq relations scalaires fondamentales:

r1u~lo(=rou~

(17.1)

(17.2)

IhC\l )2 . 1 c< a

2

10( =a

Ih \2 (hO< lo'i 0 1 - --= 2 2 r1 a

Ih ,2 0

r

2 0

= II

- 153 Lichnerowicz

(17. 3)

(17.4)

2 2 2 2 c a c a q - - - f'( =q - - - 'L = e 1 {l( 10( 1 0 10( 10( 0 2 2 2 2 2 2 2 c a 2 ).tH 2 2 c a 2 c f - - ' 1 : ' +2 ~'l:' X - -'X =c f - - ' 1 : +21LX T1 10\ Ie( 1 I 1 1 c2 1 0 leX 10( 0 roo 2

-~ C

(17.5)

1)1

Iv

1

0{

2 1

='V

.IIi 0

2

0<

2 0

tV

=

K

IVO

=L

ainsi que 1a relation onon independante des precedentes:

(17. 6)

Nous etablions que, sous des conditions convenables, les systeme precedent definit d 'une maniere et d 'une seule les valeurs apres Ie choc des deux variables theormodynamiques et des trois scalaires 2 Ihll ,u~ 10(' h~ 10(' Il est clair que si (17.3), (17.4), (17.5) definissent les valeurs de Iv (ou de Ih1 2) et des variables thermodynamiques, (17. 1) fournit alors la composante normale de la vitesse et (17. 2) ou 07.6) celle du champ magnetique. D'autre part, d'apres l'invariance de

V~

et W J5 , les composantes

tangentielles de la vitesse et du champ magnetique verifient:

(17.7)

- 154 -

Lichnerowicz Le determinant des premiers membres de (17.7), (17.8) aux inconnues vf,tf s'ecrit:

Si

eX1fo, (17.7), (17.8) determine v

t, t~

ep fonction des quantites de-

terminees par Ie systeme des 5 equations scalaires. b) Supposons d... 1 =0 en un point x de L... ; l'hypersurface est alors onde d'Alfven en x pour l'etat posterieur au choc et elle est orientee dans Ie temps (lo(lo(<:!l). La relation (17.5) donne 0(0=0 ou bien

ry <x2 /\"0 0

=0 et ou bien

Xo=O.

= Oi =0 L definit un choc d'Alfven. Les cas o 1 ' 0< =0 et 0< =0 0( fO, sont dits des chocs 1 0 ' o singuliers. Nous etablirons qu'ils sont incompatibles avec les ondes Dans

me cas A0 =0,

01

d'Alfven infinitesimales et que, par suite, il n'ont pas de realite physique. On a TMoreme: 1) Si dans un choc 0(1= 0< rO, Ie choc est nul. -0'--'--------'---'2) Dans un choc d'Alfven (eX 1= d =0) les variables thermodynamiques -0

lhl 2 , (uO(lcX), (hO( lo() sont continus si l'hypotMse 1) Si

[0(]

=0, on a

['1:J=O Sous notre hypothese

Xl = 'Xo

et par suite:

'1;'

p

<0 est satisfaite.

- 155 -

Lichnerowicz D'apres (17.3), on a [qJ=O et par suite

[pJ

=0

Dans K donne par (16. 10), les trois derniers termes sont invariants au cours du choc. On a done

[rJ

[r J =0

=0

et d' apres (17. 1) et (17. 6)

L.

n'etant pas onde d'Alfven, il r~sulte de (17.7) et (17.8) que Ie choc

est nul. 2) Pour un choc d' Alfven, on a toujours

[t:]

=0 et il resulte des

equations de choc, K etant donne par (16. 10): [ c 2f 2+flh\ 2:] 'tj =0

~

r: 2rf+}Llhl 2 LC

=0

(17.9)

Examinons l'independanca des deux variables theormodynamiques:

tf

2 =c rf-2p

- 156 -

Lichnerowlcz Par derivation de

De

m~me

cr

en p il vient; d'apres Ie

§ 6:

en derivant en S: 2

2

rS

Cl)'=c fr' +r9 =r(c f - + @)

r

ISS

Or:

2 ,f";\ 2. c 1:' = 101 V+c fV' S

S

et par suite: 2

c r't:' = S

®

rS

2 -c f r

II en resulte:

Le jacobien de

c.p

et

d(CP '1:) dip, S)

"l;'

--,--'--'-'~=c

c'est-a-dire

par rapport

ap

et S a pour valeur:

2 2 2 r '1:" 't' -r(2®-c r-c')"C' p SSp

- 157 -

Lichnerowicz d(

d (p, S)

=2-2r8 X' P

Le jacobien etant different de zero dans I 'hypothese riables

a la

'f

et

'1:'

p

0, les va-

'l: sont independantes. Ces variables etant continues

traversee de

r..,

il en est de m~me pour toutes les variables

theormodynamiques. En particulier

[r] = 0,

[r] =0,

[p] =0. D'apres

(17.1) et (17.6), on a:

et

[I h1 2]

=0, ce qui demontre Ie tMoreme.

c) Dans un choc d 'Alfven, la vitesse tangentielle du fluide apres Ie choc demeure indeterminee, la direction du champ magnetique tangentiel etant alors determinee par example

a I 'aide du resultat

qui suit. Considerons un choc d 'Alfven de type A; B

trajectoires du champ A

ainsi que:

De (17.8) on deduit:

et 1'on a

L... est

engendree par les

- 158 -

Lichnerowicz

c 'est-a.-dire:

[R]

Comme Ie choc est non tangentiel

=0. II vient:

Theoreme. Dans un choc d'Alfven d'espece A (resp. B), Ie vecteur

PI>

(resp. B~) reste invariant dans Ie choc.

V. FONCTION D'HUGONIOT ET ORIENTATIO"!\; DES O]';1)ES DE CHOC.

18. Relation d'ITugoniot relativiste. La relation (17.4) peut s'ecrire:

.)

c 2 [r2] -(T o + 'r 1)

(18.1)

[qJ +2f[X'L]- ~~II [X]

c 2[r2] -(t'o+'r1) [p] -

=0

c

.)

t~('Co+T1)[XJ +2f[~rc]- }t~l [XJO c

- 159 -

Lichnerowicz Nous nous proposons de substituer a) Examinons Ie terme

[X~J

a

(18.1) une relation plus maniable.

qui peut s 'ecrire:

- X' 0 ) 'r1+ X (t' - 't ) [X.'t1J = (X 1 010 ou

[ X. rr], = Par addition membre

a

"V

('t' 1

~1

-r0 )+(X 1-X0 )t' 0

membre il vient:

(18.2)

En reportant dans (18.1), on obtient aprus simplifications:

c2

[r 2J '-( 't'o+ 't'l)[P] + tf('t

O+'t1 -

¥)[X] + f (Xo+X 1l['t} 0 c

soit:

b) Proposons-nous de transformer la somme des deux derniers termes. On a d'abord, comme dans (18.2):

1.2

(0( +0< 0

)['\11 + 1.2 (X 0 +}:: 1)[o<J = Lrx,o<J

1 I'v

Or d'apres (17.5), on peut ecrire:

- 160 -

Lichnerowicz

[x«] = X1 ot 1-X 0 c<0

Xo o(~

=-0(1

Xo <Xo

-X 0 0( 0 =- -0((~-OJ. ) 1 1 0

La relation (18. 3) devient ainsi:

(18.4)

Xo

2 2 2 1 0(0 c (f -f H't;'+7,;')(p -p )+(1:' -7:) -u.(X +X -2--)=0 1 0 0 1 1 0 1 0 2I 0 1 ot1

Nous appelons relation d 'Hugoniot cette relation que nous substituerons desormais a ((7.4) dans Ie systeme des cinqu equations scalaires. c) Un etat initial

Y du fluide etant donne, nous considerons o jusqu'a nouvel ordre l'ensemble des etats Y verifiant les conditions:

(18. 5)

L(Y)=L(Y )=L o

H(Y)=H(Y )=H o

de telle sorte que, pour 'l:' T./.

ltH c

2'

on a:

L

X= - - 2 2 (c 1:: - fH)

2 2 (c't' -jUI)

II est commode de substituer a q la variable:

q

1

= p +-2fX

1

a2

2

loll

=q+-)A--- H d.

La relation (17. 3) peut ainsi s 'ecrire sous la forme:

- 161 -

Lichnerowicz

_

(18.6)

_

2 2 c a (_

q -q = - 1 0 10( 101.

.... - 't:

1

)

0

Sous les conditions (18.5) un etat thermodynamique ('C', p) du fluide definit (pour

1::

t ~H )

un point Z du plan ('L, Cj') et inversement.

Entre les variables c 't', S et q, nous avons la relation fonctionnelle: 1

(18.7)

q=

2'

L (c2t:- P.H)2 + P (

't;,

S)

OU p( 't;' ,S) est definie par inversion de 1'equation d'etat.

Nous sommes conduits

a introduire la fonction d'Hugoniot ~(Z ,Z)

o de la magnetohydrodynamique, consideree comme une fonction de Z

pour un point initial donne Z : o (18.8)

d'~(Z

o

2 2 2 1 'XotXo ,z)=c (f -f }-('t'+'t' )(p-p )+('t'-'t')- u..(x.,+X - 2 - - ) 0 0 0 0 2/ 0 0(

On a manifestement

'jl{.(Zo'Zo)=O et (18.4) peut s'ecrire ~(Zo,Zl)=O.

C 'est 1'etude detaillee du comportement de la fonction

%

qui permet

d'etablir la plupart des tMoremes chercMs.

19. Differentielle de la fop.ction En differentiant (18.8) sous les conditions (18.5), il vient:

- 162 -

Lichnerowicz Or:

2 c fdf = f

e dS+ 'Cdp

2 doc. =c d'Z:'

On en deduit:

M'

1 21

Xoo(o

d-dl'O =2f (9dS+('t"-'t }dp-(p-p }dt:+-~(X+X -2--}d't'+

o

0

1 +('1:- 't" ) -

ct

2 'XoCX 0 d,2

u. (dX +2c - - d't')

2I

En introduisant la variable

0

q a la

place de la pression p il vient:

Soit apres simplification:

~J

cd-J'!\o=2f@)dS+('t-"C }dq-(iHi }d'C+ IJ,.(%o 0 I

Ce qui s'ecrit, compte-tenu de (17.5):

(19. 1)

2

Xo~o 0(

2

}dt'

Lichnerowicz Dans Ie plan ("t,q) introduisons commme parametre la pente m de la aroite joignant Z

o

a z,

On a:

q-qo =m

(19,2)

('r-"C) 0

En differentiant il vient

dq=md?: +( -C

- 't:

o

)dm

et en multipliant par ('!' - 1: ): o

La relation (19,1) peut done s'ecrire:

(19,3)

20, Differentielle de S Ie long de la droite (Z ,Z 1). 0-

Considerons en x E l. un choc Yo -+ Y1 , L 'etat Y 1 posterieur au choc verifie manifestement les conditions (18,5). Nous nous proposons d 'evaluer la differentielle de S pour une famille d 'etats telle que Ie point Z correspondant decrive la droite du plan (re, <j) joignant Z Zl' D'apres (17,3), cette droite a pour pente:

m

2 2 c a

o

a

- 164 -

Lichnerowicz a) De:

on deduit: /'f -1 1:' dS= ~ dp+d 'C S 2 2

c r

Or, Ie long de la droite envisagee:

et en differentiant

X 0( 2=const.:

On aen deduit:

Ce qui peut s 'ecrire, compte-tenu des valeurs de

0(

et X

(20. 1)

b) Le second membre de (20. 1) peut s 'evaluer aisement en fonction de P(l). La relation (17.1) definissant P(l) peut se mettre sous la forme:

- 165

Lichnerowicz

ce qui peut s 'ecrire:

II en resulte:

P~) a

-

if (c2a2~+r-lh,2

10/ loi)+~r 10(

r

On deduit ainsi de (20.1) la relation importante: Ie long de la droite (2 0 ,2 1):

(20. 2)

, ....J dS 't"S...... ~ =

d"" = ~ dq.. 2 4 a flo( c a 2

P(l)

21. Orientation dans Ie tempos des ondes de choc. a) Considerons au point x ~!. un choc qui n'est ni nul, ni d'Alfven. Nous nous proposons de montrer que sous les hypotheses (H 1) ~ compressibilite ('t:' -< 0, 'C s' :> 0) que nous postulons, l'onde de choc p--

magnetohydrodynamique Lest orientee dans Ie temps. Considerons la famille des etats Y verifiant (18.5) et telle que Ie point 2 correspondant decrive Ia droite (2 0 , 2 / II resulte de

- 166 -

Lichnerowicz (19.3) et (20.2) que lIon a dans ces conditions:

(21.1)

d% =2f

e dS

et

(21.1-)

..,s

n-I

b) Si {i. 10( est> 0,

que P(l) est

>

N 1,1\

dS

l-

=~ 2 4

dq-

c a

est> 1 et on deduit de (12.2), soit:

O.

Si 10(10(=0, on a P(1)=c 2rf(1-1)(Uo(la)4> 0 Aussi si le(lQ(~O, on a P(l)O

0< est

>

et d'apres Ie Iemme 2 du

§ 16,

O. 11 resulte de (21. 2) que dS/ dq est> 0 Ie long de la

droite Z0' Zl' D'autre part, Ie long de cette droite, et la fonction ~

cJlb (Z o,Z 0 )=0, 1&z 0 ,Zl)=O

est stationnaire en un point au moins du segment

(Zo' Zl)' D'apres (21. 1), il en est de m~me pour S, ce qui est en contradiction avec dS/ Uq )0. C lest pourquoi 10( 10« 0 et

E

est oriente dans Ie temps. Nous

enon<;ons: Theoreme. Sous les hypotheses (H1)-.tt~<.0, 't's).O), toute onde de choc magnetohydrodynamique est necessairement orientee dans Ie temps. Si v; ~v~ sont les vitesses de

L

par rapport au fluide avent et

- 167 -

Lichnerowicz apres mle choc, on a v

r < c, v,! <: c.

'0

D'apres Ie lemme 2 du

1

§ 16, % est;? a et

X =k

nous posons

nous pouvons, pour un choid convenable des signes substituer

2

;

a la

seconde condition (18.5) la relation k=04=k eX • On a alors: o 0 2

k +k

2

-2kk =(k-k )

2

000

La fonction d 'Hugoniot (18. 8) peut s 'ecrire:

(21. 3)

A,P 2 2 2 1 2 'J1I,:J(Z ,Z)=c (f -f }-('t'+"t )(p-p )+('t"- 't:) -u(k-k)

o

0

0

0

0

21

0

et la relation d 'Hugoniot:

(21. 4)

2 2 2 1 2 c (f -f H't" +'t')(p -p )+(1:' -"C )-LI,..(k -k) =0. 1 0 1 0 1 0 1 0 21 1 0

22. Approximation classique de la magnetohydrodynamique relativiste. Nous nous proposons de deduire les equations de choc de la magnetohydrodynamique classique par approximation

a partir de celles cor-

respondant au cadre relativiste. Nous nous limitons aux chocs non tangentiels (arO). Nous posons dans la suite: uti 1 ol

=

w c

j=rw,

de telle sorte que a=j/c. Cherchons les parties principales, relativemente

-2

a c , des equa-

tions de choc (17.1), (17.2), (17.3), (17.5), (17.6). On a d'abord:

- 168 -

Lichnerowicz

Dl = [rw] =0

(22.1)

L'invariance de b=fhcx loe donne l'~quation:

On en

d~duit

qu'a des termes en c

-4

pres:

(22.2)

On retrouve qu 'a l'approximation classique: (22.3)

De (17. 3) il

ce qui peut

(22/4)

r~sulte:

.s'~crire

a l'approximation classique: 2

rw +p+

1

2ft Ihl

2

Compte-tenu de (22.2), l'invariance de

=0

- 169 -

Lichnerowicz donne alors

a des

termes d 'ordre superieur pres: 2 jh \2 _c_ (hO( I }2 _0_ .2 0 ~ 2 J r

o

n

en resulte:

t

(ho(I0(}2

2i

+

_0_ _

.2 J

Ihrl: ]

.Q

soit, d'apres les relations precedentes:

(22.5)

La relation (22.5) peut encore s'ecrire:

soit

(22.6)

Considerons enfin l'invariance de L. Tout d'abord:

c

-2

0(

Ii = - (1+-)-

r

c

2

c(

(HIed

2

2

u. ( . _ - -~)

J

.2 J

2 2

c r

- 170 -

Lichnerowicz est equivalent

1

a -r - ru.

(ho( 1N)2

.2 J

a l'approximation

classique. D'autre

part:

est equivalent

a I q 2.

Ainsi:

(22.7)

Nous avons retrouve en (22.1), (22.3) (22.4), (22.6), (22.7) les equations de choc de la magnetohydrodynamique classique, ecrites dans un repere lie au choc.

23. Thermodynamique des chocs. a) p et 8 etant prises comme variables theormodynamiques de base, f(p, 8) verifie d'apres (5.3):

(23. 1)

2 c f' = V:>O

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)

<

Ie mnumerotage des etats Z

on deduit

J

P1

Po

~(p,

S )dp 0

II en resulte d' apres la condition de c onvevite (H 2): 2 2 2 c (f 1 -f ) «P1- P )("C(p ,S )+ 't'(P1' S )) o 0 0 0 0

soit 2 2 2 c (f -f}-('1; +'l:;)(p -p ),0 1 0 1 0 1 0

On deduit Ide la relation d'Hugoniot que

"t" 1

> '00

ce qui implique

contradiction, On a donc PI =p 0 et Ie choc envisage ne peut Hre que nul ou d'Alfven, Theoreme 2, Pour un choc qui n 'est ni nul, ni d 'Alfven, on a sous

- 172 -

Lichnerowicz

p >i'=p 1

0

En particulier toute onde de choc est une onde de compression et

:Y 1
Supposons en effet

L.

De (23.2), on de-

duit: c 2[ f 2 (p

o

,s

0

}-f 2 (p l' S) } = 2 0

fPo

PI

1;' (p, S )dp 0

D'apres la condition de convexite (H 2), il en resulterait:

Comme So< SI' on aurait a fortiori puisque fS>O,

"t'S»O

soit: 2 2 2 c (f -f)-( 't' + 't )(p -p ) > 0 1 0 1 0 1 0 La relation d 'Hugoniot donne alors

't'1 <

~ o'

D' apres (HI) cela est

contradictoire avec Pl~Po' Sl>So' On a donc PI"? Po et, d'apres (23.1), f 1>fo ' Pour etablir '<='1 <'to' on part de:

- 173 -

Lichnerowicz

n

en resulte puisque

~'
p

ou a fortiori": 2 2

2

c (f 1 -f }-2~1(P -p ) o 1 0

>0

De la relation d'Rugoniot il resulte alors:

('t' - '?:

1

0

}{p 1-p0 + 1..2/M.(k 1-k 0 }2l<0 j

soit 't' 1 < 't'o' ce qui demontre Ie tMoreme. On a par suite

0(1 < 0(0'

VI. ONDES DE CROC ET ONDES D'ALFVEN.

24. Ondes de choc et ondes d'Alfven. Considerons Ii la traversee de

L.

un choc non tangentiel qui ne

soit pas choc d' Alfven a} Nous nous proposons Lemme.

n

d'~tablir

Ie lemme suivant

existe toujours au moins une dicection n orthogonale

u ,u1,h ,h , . '--()-l

-0 -

Examinons les differents cas possibles:

a 1,

- 174 -

Lichnerowicz 1) Si

01. 0 011 f 0, il existe une direction n au moins orthogonale

aux vecteurs 1;, V et X avec:

(24. 1)

De (24. 1) il resulte que n est orthogonal

a

v

o

et

v

l'

donc

a u0

et u l . Comme:

(24.2)

nest aussi orthogonal 2) Si

ex

~

=0,

--0---.-

a h 0 et hI on a

'\I

/Yo

=0 et il resulte du lemme 2 du

So. 16 que 1 est dans Ie 2-plan (u ,h ). Le vecteur Vest orthogonal

:5

0

0

a 1 dans ce 2-plan. II existe une direction n au moins a 1, V et u l . Cette direction est or~ogonale au 2-plan etant orthogonale a V es u l est orthogonale a hI. 3) Si

0(

b) En xE

o~c(lfO,

L,

orthogonale (u ,h ) et o

0

il suffit d'echanger Ie rt'lle des indices ·0 et 1.

introduisons une perturbation infinitesimale de l'etat

anterieur au choc. II en resulte une perturbation infinitesimale de I'etat posterieur au choc reliee

a la precedente par les relations ob-

tenues en differentiant les equatinns fondamentales de choc. II vient ainsi:

- 175 -

Lichnerowicz

[b rue/. +rbucAJ (24.3)

[~(ho/

lo<)ue> +(hO(

l()(

=0

lotl~u,B -cS(~)h~ - ~ q)~J =0

[6(p2 ;)uf.> + ~2 ~bl' _y- ~ql> _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 <0. Supposons, pour fixer les idees, 0(

0(

b")'

0

et par suite jUolo(>O (resp.hl~ >0). D'apres (24.8), Ie vecteur B o (resp. B 1) est oriente par rapport a L du m~me ctlte que I. Quant

- 177 -

Lichnerowicz au vecteur Ao (resp. AI), son orientation par rapport de I ou l'orientation opposee selon que au negatif. Pour

0(

nul, A est tangent

a L.

est celle

Of 0 (resp. eX 1) est positif

a L

Si b etait

<

0, les

rOles des vecteurs A et B seraient simplement inverses.

25. Compatibilite d'une onde de choc avec les ondes d'Alfven. a) Nous examinons d'abord les cas OU Ie choc choc d'Alfven, est tel que

D(

1

L

envisage, non

01. 0 0(1=0. Supposons en premier lieu

=0

0<

0

>0

Dans l'etat

Y;, les ondes d'Alfven de type A et B qui abortissent o en X6 peuvent creer en ce point une perturbation ( C\U 3 , ~h 3) o 0 arbitraire et, dans (24.4), (24.5), on a puisque eX 1=0

z..

a

2

0( 2 -(h 10\.) =0 1

2 r1

Pour que ces relations (24.4), (24.5) admettent une solution quels

~u o3, bh 03, il faut rapport a ces variables:

que soient par

soit:

(25. 1)

et il suffit que lIon ait identiquement

- 178 -

Lichnerowicz

(25.2)

Si nous mettons (25.1) sous la forme: 2 -a rl

0(

ftl

~!

-

a r

hI IQ{

I. hoe I o CI.

0

il vient d'apres (25.2):

~o<

o

=0

soit

~ =0 ce qui est impossible.

choc

L.

n y a incompatibilite de I 'onde de

avec les ondes d'Alfven

b} Supposons maintenant:

0( =0

o

ainsi

2 f-H=c 1:

(25.3)

0

est> O. On a:

;>0

Avant Ie choc, une onde de type B et une onde de type A tangents

L

peuvent abortir en x E L

a

creant une perturbation arbitraire

( bu 3, ah 3). Mais peuvent s'eloigner de x d'une part l'onde de type A o

0

pour I 'etat Y port ant une perturbation (

J u3

,

J' h 3

) verifiant:

A A 0 0 0

- 179 -

Lichnerowicz

(25.4)

d'autre part une onde d'Alfven de type B pour l'etat Y1 portant une perturbation (

6u~B' dh~B)

verifiant

(25. 5)

Les relations (24.4), (24.5) relient cette derniere perturbation

a une

perturbation ( Ju 3, Jh 3) anterieure au choc 'l.vec o 0

Ainsi pour qu 'une perturbation ( travers Ie choc

L

l uo3, J"h 3) puisse 0

t!tre transmise

a

et s 'eloigner, il faut et il suffit que:

(25.6)

Nous etablirons dans un instant que:

f31 L'onde de choc

L.

a

ro

+ holol..J

""

f

0

sera compatible avec la perturbation arbitraire

( ~ u 3, 6 h 3) s'il existe toujpurs une decomposition o

0

- 180 -

Lichnerowicz

avec: r 3 oh

oA

=

-13 C\U fo

3 oA

ou l'on a pose:

TI

B ~ +h~ I

r1 On est ainsi amene

ro

0

'"

a etudier Ie systeme lineaire:

Le determinant de (25.7) est donne par:

1T+~= o

131hQl0 IeX +V(.2, 02

~ + r

0

(l,

J-' 0

((.J,

J~ 1

Po ~ + hoi I n ro O d

a +ho( I ) roo( 0

( A ~ +hoi I )=0 ror 0 0< o

0

4tH

- 181 -

Lichnerowicz Ainsi (25.7) n 'admet pas de solution pour toutes valeurs des seconds membres et

L

n'est pas compatible avec les ondes d'Alfven.

II nous res t e a' mont rer que

rf'.. 1 r_a +licx.0 l ctJ40

. S'1 non, on aural't

o

ce qui peut s 'eerire:

!h 12

220 r 0 (e 1:0+f -2- )

r

o

ou

II en resulte:

1

ttl soit en divisant par

0<

e2

('{; 0+

( _1_

0(

1;2

1

1

"C!

Ie 2=

o

"Z: - 7:

1

1: 1)= - - =H't' 09

0

(h~ 10/ 2

a

't;2

1

- 182 -

Lichnerowicz Apres simplification il vient: (heX I )2 1 d. 'C' =0

a

Cette relation implique

2

1

h1 =0 donc H=O, ce qui est absurde.

Nous pouvons enoncer pour un choc non tangentiel. Theoreme 1. Si

z:

est une onde de choc telle que

est incompatible avec les ondes d 'Alfven sponde

a un choc d'Alfven

a moins

0( 0(

o

1=0, elle

qu 'elle ne corre-

(01. = 0( =0) . 1-

.-....-----------'- 0 -

Les chocs singuliers

n 'ont pas de realite physique.

c) Supposons maintenant que pour Ie choc

L

envisage, on ait

Les ondes d 'Alfven de type A et B dans l'etat

Y creent en x o 3 3 une perturbation ( ~ u , Jh ) arbitraires; <0(1 etant fO, Ie o 0 systeme (24.4), (24.5) admet une solution unique definissant une perturbation de l'etat Y1 pouvant s 'eloigner de x selon les ondes d'Alfven A1 et B1 correspondant

a cet etat. II y a compatibilite

de l'onde de choc avec les ondes d'Alfven d) Supposons que lion ait

eX 1

< 0 < C
Les ondes d' Alfven de type A et B dans l'etat Y creent toujours o en x une perturbation arbitraire (~u 3 6h 3) o 0

- 183 -

Lichne rowic z

Mais seule s'eloigne de x, dans l'etat Y l' une onde d'Alfven de tipe B port ant une perturbation verifiant

De (24.4) et (24.5) resulte:

11 est necessaire que

(25.3)

(J u 3, o

5h03)

verifie la rEllation:

A.2 ~ + A hOI 1 ) c\u 3 _( A ~ +holl )c5h 3 =0 r rIo 0( 0 ''''I r o d < 0 o 0

J'" 0

Cette relation ne peut Nre une identite en (du 3 , Jh 3 ), sinon l'on o 0 aurait

ro (l..

~ +ho( 1 =0 roO( o

- 184 -

Lichnerowicz et par suite choc

L

0< =0. Il yaa

o

ainsi incompatibilite de l'onde de

avec les ondes d'Alfven

e) Supposons enfin que l'on ait:

Avant le choc, seule une onde d 'Alfven de type B aboutit en x creant une perturbation (

0 u!B'

3 ,3 Ou oB - oh oB

t;

ro

hh!B) verifiant =0

Mais peuvent s'eloigner de x une onde d'Alfven de type A pour l'etat Y

et une onde d 'Alfven de type B pour l'etat Y 1 qui avant o

le choc correspond

cS u

3

oB

3 =&- u 03 + ~ u oA

avec, d'apres (25.3):

ou:

-

3

-

3

a une perturbation ( ~ u o , cS h 0 ). Nous posons:

- 185 -

Lichnerowiczz NOBS sommes amenes comme au -

3

b

3

a resoudre les equations: 3

~u o +~u 0 A'" =~u oB

3 'IT -~ u o3 - ro t2. ~ uo u A=

3

""o 0u 0B J-

ou Ie determinant

Si

1'1: II y

g

+h~lo(

=0 on obtient aussi trivialement une solution.

compatibilite de l'onde de choc 2: avec les ondes

d'Alfven. Nous enon90ns Theoreme 2. Pour qu 'un choc non d' Alfven soit compatible avec les ondes d' Alfven, il faut et il suffit que

0(

0~ 1

>

0

f) Les chocs envisages peuvent done ~tre decomposes en

chocs lents tels que

et chocs rapides tels que: 0<0( <0( 1 0

Pourtant de la relation fondamentale:

- 186 -

Lichnerowicz

Pour un choc lent

0< ~

>.o
k~ < k!

donc

et par suite

'''' l't'" '" 1 "'s sont 'mvers",es, lh 1 2< Iho\2, P our un ch oc rapl'd e, I es m",ga ' On a donc: Proposition, Dans un choc lent, la grandeur de champ magnetique diminue; dans un choc rapide elle

augmente,

VII, VITESSES DES ONDES DE CHOC ET THEOREMES FONDAMENTAUX,

26, Entropie et courbes isentropiques, a) Un etat initial Y

o

etant donne, nous considerons la suite des

etats Y verifiant

(26, 1)

H(Y)=H(Y )=H o

kg( =k lX

o

0

(avec

~O(

et etudions les points representatifs correspondants Z dans ('t:', q); si H est

plan ('t',

eil,

> 0 la droite 't' = f

la courbe isentropique

j

(26,2)

correspondant

2 ~ LLk 0( roo ----2 2 (c't: -}'lH)

En derivant (26,2) Ie long de

~

~e

plan

HI c 2 est interdite, Dans Ie

S de l'eutropie est definie par:

1 p('t',S)+'2

'> 0)

0

, il vient:

a la

valeur

- 187 -

Lichnerowicz 2 2 2 c LLk 0( roo

0<.3 et en derivant line seconde fois:

On en deduit d'apres (8.5)

2_

(~) =

(26.3)

d~j'

1

-~

M

P

ou l'on a pose:

(26.4)

qui est strictement positif

en vertu des hypotheses (Hi), (H 2).

Ainsi les courbes isentropiques

Notons que la fonction

(26. 5)

~

s

= -

91 1:'

S

1:'p

-;J

du plan ('t, cD sont convexes

('l:,S)definie par (26.2) verifie

- 188 -

Lichnerowicz b) Soit

A

une droite de pente m du plan ('t', q) issue du point

Zoo On a, Ie long de

q-q

o

A

= m(t -

Lemme. En tout point Z

or0 )

de

'!

A

ou S est stationnaire, qn a:

I

-( e,2

,

p 1:S

d 2S

A'lssi (-2 ~ est

-< 0,

M)p ~

sous les hypotheses (HI )~H2l.-

Par1:fns de la relation fonctionnelle:

q= 9' (t, S) A

et derivons la Ie long de

par rapport au parametre 'l:.

vient: CD' 11;

+

dS

CD' ( - )

IS

d'C~

=m

et en derivant une seconde fois:

h

dS dS2, 'f~2 +2q>~'t: ( d"ClA +
~

de

/1

ou (dS/d't:1.., -.,

(
On en deduit d'apres (26.5)

+(CP's)1

d2S _ ( d'C.1 ~ -0

s'annule, on a: 2

(~~

=0

Il

- 189 -

Lichnerowicz

ce qui demontre Ie lemme. c) Considerons dans Ie plan (t', q), la courbe d 'Hugoniot cjG definie par l'equation ~{z, Z)=O. On deduit de (19. 1) que Ie o long de ~ : (26. 6)

2f

dS

® {d,,.)',1 .. '11'0

+("t -

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 enon<;ons: o

Theoreme. La courbe d 'Hugoniot et l'isentropique S=S du plan (t:, ti) o

ont au point initial Z

un contact du second ordre. Pour un choc faible,

o I 'accroiseement d 'entropie est du troisreme ordre par rapport

puissance du choc mesuree par I 'accroissement de pression

a la

- 191 -

Lichnerowicz 27. Conditions necessaires sur les vitesses des ondes de choc. a) Considerons un choc qui, au point x de

a Zl'

Nous avons vu que Ie long de la droite

la fonction

11<,

L , fait passer de Z o A joignant Zo a Zl'

(Zo' Z) est stationnaire en un point au moins

Z~ situe

entre Z 0 et Z 1 et que d 'apres (21. 1), il en est de m~me paur S. Du lemme du

§ 26,

il resulte que Ie point Z-5 de

tionnaire est unique et correspond La variable 't

a un

decroissant Ie long de

b

ou S est sta-

maximum strict de S sur

/J..

orientee de Z 0 vers Z l'

on a:

dS

d't:

'> 0

pour 't .:::.. "r:-t

et aussi:

(dS)

d't:

0

<

0

On en deduit d'apres (20.2):

(27. l)

0< P(l) ') 0

o

0

b) La relation (12.5) peut s'ecrire:

r.

2

(27.2)

b

a

2 2 ~ 2 ot (c rf+JPhl )((u 1«) -1 Ie*;)

oI..=(v)

c

2

2

p.) (vr

c

2

- 192 -

Lichnerowicz Considerons un choc rapide (0

v

L.

L

A

>v0

o

v

De (27.1) il resulte que Pill

m~me

en est de

< 0( 1 < do).

pour

TT

1

>

D 'apres (27.2), on a:

A v

1

2)0 et Pill <0; d'apres (12.2), il 1 (L.:--i). On en deduit c

o

est

z:

Considerons maintenant un choc lent ( IXI

(.0(0

<0). D'apres (27.2):

De (27.1) il resulte que P(l)o est <0 et P(I)l/'O .. On en deduit de m~me

v

l: o

Nous obtenons: Theoreme.

Sous les hypotheses de compressibilite (Hl)~), les

r.

I.

vitesses v 0 et par _ rapport au_fluide d_ 'une onde de_choc, ----- - -v1 -'-_ .!....L_ _ ___ __ __ _ respectivement avant ou apres Ie choc, verifient vnecessairement les inegalites suivantes: 1) pour un choc rapide ML

v n

A

< V0 <

MR!.

v0

<

vVo

ML A L VI <.v 1 <;: v 1

MR

< vI

- 193 -

Lichnerowicz 2) pour un choc lent: ML !. v o
-< v A
0

ou interviennent les vitesses

magn~tosoniques

les vitesses d'Alfven avant et (27. 1)

r~sume

apr~s

lentes et rapides et

Ie choc.

la situation.

28. Theoreme d'existence et

d'unicit~.

a) En inversant la. fonction 't' ="2:' (p, S), on obtient une fonctioR S=S( p, 'Z;) exprimant l'entropie en fonction des variables En raisonnant comme au

't;'

p

Les conditions de

§ 8,

p et 't: .

on obtient:

~

- S'

't:

compressibilit~

't;' =

S

1 S!c

(H 1) se traduisent par les

in~galit~s

S~>O, 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.

<

- 194 -

Lichnerowicz b) Considerons dans Ie plan ('t;, q) la branche de l'istmtropique correspondant de

-5 , nous

a S=S qui est telle que o

"C'~

"'C avec 0

> O.

CXO( 0

J

Le long

avons d'aprus (19.1)

En derivant une seconde fois, il vient:

soit d'apres (26.3):

Aussi sur la branche consideree, (diG /d't"~ est» 0 et est

1fe, (Zo~

<.0. c) Soit

11

que Ie long de

(28. 1)

Soit m

une droite de pente m issue de Z . Il resulte de (19.3) o

A: ( d~ ) d't:

0.( dS ) d't'

!::. =2f ~

la pente de l'isentropique

-&

t.

en Z . De l'hypothese globale

o 0 faite, il resulte que Ie long de la branche envisagee de

~ , q peut

prendre des valeurs arbitrairement grandes. Or la courbe convexe. On en deduit que si m <m ; la droite ---0

A

--!

est

rencontre la bran-

- 195 -

Lichnerowicz che consideree de

Mais Ie long de

~

en un point unique Z AfZ 0 et pour ce point:

A

, S(Z ):S(ZA)=S et S(~) est necessairement o 0 stationnaire entre Z 0 et Z A en un point Z~ necessairement unique

d'apres Ie lemme et qui est un maximum strict pour S sur Nous avons donc (dS/d't)b.

< 0 en

Z . Quand Z decrit

<

0 en Zo et d'apres (28.1), (dlie/d~)A

t::. de Z

vers zA,1tG (Z ,Z) commence

0 0 0

par ~tre positif et il existe donc au moins un point Z Z

o

et Z

A

pour lequel ~ (Z ,Z1)=0. 0

Nous savons que sur

Il, S

croit de Z 0

maximum en Z~ et decroit constamment a Ie

m~me

/l

a Z-&~

1

sur

A

entre

passe par un

ensuite. D'apres (28.1),1-'

comportement et Ie pojnt Z 1 est necessairement unique.

a tout nombre m <m , correspond un point unique Z 1 -de "--"--------'-0 la courbe d'Hugoniot de Zo avec 't'1< 'to ~o( 1 ~o ';> 0) tel que Aussi

m=q1'~)rt' 1-~' Nous voyons que m fournit une parametrisation simple de la branche consideree de la courbe d 'Hugoniot. Nous notons qu'il reesuHe de(19.3) que Ie long de

.-x:

et que par suite S croit quand m decroit. c 2a Z ~,P Pour m=-(J{--(O, Z1 satisfait (17.3) et 'l'I'<>(Zo,Z1)=0; Z1 etant connu,

t 1 letlOllPl Ie soit et f1 est donne par exemple'par la

relation d 'Hugoniot. De (17.5), on deduit la valeur de

I

hl,

de (17.1)

- 196 -

Lichnerowicz celle de u

~Id

et de (17. 2) la valeur de

~ I 01. (qui

a Ie signe de

h~lo<)' Les equations (17.1), (17.2), (17.3), (17.4), (17.5) ont aussi

DC 10(0

une solution unique telle que

>0

d) Composons la tcondition

et la condition necessaire trouvee au

§ 27:

~ ~O, on a:

D'apres les resultats du

(28.2)

Or 2

c't'

En divisant (28.2) par "t"

Or d'apres

Ie

~ 26 a):

p

p

0(,

-- ~ 2 r

il vient:

- 197 -

Lichnerowicz

1 c 2}Lk 2 ~0( P

1 --'7:;' p

On obtient aussi: 1

(28.3)

Au point Z

o

il vient:

2 2

=

m -

c a

o

2 2 0( On voit aussi que c a /1 10( < m 0 est )equivalent

a ()( 0P(l) 0> 0,.

Nous

enon9 0ns Theoreme. Si la fonction S( p, 't') satisfait les hypotheses (H 1).JB 2) pour 'to" '- ~

et pour des valeurs arbitrairement grandes de

>0

a tout

etat

P(l) correspond une solution unique non triviale ·0-0 des equations de choc telle que Oi. 1 ~ > O. y

--0

verifiant

p ,

0(

Les inegalites figurant dans cet enonce s'interpretent immediatement, comme au

927,

en termes de vitesses de l'onde de choc et des vitesses

magnetosoniques et d' Alfven.

- 198 -

Lichnerowicz

VIII. RETOUR AUX RAYONS MAGNETOSONIQUES.

29. Etude de la direction N ~. a) A partir des equations fondamental de la magnetohydrodynamique, on deduit que pour une onde infinitesimale lignes de courant (uO{ 10(

L non engendree par des

fo), on a d'abord:

Aussi on a l'invariance par

~ du scalaire et des deux vecteurs:

V~

0(

a=ru 10( fO

Les invariants scalaires de choc mis en evidence aux ~ ~ 16 et 17 sont donc aussi des invariants pour l'operateur

b

de discontinuite

infini te simal e. Nous supposons l'onde envisage non pas d'Alfven (0. fO), mais magnetosbnique de telle sorte que

P(l)=O

D'apres les ca1culs du

(29. 1)

r

~ 20, ceci peut s'ecrire:

-21 (c2a21::+f\hI210"\)I)+oIloilcx. =0 (c 2a 2-c +)-,-lh,2 lo(lo(f O)

r b) Transformons de

m~me

l'expression du vecteur

la direction du rayon (voir (14. 3))

NP-> 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)

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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

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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

® = ----4

Ig 44 ax

and

®'

=

(9.8)

- 236 -

10.

Taub

The Lagrangean For A Perfect Fluid We shall define this function as 00 .1)

SS)

where p,

E,

® and

S are defined as above.

Then the

variational principle discussed in Lecture I may be written as

1'(0) I

=a

where

= I g - 2Kl r

00 .2)

IRA

00.3)

with Ig

=

d\

V

If

=

Ip(C 2

+

E

-

4

d x

«I)S)A

00 .4)

It then follows from the results of the preceeding section that the equation

is equivalent to 2 I'

r

=

I[T\.IV g , r \.IV

2pSU\.la' Jr-g d 4x , ,\.1

00.5)

where T\.IV

r

_ pg\.lV = pCc 2 + E + £)U\.lUV p

and use has been made of equations (9.3), (9.6) and (9.8).

00.6)

We

may integrate by parts the last term in equation (10.5) and obtain 21'

r

Q , = I[T\.IV r .:l\.lV

+ 2CpSU\.I)

;\.1

a'J;:;g d 4x (10.7)

- 237 -

Taub

We now turn to the calculation of r-gR

I'g •

We have

]..Iv = r-g g R]..IV

and R]..IV

= rO ]..IO,V

rO + rP rO ]..10 pv ]..IV,O

P rO r ]..IV po

(10.8)

where P r ]..IV

1 pI..

= 2"g

(gA]..I,V + g AV,]..I - g]..IV,A ).

(10.9)

Hence ,P 1 PAC , r ]..IV = 2"g g A]..I;V + g'A]..I;V - g']..IV,A )

(10.10)

where the covariant derivative is taken with respect to the metric g]..lv' g ']..IV

In deriving this result we use the fact that

= -gOT 'g0]..l gTV ,

(1o.n)

as follows from the equations

= 6]..1 o· It is a consequence of the above that rIO

]..IV;O

)

=

g

]..I P va), ] g g ]..IV;P ;0 (10.12)

Therefore I'

(10.13)

where G]..IV

= R]..IV

I

2" g

]..IV R •

The Euler equations are obtained by requiring for arbitrary

,

g]..lv(O)

and

u'(O)

of the region of integration.

1'(0)

=0

which vanish on the boundary

Since the second integral in

- 238 -

Taub

equation (10.13) has an integrand which is a divergence of a vector field, it vanishes for such variations and the Euler equations become

=

o

(10.14)

=

o

(10.15 )

and

where

T

].1V

F

is given by equations (10.6).

In view of the definition of

and the definition of

specific entropy, equations (10.14) and (10.15) are equivalent to (10.14) and (10.16) This is so, because equations (10.14), together with the Bianchi identity G].1V ;V

= 0

imply that T

].1V

F;v

2 V = [p(c + e: + £)U].1U p

pg].1v] .

'v

= 0

(10.17)

Hence

= o

= In view of equation (9.5) we may write this equation as v) + pU v 8S Cc 2 + e: + £)(pU p ;v ;v

= 0

or as (c 2 + e: - ®S + ~) (PUV);V +

®( PUvS)

;v

= 0

Hence equation (10.15) implies equation (10.16) and conversely.

- 239 -

Taub

Equations (10.14) and (10.16) are the Einstein field equations for a self-gravitating fluid when matter is conserved. Equations (10.17) are the equations of motion of the fluid. Equations (10.16) and (10.17) reduce to the special relativistic equations of conservation of matter, energy and momentum when the space-time is Minkowski space.

They may therefore be

properly called the generalization of these equations to general relativity.

The special relativistic equations re-

duce to the Newtonian ones in case the fluid particle velocities are small compared to the velocity of light. It should be noted that through the use of the comoving coordinate system in which both the unperturbed and perturbed particle paths are represented by the curves

x

i

= constant.

The variation of the particle paths does not appear in the evaluation of

Thus the field variables corresponding

I'.

to the particle paths do not appear in this integral and there are no field equations for these variables.

The equations

describing the particle paths in a general coordinate system are of course derivable from equation (10.16) and (10.17). The latter ones may be written as equations (10.15) or as pS

U~ ,~

(10.18)

= 0

and p(c

2

+

')UvU~ ~ ;v

_ = P ,v (gV~

U~Uv)

(10.19)

where i

=

E

+

E p

(10.20)

- 240 -

Taub

LECTURE III SINGULAR HYPERSURFACES 11.

The Existence of Shock Waves It has been shown [1, 2] that solutions of equations

(10.14) for

g

~v

,p, p, and

u~

may be found which reduce

to solutions of problems in fluid flow in the special theory of relativity when

K = O.

If we further take the limit of

c

• , we obtain solutions to problems

these solutions when

~

in classical hydrodynamics.

Further, the approximation

method which was used is such that any special relativistic solution of a plane-symmetric problem in hydrodynamics could be obtained by such a reduction process. In particular those solutions of the special relativistic: equations describing the motion of perfect fluids which are physically unacceptable in certain regions of Minkowski (flat) space-time have their counterpart among the solutions of equations (10.14).

Such solutions are associated with the

formation of shock waves in special relativity and in classical theory [3].

For this reason it was suggested that the theory

of a perfect fluid in general relativity must allow for the existence of shock waves.

That is, we must contemplate the

existence of three-dimensional hypersurfaces in space-time across which there may be discontinuities in the stress energy tensor, the

g~v'

and their derivatives.

If such hypersurfaces are to be considered, then we must consider equations (10.14) as holding on each side of such a hypersurface, and we must supplement these equations by

- 241 -

Taub

conditions which relate the values of the g

~v

,

the

derivatives of these quantities, and the stress energy tensor on both sides of such a hypersurface.

The relations that must

hold between the components of the stress energy tensor across a hypersurface of discontinuities must be the generalization of the Rankine-Hugoniot equations of classical and special relativistic hydrodynamics [3J. It is the purpose of this lecture to derive and discuss a set of conditions of the type described above for a spacetime in which the matter present is a perfect fluid.

The

method used is based on the existence of the variational principle discussed previously. If the field equations and the equations of motion, the conservation equations, can be derived from a variational principle, then we may generalize the variational principle by allowing the region of integration involved to include regions of space-time which contain hypersurfaces across which the matter distribution and the metric tensor and its derivatives are discontinuous. hypersurfaces.

We may even vary these singular

In the classical theory such a generalization

is known to give the classical Rankine-Hugoniot equations [5]. It will be shown below that such a generalization of the variational principle leading to the field equations and the equations of motion for a space-time containing a perfect fluid leads to a general relativistic generalization of the Rankine-Hugoniot equations which reduces to the appropriate equations in the special relativistic and classical limits. shall also obtain conditions that must be satisfied by the

We

- 242 g~v

Taub

and their derivatives across hypersurfaces of discon-

tinuities.

Such conditions have been discussed by S O'Brien

and J. L. Synge [6] and by A Lichnerowicz [7].

The results

given below are obtained in a general coordinate system. When the coordinate system is chosen to be that used by O'Brien and Synge, the equations obtained reduce to those given by them. A transformation of coordinates with a discontinuous second derivative may be show [8l

to reduce these conditions

to the requirement of Lichnerowicz, namely that the derivatives of the

g~v

and

be continuous in the new co-

ordinate system. The equations governing the behavior of thin shells of material discussed by W. Israel [9] and A. PapapeTrou and A. Hamoni [10] may also be derived from a variational principle.

These equation differ from those describing shocks

because in the former situation, the integrand occurring in the integral defining the variational principle is such that its evaluation involves an integral over the singular hypersurface.

In the case of shocks no such hypersurface integral

occurs directly in the definition of the variational principle.

12.

The Generalization of the Integral I The variational principle we shall now consider will

involve the integral I defined by equation (10.2).

However,

we shall assume that there exists a three-dimensional hypersurface

L which divides the region of integration

two four-dimensional regions

VI

and

V2

V into

in each of which

- 243 -

Taub

the integrand exists and is integrable. be a contribution to

I

There mayor may not

involving an integral over

r.

Thus I

f

=

VI

(R - 2K£)r-g d 4x + § +

V2 02.1)

where IA

=

f

2K£) r-g d

(R -

4

x,

(A= 1, 2)

02.2)

VA S

dr

=

I'd r ,

02.3)

is the invariant three dimensional volume measure on the

hyper surface

r

and

I

is a function of the

field variables of the fluid evaluated on

g).1V

and the

I = 0,

In case

r.

we will see that the singular hypersurface is a shock wave or a boundary between two regions of space-time across which no matter flows (slip-surface).

In case

f *- 0

the singular

hypersurface will be shown to be a thin shell. In a general coordinate system with labels family of hypersurfaces

a

may be defined by the para-

~(e)

metric equations

= x *).1 (Ct,B,y;e) where the parameters

Ct, B,

surface and the parameter The vector field

=

y

e

02.4)

define a point on the hyperlabels particular hypersurface.

- 244 where

Taub

a, Band yare kept constant in the differentiation

is then defined on the hypersurface

~(e).

We observe that in the comoving coordinate system we have

where

dx ll ae-

is evaluated for fixed

a, 6, y

and

e

by

use of

equations (12.4) and the inverse of equations (8.4). The coordinate system used to write equation (12.1) may be an arbitrary one in space-time.

In case it is the co-

moving one the variation in I due to a variation in the singular hypersurface is given by (12.5)

= where

(12.6)

and we have used the notation [h]

= li m {h(x ll _ E=Il) - h(x ll + E=Il)}

(12.7)

=

E+O

Equation (10.13) which holds in each of the regions

Vl lS

and

V2 may be used in the evaluation of

given by equation (12.1).

variables

gllv'

II

where

I

Under a variation of the

the fluid field variables and the singular

hyper surface we then have

- 245 -

Taub

I'

+f[{;:g(g~VgPO

-

gP~gVO)g~v;P

+ 2Kpa'SU o ;:g}AO]do (12.8)

+f::~=~d~ +f[f:g(R ~

2K£)=OAO]do +

~

f 1~Vg~vd~ ~

In this equation the last term arises from the variation of S in equation (12.1) with respect to the field variables. The third and fourth terms come from the variation in to the variation of the hypersurface

I

due

and the first two

~

terms come from the application of equation (10.13).

The

divergences occurring in that equation have been integrated under the assumption that VI

and

13.

,

g~v

vanishes on all boundaries of

V2 other than the hypersurface

~.

Shock Waves We shall first discuss the conditions I' (0)

in the case

f = I

1

~v

=

O.

= 0

That is the case for which

= II + 12

=

f V +V l

4

(R - 2K£)!=g d x. 2

In equation (12.8) we may set

,

g~v

where

g~v

=

represents the variation in

g~v

due to the

- 246 -

Taub

x*~

variation of the metric tensor at a fixed second terms represent the variation in

and the due to a

g~v

variation of the world-lines of the fluid particles.

After

an integration by parts, equation (12.8) may be written as = -f {(G Vl +V 2

I'

+

+

~v

~v • + KTF )g~v

I[r-g(g~VgPO

f[1=g{R(=o

vo • - g P~ g)g

~V;P

Ao]do -

I[I=gT o~· F

~

~

Ao]do

+ (13.1)

~

We now require that

1'(0) = 0

which vanish on the hypersurface trary.

~

for

a'

and are otherwise arbi-

It then follows that in region

equations (10.14) and (10.15) hold.

and

g~v '

VI

and in region

V2

It 1S a consequence of

the first set of these that T~v

F;v

1n regions

VI

and

V2 .

= o

(13.2)

Thus no additional conditions are

obtained by setting to zero the coefficient of

~~

in the

first integral on the right-hand side of equation (13.1). We restrict the variations of

a'

on the hypersurface

(13.3)

In view of the fact that equations (10.14) hold we may write this equation as

- 247 -

Taub

<13.4) This equation, which relates the variations of the hypersurface, the variations of the particle paths and the variations of the temperature is the general relativistic analogue of the relation which was previously discussed in the classical theory [5J.

It arises because in the theory of shock waves,

one does not have a conservation of entropy flow across shocks and hence one does not have the generalisation of equations (10.15) holding.

Instead one requires a generalisation of

the equation describing the conservation of mass. If equation (13.3) holds, it follows from equation (13.1) and the requirement that

I'(O)

= 0 for arbitrary

~~

that

we must have (l3.5)

These equations give the conditions that must be satisfied by the discontinuities in the stress-energy tensor of the fluid across the hypersurface

~.

They are a natural generalization

of equations (13.2) and may be derived from them by rewriting the latter equations as

= for arbitrary vector fields

f~

and integrating these equa-

tions over an appropriate volume in space-time containing the hyper surface

~.

These equations have been discussed in some

detail in reference [3J.

- 248 14.

Taub

The Conditions on the Metric Tensor Across a Shock We now turn to a discussion of the second integral in

equation (13.1).

In order to evaluate this integral we

introduce coordinates adapted to the hypersurface

r.

We

shall treat the case for which

o

<

Cl4.1)

and indicate the changes that must be made if this condition does not hold. in the region

We shall assume that 1n the region

Vl

(and

is defined by the

V2 ) the hypersurface

equation

where

x

o

1

1

x1

= x0

i x

= xi(oj)

Cl4. 2)

a, i, j = 1, 2 , 3

is a constant.

The induced metric on the hypersurface =

lJ

i

V

g llV x I1.Y IJ.da da

where we may choose the

ai

j

=

y .. da

r i

1J

is then da j

Cl4. 3)

so that Cl4.4)

are three vectors tangent to the hypersurface.

The unit

normal to the hypersurface is

=

Cl4. 5)

- 249 -

Tauh

where N2g11

= -1

(14.6)

11

= 0

(14.7)

and n Il x l'1 yij

If we define

so that

ij Y Yjk

= 6ki

(14.8)

we then have gllV

= _nlln V + yijxll"xV" 1

(14.9)

'J

where nil

= gllVn v

= Ng ll1

(14.10)

Equations (14.6) and (14.7) may be written as gIlV x Il I,n 1

v

= 0

and g

IlV

nlln v

= -1

These equations determine the components of n

1

1 = if 1yij

ni

= -N

N2

2 = g 11 (-1

nil

to be

= - Ng ll

g1j

= Ng l i + y

ij

g4i g4j)

2

It follows from equation (14.9) that

(14.11)

- 250 -

where

is the determinant of the

y

Taub y ..•

1J

We note that under the transformation of coordinates given by the equations

where

we have on the hypersurface _1 n -].

n

x

1

= =

as follows from the equations

= n nl

Thus if in

VI

*0

cr3x ll

we may always choose the coordinate system

so that nl _ 1

=

=

o

That is, so that

= 1. =

o.

Such coordinates are of course Gaussian coordinates based on the hypersurface

r.

We shall restrict ourselves to such

coordinates in the remainder of this section. In Gaussian coordinates, we have

= and hence

= o

(14.12)

- 251 -

Taub

Therefore \! = n n)J;\!

=

o

(14.13)

The second fundamental form of the hypersurface

is

~

defined as S"l, ,

lJ

=

=

04.14)

S"l ..

Jl

The last of the above equations holds in view of equations 04.12).

It follows from the definition of

and

S"l"

lJ

n

)J

that in the Gaussian coordinate system S"l ..

lJ

=

-n f~, p

=.

lJ

Since the four vectors

1

lJ

n)J,

04.15)

=

-f"

xli

are linearly independent

we may write )J

p

= n \! (n ; pn) +

S"l

ij)J p x "x "g • 1 ] P\!

In view of equation (14.13) this becomes =

ij v S"l

}I,' x''J g p

l

04.16 )

PV

where

Hence =

S"l

.. = Yij S"l lJ

We now apply the above results to the evaluation of the integral J

=

I Vl

;:g(g)JagPa

P va' g)Jg)g

)J\!;pa

d 4x

- 252 where

Taub

is an arbitrary tensor defined in

that it vanishes on all boundaries of surface

We may evaluate

~.

system based on J

and such

except the hyper-

in an arbitrary coordinate

in particular in the Gaussian coordinate

VI

system in

J

VI

VI

~.

Then we have

=

where equation (12.6) defines normal to the hypersurface

and

r and

d~

is the unit is the invariant

volume element of this hypersurface, that is, we may evaluate J

in the Gaussian coordinate system based on

~

where we have

;:

=

(nPx lJ .x\)_ 11 fJ

=

Y

-

v I' P ij' n X1.xl.)y (g llV P 1 .)

• A gA/llP)

• ij ij' ij Y P . . + gun - goon goo 1J, 1 + 1J 11J

where P.

1

P. .

11 J

= = P..

1,-J

Pk {koo} 1J

and the {k.. } are the Christoffel symbols formed from the 1J

Hence we have J

= J (y ij·g .. 1 1J, ~

Y.. o 1J

- 253 -

Taub

Thus the second term in equation (13.1) may be written as

.. 1 f[1Y(y ij·g1J,

=

~

The vanishing of

I

I

(14.17)

.

given by equation (13.1) for

g)lV

such

that ( ~)lV -g g g)lV ) +

=

but otherwise arbitrary on

= ~

o

and for arbitrary

(14.18) g .. 1

1J,

then

implies that (14.19) and that

[ If( yijn

- n ij )] = 0

(14.20]

Equations (14.19) imply that the induced metric on the hypersurface the metric in

takes on the same values when computed from

~

Vl

as when computed from the metric in

V2 •

Since we may use Gaussian coordinates based on the hyper surface ~

in each of these regions we have that in these coordinates is continuous.

the metric tensor

Equations (14.20) then

imply that the second fundamental form of the hypersurface

~

takes on the same values when this hypersurface is regarded as one in

Vl

or one in

the restriction imposed on

V2 •

.

g)lV

This result obtains without by equation (14.18).

However

that equation is the statement that the variations in volume element, as measured by the variations in the on the hypersurface

~.

;:g,

vanish

- 254 -

15.

Taub

Thin Shells The equations that determine the behavior of thin

shells may be derived by considering the conditions for which I' = 0

where

I'

is given by equations (12.8) with

We shall discuss the case for which Vw £

I'

but not one the hyper surface

= 0 inside

VI

and

V2

P

~.

=0

inside

f VI

o.

~

and

In that case we have

and equation (12.8) becomes

=

when we set

=0

= 0 and f = O.

It then follows by using the argument given above that. in the regions

VI

and

V2 we must have

GlJ V = 0

(15.1)

Then on setting g

lJV

+,

'>lJ; v

+,

'>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
=

+

KT~V)g~V;:g

d4x (19.3)

+

I< g~V g po

H - g ~p g va), g ~v;po -g d 4x

where T~v

F

= (w +

p)U~Uv

_ g~Vp.

(19.4)

This variational principle then leads to the field equations G~v

(19.5)

=

with equations (19.4) holding. p

= pew)

In the latter equations

and hence one of the consequences of the Bianchi

identities

= is that

where

a

is a function of

w (or

p)

defined by equation (17.7).

- 266 -

Taub

LECTURE V

A VARIATIONAL PRINCIPLE FOR CHARGED FLUIDS

20.

Introduction It is the purpose of this lecture to discuss a variational

principle from which the equations governing the motion of a gravitating charged electromagnetic fluid with dielectric permitivity, magnetic permeability and conductivity are derived.

In case the conductivity is zero or infinite, we

will have a 'holonomic' variational principle.

In other cases

the principle will be 'non-holonomic' in that not all terms in it will be de~ivable from the variation of a function of the dependent variables which enter into the problem.

This is to

be expected, since for non-vanishing and finite conductivity one is dealing with a non-conservative problem since ohmic heat is involved and for such problems holonomic variational principles do not exist. We begin with a discussion of the

electromagnet~fields.

We shall use the notation given by A. Lichnerowicz [1] for the Minkowski formulation of the Maxwell equations governing

- 267 -

these flelds. H~V

and

Taub

They are descrned by two antisymmetric tensors If u~

G'lV'

is the four-velocity vector of the

fluid satisfying (20.1)

Then the electric field the magnetic field

eu '

the electric induction du'

and the magnetic induction bu as

ha

measured by an observer whose world line is given by a solution of the equation dx~

(fT"" ::

U

~

,

that is, one who is at rest w1th respect to the fluid, are given by (20.2)

(1IJ

ibf) = u '!lUf3 where

1::

J:I

and 1

v

Huf3 = ¥u:3~5H

~5

(20.3)

Haf3 _

lEaf3~5HIJ - 2" ~5 '

E(((3~5 and EUf3~5 are pure imaginary tensors defined by the equations

EUf3~5 :: -rgEUf3~5 EUf3'Vr 5 and the \'Ie

E'S

::

-1. af3'V5 .{gE r

(20.4)

are the Levi-Civita alternating tensor densiUes.

have introduced the pure imaginary quantities in order to

preserve the commutativity of various methods for manipulating

- 268 -

indices.

Taub

It follows that

It follol-IS from equations(20.2) and(2o .3) and the prop-

erties of the

that ibf3Ef3AIlV = _u"R llv _ u~vA _ uVHAIl EIS

Hence HIlV

= ulle V - uVe ll - ibf3UAEf3AIlV

(20.5)

Similarly we have GIlV = u~Ldv

_ uVd ll - ihf3UAEf3AIlV

(20.6)

For the purpose of simplifying the ensuing discussion we shall assume simple constitutive equations, namely

where

A,

the dialectric permitivity of the matter and

Il,

the magnetic permeability are scalar functions which may depend on the coordinates.

The situation in which

Il are functions of the vectors

ea

and ha

A and

respectively

and of other variables characterizing the matter may be dealt with in a manner analogous to the discussion given below. It follows from equations (20.5), (20.6) and(2o .• 7) that ( 2o.• 8)

where AIlVa'T _- -.l ( all 'TV ov 'Til) + 2 1 (A 21l g g -g g

-

1) (u ~Lu ag 'TV -u Vu ag'!jl

~

- u Ilu 'T g av + u vu 'T g all)

(20· .9)

- 269 -

Taub

Hence (20.10) It is a consequence of equations (20.5) and(20.6) that

H,LV H !-tv

= 2(e Ve v-b Vbv )

!-tV ,I H H!-tv

a~LVa

!-tv

= 2(d vdv_hvh v )

a!-tvG

a 11VH!-tv

= 2(e vdv-h Vbv )

J G!-tv H V V !-tv = 2i(b d v+h e v )

v

!-tv

= 4ie

v bv

= 4id Vh v

oo.n)

- 270 -

21. The

Taub

Equations

l'lax~leJl

These equations are v

IlV

Hj v = 0

(21.1)

= JIl

(21.2)

and a[J·V

iV

where the semi-colon denotes the covariant derivative.

The

vector JIl defined by equations (21.2) is the electric current and as a consequence of equations (2L2) satisfies the conservation equation JIL

ill

= 0

\ole may write

where

E

is the proper electric charge density and

is the conduction current.

III

Ohms law then is given by the

statement that

"lhere

C5

is the conductivity of the fluid.

Equation (21.5)

is the third constitutive equation for the matter.

It may

be replaced by a more general (non-linear) lal'l without affecting many of the results given below. Equations(21 .1) are equivalent to the statement thnt HAlliV + HIlViA + HVAjll = 0 •

(21.6)

These equations imply of course that there exists a fourpotential such that

- 271 -

Taub

(21.7) The Minko\'lslci stress-energy tensor is defined as 1"IlV = HllpGPV + .kllvHPO"G lj~

(21.8)

pO"

and satisfies (2L9)

It follows from equations(20.5) and(20.6) that 1"

IlV

=

(21. 10 )

where Vv = le Ph O"u 1"EpcJ'rV

(21.11)

= id(lb CYu1"E PO"1"V

(21.12)

w.v

and hence satisfy ullv

Il

= ullvi Il = 0

When the constitutive

(2L 13)

equation~(2Q.7)

hold we have

and <21.14) It fo11ov/s from the Maxwell eoua tions, when these are

written as equations(21.2) and(21.6), that ,-IlV = HllpJP + aPO"gllV (H + ~l ) ;v VpjO" 2 pO"jV

+ .kIlV(H GPO" _ G nPO") lj~

pO"

jV

po' ;v

Taub

In view of equations C21

.6) and(2D.8) we may write this equation

as ~v = H~ JP + 1 ~vAPoa(3H H 't" j V P "IF j v Pcr-u(3

On making use of equation(zo.9) we may in turn write this equation as (21.15 )

+ u A gP~(w -v) j

P

A

A

If we define W= -

1(e 0d +bPh ) 2

P

(n.16)

P

and JlV = +

(g~V_ul!uV)W

+

e~dv

_

h~v

(21.17)

then equation (2.10) becomes 't"~v = Wu~u V _ u~v v _ wlll- ollV

(21

.18)

and

{21.19) Substituting from equation(21.19) into the left hand side of equation(21.15) and multiplying the resulting equation by Ull we obtain

(Wuv-VV).v + aIlvu ,

~jV

= (0+

1 1.. uD)eOe 2 jp 0"

(21.20)

This is in the form of a conservation equation and relates the change in ii, the energy density of the electromagnetic fields as measured by an observer at rest with respect to it, the divergence of v Il and the electromagnetic stresses

crltv.

The vector w~ is related to the momentum of the electromagnetic field.

- 273 -

22.

A Variational Principle for the

Taub ~1axwell

Equations

Let us assume that the space-time with the metric tensor 1s given and is not influenced by the presence of the

g~v

electromagnetic fields described by

H~v

and

G~v'

This is

the case in special relativity where the space-time is the Minkowski space-time and in a galilean coordinate system the

g~v.

are constants.

Vie

shall not restrict ourselves

to galilean coordinate systems nor require that the curvature tensor of space-time vanish. Consider the integral over an arbitrary region of spacetime defined by IE = where

H~v

J fxrLvH~vJi d4x

=

J *A~vaTHaTHlJ.vJg d4x

is assumed to satisfy equation(21:6), that is,

it determines a four potential ~IJ.

(22.1)

~IJ.'

We shall assume that

in addition to depending on the space-time coordinate,

also depends on a parameter e. that the

glJ. v and

For the present, we assume

u~ are independent of

e.

Then

where ~,

~

=

If we require that

( d~~) de

e=O

~~

vanish on the boundary of integration

then we have by integrating by parts

- 274 -

Taub

The variational principle which requires that (22.2) where Jil is given by eruations(21.4) and(21.5) is then equivalent to equations (21.2).

In case

III is absent from

the latter equation, equation(22.2) may be derived from looking for the extrema of the integral

The second term on the rlght hand side of equation(22.3) represents the interaction Lagrangean. It is l'lell known that if the Lagrangenan of a variational

principle depends on the coordinates only because of its dependence on the field variables being varied, then there exists a second order, non-symmetric tensor t~ satisfies a conservation theorem.

which

For the Lagrangean

~ = bllVHllv we have

and or

t~ where

1"

= -

1"~ +

(GIlO'cpp);1l -

G~~CPp

is the rUnkowski tensor.

(22.4)

In case Jil = 0, and the space-

tlme is flat the divergences of the tensors

t and

T

are equal

and equal to zero as a consequence of the field equations.

- 275 -

23.

Taub

The General Variational Principle We now turn to the discussion of a variational principle

from which we propose to derive the equations governing the motion of a perfect fluid which has a dielectric permitivity, a magnetic permeability, is self-gravitating and is subject to and electromagnetic field. conductivity vanishes.

We shall have to assume that the

If it does not do so we will have to

resort to a non-holonomic variational principle.

In the sub-

sequent discussion we shall introduce as field variables, the metric tensor

g~j\)

of space-time, the particle paths of the

elements of the fluid, the density of the fluid, a variable related to the temperature and the four-vector potential. The variational principle will be described in terms of I

where II

=

(23.1)

is the Einstein constant of gravitation,

K

IE

and

are given by equations (22.1) and (22.3) respectively, Ig

where

R

tensor

= f

r-g

Rd 4 x

(23.2)

is the scalar curvature determined by the metric and

gil\! '

(23.3)

= with

p

energy, entropy.

the rest density,

£

the rest specific internal

® the rest temperature and IF

discussions.

S the rest specific

is the integral which entered into our previous

- 276 -

Taub We have seen (cf. Lecture II!) that

and if this is to vanish for arbitrary

g~v

and

at

which

vanish on the boundaries, we obtain as the Euler equations (23.4) and (23.5) Equation (23.5) may be shown to be a consequence of the fact that the conservation of mass holds and eJ..lV

;V

=

o

(23.6)

as a result of equation (23.4). We now turn to the evaluation of

We have evaluated the contribution to in the preceeding section.

of the

IE

from the variation

Thus after an

integration by parts II

E

=

where J..l

J[_G].lV,.1 ;v'i'].l

AJ..lV01

and

A

1

is to be computed from equation (20.9), with

assumed to be independent of

It may be verified that I

I

E

=

e.

- 277 -

Taub

where = (23.7)

= E~v

The symmetric tensor

is similar to the Minkowski tensor.

It is known as the Abraham tensor [2].

In fact we have (23.8)

Hence on evaluating I'

=

I'

=

we find that J{[(R~v _ l2g~vR) + KT~v]g'

~v

+ 2K(pSU~)

;~

a' + 2K(G~V - J~)~'}~g d 4x ;v ~

where

=

e~v

Thus the requirement that and R~v

~'~

+ E~v • I' = 0

(23.9)

for arbitrary

g'

~v '

a',

leads to the Euler equations

h~vR + KT~v (pSu~)

;~

G~v

;\1

= 0

03.10)

= 0

03.ll)

=

J~

=

£u~

(23.12)

The latter equations are of course the Maxwell equations. Equations (23.10) are the Einstein field equations with the source of the gravitational field given by both the matter present and the electromagnetic field.

Equations (23.11) are a

- 278 -

Taub

consequence of equations (23.10) and (23.l1hlhen matter is conserved.

That is, the equations

hold and there are no ohmic losses.

For, it is a consequence

of equations {23.lq)and the Bianchi identity that T~V = 0

(23.14 )

jV

These four equations which are the generalization of the Lorentz pondermotive force equations contain the definition of the pondermotive force acting on the matter and a statement concerning the conservation of energy.

The latter

statement is equivalent to the statement about the rate of change of rest specific entropy along a world-line of the fluid vanishes.

- 279 -

24.

Taub

Summary The general variational principle given above, from

which one may derive the Einstein field equations, the Maxwell equations, and the equations of hydrodynamics may be applied to the problem of general relativistic magnetoe U = O.

hydrodynamics by setting

a =0

The restriction to

may be removed by using a non-holonomic variational principle, that is by replacing ving

u'

and

II

by an appropriate expressioninvol-

¢'

It is worthy of notice that the pondermotive force acting on the charged fluid with electric permitivity and magnetic permeability is not given by the divergence of the Minkowski tensor.

Rather, it is given by the divergence

of the symmetric tensor

E~v the Abraham tensor defined by

equation (23.7) related to

t~V

by equation (23.8).

A similar

result holding for less general circumstances has been given by Penfield and Haus [3J.

It is reasonable to expect that the

pondermotive force is to be derived from the symmetric tensor

E~v

even

in non-conservative cases where the varia-

tional principle does not apply in the form discussed above in detail.

This would resolve the old controversy over the

appropriate stress-energy tensor to be used in describing general electromagnetic fields in moving bodies. From the discussion of the variational principles it is clear as to why

E~V

replaces

t~V

in the calculation

- 280 -

Taub

of the pondermotive force.

This tensor describes the energy

of the electromagnetic field alone as follows from the discussion at the end of section22 above. A~vaT

depends on u~,

Lagrangean

~

However, because

as well as on A and

~,

the

contains interaction terms between the fluid

and the electromagnetic field, even in the case of an uncharged medium.

The presence of these terms then is

responsible for the additional energy which changes

T

~v

into E~v We conclude with the remark that the methods given above will apply to mare general situations then those treated.

Thus

we need not assume that the electric properties of the matter are isotropic or that that matter is a fluid.

All that is

required is that a Lagrangean exists that is a function of the field quantities describing the various properties of the medium.

Thus if in general

and i f :-. ~v _, ~vaT 00

~="

aT

,aT~v

="

,

we may define a Lagrangean for the electromagnetic fields. Similarly if the matter has a more general stress-energy tensor than that given by a fluid, but involving no viscous or similar forces, it too may be described by a more general Lagrangean.

- 281 -

Taub

References 1.

A. Lichnerowicz, Relativistic Hydrodynamics and Magneto-

2.

W. Pauli, Theory of Relativity, Pergamon Press, New York (1958), p. 110.

3.

Paul Penfield, Jr. and Hermann A. Haus, The Physics of Fluids 9 (1966) 1195-1202.

hydrodynamics, W.A. Benjamin, Inc., New York (1967).

- 282 -

Taub

LECTURE VI STABILITY OF GENERAL RELATIVISTIC GASEOUS MASSES AND VARIATIONAL PRINCIPLES

25.

The Second Variation In this lecture we shall derive and then apply the well-

known result that if a set of equations are the Euler equations of a variational principle based on an integral

I(e)

then the perturbations of solutions to the Euler equations satisfy equations which may be derived from another variational principle.

The latter principle is given by an

integral equal to

1"(0)

where the prime denotes the

derivative of I(e) with respect to principle based on

e.

The variational

1"(0) is called the second variation.

We shall apply these results to the discussion of the stability against radial perturbation of spherically symmetric static solutions of the Einstein field equations for a self-gravitating fluid which obeys an equation of state.

In this case we may define

by equation (19.2).

I

to be that given

We shall show that the second variation

problem for this integral is the same as the variational principle given by Chandrasekhar [1] for this problem. Let

£(~A;~~~)

be a scalar density formed from some

scalar or tensor fields

$A

and the derivatives of these

fields with respect to the coordinates in space time,

=

1, 2, 3, 4

A = 1, 2,

, N.

- 283 -

Taub

Then (25.1)

where the integral is carried out over an arbitrary four volume in space time determines a variational principle in the following sense. x~

of the

We assume that the

and a parameter

e,

thus

of

e

¢A

are functions

A

= ¢ (x;e).

Then

I

is also a function

I' (0)

=

drl de e=o

=

and we may require that

o

for arbitrary A

=

(d¢ )

de e=o

We have rICe)

= (25.2)

rICe)

=

since =

On integrating the above expression for rICe) by parts we obtain

r' (e)

= (25.3)

- 284 -

Taub

where C25.4)

=

The second integral in Eq.C25.3) may be written as an integral over the hypersurface bounding the four-volume of integration. The requirement that

=0

I'CO)

for arbitrary

¢,ACO),

in particular, for those that vanish on the boundary of the region of integration then leads to the Euler equations

=

o

=

(25.5)

The equations satisfied by the difference between two "almost equal" solutions of these equations, or the equations satisfied by perturbations of solutions to these Euler equations are

o

= where FA(O)

FA(e) = [

is given by Eq. (25.4).

a2.c ¢,B a 2¢Aa¢B .

+

a2.c

¢,B ,lJ

3¢Aa¢B

,lJ

- ( a¢Aa2.ca¢B ¢IB + a¢A a2a¢B.c ,lJ

Thus

,ll

v

¢,B) ] , v , II e=o

(25.6)

= 0 These are a set of linear equations for the variables A whose coefficients depend on the ¢ (x;O) and their derivatives.

The

¢,A(O)

¢,A(O)

are called the perturbations and

- 285 -

the

A ¢ (x;O)

Taub

the unperturbed solutions.

Now it follows from Eg. (25.2) that I' (e)

=

fF A¢"Ad 4x

+ (F,¢,A d 4x + . A

f (~ ¢"A) 3¢A

,11

,11

d 4x (25.7)

32£ + f[ 3¢A3¢B

32£ ¢,A ¢,B] d4x ¢,A¢,B + ,v ,11 3¢A a¢B ,11 ,v ,11

or I"(e)

=

fF A¢"Ad 4x

:0 2£

+j[

+ f (~¢"A) d 4x a¢A ,11 ,11

¢,A¢,B +

a¢Aa¢B

(25.8)

2 a2£ 2a £ ¢,A¢,B + H,A¢,B d 4 ,11 ,v' x. a¢Aa¢B a¢A a¢B ,11 ,v ,11

From Eq. (25.8) we have 1"(0)

= f[

32£ ¢,A¢,B + 23 2£ a¢Aa¢B a¢Aa¢B

¢,A¢,B +

,11

,11

32£ a¢A a¢B ,11

¢ ,A¢ ,B] d 4x ,11 ,v

,\I

(25.9) when the ¢A(O)

¢A(O)

that is the

are unperturbed solutions of the Euler equations

associated with

I,

¢a)

¢"A(O) = 0, on the boundary of

and the

the region of integration. the

FA(O) = 0,

are such that

functions of

x

If we now consider the and a parameter

J(f)

f

¢,A

(not

we may define

= 1"(0)

and examine the Euler equations resulting from the condition

=

o

- 286 -

This is the "second variation problem."

oJ

=

2JF'Ao~ ,Ad 4x + 2J (

~

2J (

a 2£

d~A a~B ,]1 ,v

Taub We find

a2£ ~,Ao~,B) d 4x ,]1 a~Aa~B ,]1

~ ,Ao~B) d 4x , ]1 ,v

Hence for variations of the

~ ,A

the boundary of the region of integration, extreme values when the

~,A

~,A

vanish on

1"(0)

takes on

such that

satisfy the equations

the equations satisfied by the perturbations. Thus the solutions when considered as ~ ,A

~A(x;O)

are such that

FA(~) = 0,

I' (0) =

o,

for

which vanish on the boundary of the region of integration,

and the solutions the

~A of the Euler equations

~ ,A

of the equations

FA(~;~')

= 0 , where

~A satisfy the Eurler equations and are coefficients

in the linear differential equations, are such that

1"(0)

takes on extreme values.

26.

The Spherically Symmetric Case In a spherically symmetric space-time we may write the

line element as = e 2~ dt2 - e 2~ d r 2 - e 2]1 dn2 "

(26.1)

where . 26 dX 2 d6 2 + Sln

(26.2)

- 287 ¢, W and

Taub

are functions of r, t and a parameter

~

e

and

these are comoving coordinates for each value of the parameter

It then follows [2J that the non-vanishing

e.

components of

G~

are

v

2~ W ) + e-2~ r r

(R 2 2

~) 2

=

- (R 33 _ ~)2

= e- 2¢[W tt

(26.3)

+

~tt

+ ]It2 + w2 - W ¢ + t t t

~t~1jJt

- ¢t)J

-W)J - e -2¢[ ¢rr + ]lrr + ]lr2 + ¢2r - ¢r1jJr +]l(¢ r r r -2¢

R4

= 2e

Rl 4

= - 2-2W[ ]l rt -

1

where the subscripts

~rWt + ~ t ]l r J

[~rt - ]It¢r

r

and

~t¢r - ]lrWt + ]lt~rJ

t

denote the derivatives with

respect to these variables. The Einstein equations become

-F =(R 4 ¢ 4

!R) + kw 2

=

0

-F =(R l W 1

!R) 2

kp

=

0

-F ]l =(R 22

!R) - kp 2

=

0

(26.4)

and Rl 4

=

0

(26.5)

- 288 -

Taub

The four equations (26.4) and (26.5), are not all independent in view of the Bianchi identities.

It may be shown that the

solution of these equations is determined by the solution of equation (26.5) and and of

F = 0 ¢

for

Fw=

0 for a range of values of

t

t = O.

The unperturbed solution we shall consider will be assumed to be static, that is

¢, W and

to be functions of

In that case it is no restrictior

r

alone.

~

will be assumed

to take ~

=

log r.

Equation (26.5) is identically satisfied and equations (26.4) reduce to

(26.6)

It is a consequence of these equations that (26.7) It is a further consequence of Eqs. (26.6) that =

(26.S)

The last equation also follows from the equation of state assumption.

- 289 -

Taub

The equations satisfied by the perturbations, ~'

and

and

are obtained by differentiating Eqs.

~'

(26.5)

with respect to

e

(26.4)

e = O.

and setting

We

then obtain from Eq. (26.5) and the last of (26.4) the equation

~r't - ~'~ t r - !~, r t where now

~r

+

!~, r t = 0

(26.9)

is determined by Eqs. (26.6).

The solution

of Eq. (26. 9) is given by ~'

where

- ~'

0

, ~o

= e~(e-~r~') r =

~'

(26.10)

(r,O)

and we have chosen our comoving coordinates so that

, ~o

=

~'(r,O)

(26.11)

= 0

This can always be achieved by a coordinate from transformation involving

r

alone.

The function

~'

may be evaluated by using the integral

of the field equations given by Eq.(17.8) which holds for all values of

e.

That equation may be written as oe ~+2~

where now the subscript zero on

fo

= oo e\jJo+2~0

(26.12)

fCr, t, e) is defined by = fCr,O;e)

On differentiating Eq. C26.12) with respect to

e,

setting

e = 0 and using Eqs. (17.6), C16.5), and (26.11) we obtain

- 290 -

(3~'

=

Taub + r~'l' - ~ l' r~')

(26.13)

or

= where

a

-2" e't'(r 3 ~'e -"'t')

(26.14)

l'

is the velocity of sound in the unperturbed fluid,

is given as above and Thus

I'

~'

and

~'

~

,o

is

~'(r,O)

for

are determined in terms

e = O. of~'

This

function may be determined by solving the equation F'

~

The quantities

~'

and

= 0 enter into this equation but may

~'

be eliminated by means of Eqs. (26.10) and (26.14).

We shall

discuss this equation in the next section. When the field equations, Eq. (26.4) and (26.5) are applied to a problem in which there exists a hypersurface in space-time across which the stress-energy tensor is discontinuous, the equations must be supplemented by conditions satisfied by the metric tensor, its derivatives and the stress energy tensor on this hypersurface.

Thus for the problem

we wish to consider, namely that of a gas occupying a limited region of space-time and bounded by a vacuum there exists the hypersurface

L defined by r

where

=

is the constant comoving coordinate of the boundary

element of the material.

- 291 -

It is well known

Taub

that the conditions referred to above

become in this case

¢, W, and

and that

L.

are continuous across the hypersurface

In addition all first derivatives of these quantities

except m~st

~

,!,

'l'r

must be continuous across L.

These conditions

hold for the perturbed as well as for the unperturbed

equations.

Hence we must have

=

=

o

(26.15)

In view of Eq. (26.14) this condition becomes a boundary condition on the function

~'.

Another condition is the requirement that for the perturbed and the unperturbed solutions the function R

at the origin.

=

e~

= r

= o

This function is the analogue of the Eulerian

coordinate of an element of the fluid which has the Lagrange coordinate

r.

Hence we must have

R'

= e~~'

= r~'

= 0

(26.16)

at the origin. Eqs. (26.15) and (26.16) provide boundary conditions for the second order partial differential equation

' = FW

o.

We close this section with a discussion of the implication of the Bianchi identities.

- 292 -

Taub

If we define

= these identities are 1

= -~ v-g

).1

(;:gK v ) ,).1

=

They hold for all values of

e.

differentiated with respect to e

=

0,

o

If the above equations are e

and if it is assumed that

and then evaluated for K).1 (x' 0)

v

'

= 0,

it follows

that 1

=

;:g where now

g).1V

= g).1v(x;O)

o

(26.17)

is the unperturbed metric, and

r ).1PV is determined from this metric and this metric satisfies the field equations. We now evaluate Eqs. (26.17) for the case considered above, when the unperturbed metric is spherically symmetric and static and the perturbed metric depends on time but is still spherically symmetric.

In that case Eqs. (26.17)

reduce to two equations corresponding to

v =4

and

v = 1.

These are (26.18) and 2rF'

).1

respectively.

=

0

(26.19)

- 293 Taub

R'41 = (R,l) =0 4 t

Hence when

as is the case when

Eq. (26.10) holds, Eq. (26.18) becomes

=

(26.20)

F~(r,O)

and Eq. (26.19) becomes (26.21) The first of these equations implies that the equation is only a restriction on the functions

o

and

It may be verified that on substituting Eqs. (26.10)

~'

o

into the expression for

-F~

27.

~'

F~

one obtains

2 -2'" = r2 (re 'f lji 0, ) r

The Equation

F~

The equation

+ k

(w + p) ~ ,

a2

0

= o

(26.22)

=0 = 0 is derived by differentiating the

F~

second of Eqs. (26.4) into which Eqs. (26.3) have been substituted setting

e

= 0,

the unperturbed solution.

F'

=

-2 , 2[ e lltt

and making use of the values of One then obtains

,

, -2lji(1 + ~ ) _ u ! e-2lji~, llr e .r 'f r 2" - r 'f r r

(27.1)

when Eqs. (26.10) and (26.14) are used to express ~'

in terms of

U',

one finds that

lji'

and

- 294 -

re2iJ!Cw + 12) F' 2 iJ!

Taub

1 2 2iJ!-2¢ Cw+p)~ + -4 p ~ - w-+p = e Pr~ r r tt 2 - e - iJ!-2 ¢[ e 3¢+ iJ!Cw + })a Ce-¢r2~) r Jr r + ke2iJ!Cw + p)p~ +

iJ!~C~ r

C27.2)

~ )_e-iJ!-¢Ce¢+iJ!¢') + r r 0 r

where C27.3)

= rll'

~

The equation

F' = 0 lji

where

F'

is given by equation

iJ!

C6.2), has a boundary condition indicated in Eqs.C26.1S) and C26.16).

It is the equation given in [2J for the case of

the radial perturbations of a self gravitating fluid when the equation of state was such that the fluid was isentropic. In that case When

cr

= p,

¢' = lji' = 0, o

0

the rest mass density of the fluid. the equation is the same as the equation

given by Chandrasekhar [lJ as may be seen by writing

= and thus defining

y.

This definition of

y

is that

given by Chandrasekhar as may be verified by writing w = NCl + uCp,N)) where

u

Cop/3w)S

is the internal energy.

If one then computes

and remembers that TdS

=

one verifies that the definition of used by Chandrasekhar.

y

given above is that

- 295 _ 28.

Taub

The Evaluation of 1"(0) In this section we shall use the results obtained above

to express

1"(0)

in terms of

observing that when

11 '

,

,/,'1'0' , and ¢'. 0

We begin by

is defined by Eq. (19.2) and when the

I

perturbed and unperturbed metrics are of the form given by Eq. (26.1), then it is sufficient for the purpose of calculating I'(e)

and

I"(e)

to evaluate

system In which Eq.

(26.1)

I(e)

in the coordinate

holds.

Thus we have 1

811 I(e)

:

~+,/,

ff{e'l'

'I'

2 + 211 ¢ ) - e -~+'/'+2" 2 + 211 ljJ ) + e'l'~-'/'+2" 'I' ~(11 'I' 'I' ~(11 r rr t tt

(28.1) Hence

~1II' (e) where

:

ff{e¢+ljJ+2 11 (F ¢' + F ljJ' + 2F 11')dr dt - See) ¢ ljJ 11

(28.2)

F¢,FljJ' and FI1 are defined by Eqs. (26.4) and (26.3) and

See)

(28.3)

=

with A

= (28.4)

B = e¢-ljJ+2 11 (_ljJ'(211 r + ¢r ) + ¢'¢ r + 211'11 r + 211'r + ¢') r (28.5) The integration in Eqs. (28.1) to (28.3) may be taken to be the region bounded by the inequalities

a

~

r <

00

(28.6)

- 296 -

Taub

Across the boundary

r

=

(28.7)

There is a discontinuity in the stress energy tensor. pressure density

p

must be continuous at

w need not be.

~', ~I,

but the energy

The requirement that

L811 I , (0) for arbitrary

= rb

p

The

= 0

and~'

which vanish together with their

derivatives on the boundary of the region given by the inequalities (28.6) and such that

~', I.l' .and ~'

r and

~"

~

rI

may

take on arbitrary values on the interior boundary given by Eq. (28.6) leads to the field Eqs (26.4) and the boundary conditions discussed in Section 26 (cf.[4]). We also have 1 1"(0) 8TI where

~,

= ~

and

~

are evaluated for

satisfy the unperturbed equations.

e = 0 and these functions In view of Eqs. (26.10),

(26.7), and (27.3) we have = e~+~r2~'F' +(e~+~r2~F') o ~ ~ r

_e~+~r2(~-~(e~r2F') r

~

On using Eq. (26.21) we obtain

r

+

i

2

re2~k(w + p)F,:). 'I'

- 297 -

Taub

This equation holds for all values of r

~

rb w = P = 0

and for

r

~

rb

r,

however for

we may use Eq. (27.2).

Hence we have 1 81i 1"(0)

(28.8)

=

where J

= (28.9) + r 2e ¢+1jJ ~ 2 (ke 21jJ (w+p)p + 4 Pr -

r

I W+'P

2 Pr )] dr dt

tl'"

=

JI

-f

fr 2e¢+1jJ[F¢¢' + F1j!1jJ~] dr dt

o

0

(28.10)

tlrb

_ r2~(e¢+1j!¢,) ] +kf J [r2e¢+1j!~1j!~(~ + bP· r r) o r o

dr dt

0

and

= (28.11)

with

F~

given in terms of

If the functions 1'(0)

= 0,

¢', 1j!'

¢', 1j!' and

~

and~'

by Eq. (27.1).

are to be such that

that is if they and their derivatives are to

vanish on the e_ erior boundaries, and if the boundary conditions on

~

are to hold ar

r = rb

and

r

=

0

we

must have

L871

1"(0)

=

J

(28.12)

- 298 -

Taub

J is given by Eq. (28.9).

where

The Euler equations of the variational principle 1 81T

or"(o)

= oJ

=

(28.13)

0

is the equation re 2lj!(w + 2

;e) F' lj! =

(28.14)

0

where the explicit form of this equation is given by equation (27.2).

This equation is equivalent to

F' = 0 lj!

and the

variational principle defined by Eqs. (28.13) and (28.9) was of course to be expected in view of the general discussion given in the introduction.

29.

The Stability Criterion The variational principle defined by Eqs. (28.13) and

(28.9) may be related to that given by Chandrasekhar in [2J by observing that if one writes ~

= sinCat +

(29.1)

a)~(r)

we have 1 871

r"(o)

where

j where

/1

=

J

tl

2

(29.2)

= ! sin Cat + a)} dt 0

rb

= !

- a 2e 3lj!-¢ r 2 (w + p)

~

2

(29.3)

dr + 11

0

rb

4Pr = J [r 2e ¢+lj! ~ 2 (ke 2lj! (w+p)p + r 0

2 + e +3¢+lj! (w+p) a (re'¢~);J dr 2" r

1

~Pr

2) (29.4)

- 299 -

Taub

The variational problem (29.5) has as its Euler equation, Eq. (28.14) with Eq. (29.3).

The functions

~(r)

equation, that is the extremal

~

given by

satisfying this Euler ~(r)

=

~o(r)

are such that

Chandrasekhar has pointed out (cf. [2]) that the variational problem given by Eq. (8.4) expresses a minimum principle for the determination of the lowest value of 0

2

and that a sufficient condition for the dynamical instability

of a mass is that

)1

= 0 for some "trial function"

~ which

which satisfies the required boundary conditions. However, if such a trial function exists we shall have

~~

0

and in view of Eq. (29.2), for this trial function L1,,(o) 811

<

0

(29.6)

Thus the sufficient condltion for instability used by Chandrasekhar is equivalent to the condition that there exists a trial function such that the inequality (29.6) holds.

The

latter criterion may be applied to discussion of the stability of general solutions of the Einstein field equations.

We

need not restrict ourselves to a static unperturbed solution and consider perturbations of such solutions which depend on the then defined time coordinate in an exponential manner.

- 300 -

Taub References

1.

Chandrasekhar, S. The dynamical instability of gaseous masses approaching the Schwarzschild limit in general relativity. Astrophys. J. 140, 417-433 (1964). -

2.

Taub, A.H. Small motions of a spherically symmetric distribution of matter. Les Theories Relativistes de la Graviation, pp. 173-191. Centre National de la Recherches Scientific, Paris (1962).

3.

• Singular hypersurfaces in general relativity, --IITinois J. I'lath. !, 370-388 (1957.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I ME)

GENERAL-RELATIVISTIC KINETIC THEORY OF GASES(1)

J.

EHLERS

Corso tenuto a Bressanone dal

7 al

16 Giugno

1970

GENERAL-RELATIVISTIC KINETIC THEORy OF GASES

( i )

Introduction The relativistic kinetic theory of gases, which will be presented in the following lectures, is of interest for a number of reasons: It

offers a sim

pIe, microscopic model for matter in bulk 'which is sufficiently general

to

provide a oasis for hydrodynamics and thermodynamics of simple and multicomponent systems. Definite conservation laws, balance equations, equations of state, transport and reactions can be derived from it, and if cross from a microscopic scattering theory are fed in, kinetic theory gives transport and reaction coefficients. As in the non-relativistic theory, the arbitrariness of the constitutive equations and the indefiniteness of the transport coefficients inherent in the phenomenological continuum approach are overcome by the kinetic theory. Moreaver, kinetic theory provides a description of gases under conditions where fluid dynamics does not apply, e. g., when collisions are rare and the mean free path is large.

This work was supported in part by NSF - grant GP 20033

- 304 -

Ehlers

Applied to macroscopic particles like stars or galaxies, kinetic theory offers a method of treating systems or the sustem of galaxies, the "gas" of cosmology. Another asset of relativistic kinetic theory is its uniform treatment of gases consisting of particles with positive mass and those having zero mass particles; its application to photons gives the cosmologically and astrophysically important theory of the transport of radiation. Specific applications of relati vistic

ki~ tic

theory to astrophysical pro-

blems which illustrate the usefulness of this theory will be mentioned later. Although the domains of applicability of fluid dynamics and kinetic theory overlap, neither of them contains the other one. Nevertheless, kinetic theory may be considered as the more fundamental of the two theories, sin ce within it one can derive from simple microscopic laws and plausible statistical assumptions and approximation methods the general forms of all the laws which are postulated in fluid dynamics; only the numerical values .of (e. g.) transport coefficients have to be changed on leaving the domain of validity of kinetic theory. Ideally, one would like to derive both kinetic theory and fluid dynamics from statistical mechanics; at the relativistic level, this has not yet been achieved. Therefore, we have to introduce the basic concepts and laws of kinetic theory on the basis of plausibility considerations as did Boltzmann. There are many unsolved problems in relativistic kinetic theory, questions concerning the foundations, the mathematical structure, and specific physical applications. We shall refer to some of them in the following lectures.

- 305 -

Ehlers

Several systematic expositions of relativistic kinetic theory exist which naturally have much in common with the following lectures, in particular those by N. A. Chernikov (1963,1964), C. MarIe (1969), J. Ehlers and R. K. Sachs (1968), and J. Ehlers (1969). The elementary aspects of the special-relativistic theory which precede the Boltzmann equation (or sidestep it) are contained in the well-known book by J. L. Synge (1957) whose geometrical spirit has strongly influended the present lectures. (More specific references will be given at appropriate places in the lectures.) In order to free equations of inessential factors, we shall use the following convention regarding physival dimensions and units: We put

c = 8 Tt G =1i = 1, where h

c

is the speed of light in vacuo, G Newton's constant of gravitation,

the quantum of angular momentum, and

k

Boltzmann's constant. All

physical quantities are then measured by pure numbers.

- 306 -

Ehlers

1. Assumptions on spacetime. Notation

Let

X

denote spacetime which we assume to be a real,

four-dime~

sional, connected, differentiable Hausdorff manifold. In addition, sume

X

we as-

to be oriented, and take always oriented local coordinate sy-

stems (x a ), a = 1, ... ,4. The tangent space to

X

is denoted as

p

d

T (x); its dual, T"'(X). PaP

T4t are (--) and (dx ), respectively. p dX a carries a normal hyperbolic metric whose signature we take as

T

Natural, dual bases in X

at p

and

+ 2. The metric tensor or gravitational potential is written gab' the Riemannian connection is r~c'

and the Riemann! curvature tensor is R~Cd'

Rab: = RC b' and the Einstein tensor by ac Rg ab , where R: Ra a · The sign of the curvature ten-

The Ricci tensor is given by Gab:

1

= Rab - 2

sors is fixed by the Ricci identity. d

=

We assume that

vd R abc'

X is time-oriented with respect to

gab' so that it is

meaningful to distinguish between future directed and past directed timelike and lightlike vectors, respecti vHy An orthonorlaal basis

and such that

(1)

e4

(e.) J

( 1)

of

T

p

is always chosen to be oriented

is future -directed.

An example of a pair (X, g b) which is not time-orientable is given in appendix I of Ehlers (1969).a

- 307 -

Ehlers A coordinate- system (x a ) is said to be inertial at

r : c Ip

(?xal

if

p ) is orthonorlaal. The physical interpretation of general relativity theory is largely =

0 and

p, p ~ X,

b~

sEld on the correspondence principle that physical laws in the presence of gravitation retain their special relativistic form at with respect to coordinates which are inertial at

p if expressed

p. This guiding pri!!.

ciple is not unambigious, however. The assumption that spaceti me is oriented is not necessary for

kin~

tic theory; it is made here only for convenience. Without this assumption several quantities appearing in kinetic theory would have to be defined with respect to oriented domains of X, and it would have to be shown that a change of orientation preserves all relevant equations. This can be done. The assumption that spacetime

is time-oriented is also not necessary

for those pArts of kinetic theory which are independent of the Boltzmann equation. Without it, some quantities would have to be defined relative to time oriented domains of X, and the relevant equations would have to be shown to be insensitive to changes of the time orientation; that can easily be done. The Boltzmann eqllation, however, can only be formula..! ed in a time-oriented spacetime, and its form is not preserved under a change of that orientation. The reason is that the occupation numbers of initial and final states enter the collision integral in a non- symmetrical manner, as will be seen later and as is known from ordinary kinetic theory. The arrow of time built into the Boltzmann equation shows particularly clearly in the Htheorem, to be derived later.

up

- :W8 -

.Ehlers 2. Some facts about differential forms and integration (1 ) Kinetic theory deals with various kinds of averages which are sed as integrals. The domain of integration is sometimes a ce in

expre~

hypersurf~

X, sometimes a specetime region, sometimes a hypersurface or

a region in phase space (to be defined below). The appropriate tools for forming such integrals-volume elements, hypersurface elements etc. are differential forms. We assume that the elements of the theory

of

differential forms on manifolds are known, and collect here a number of facts which we need later. On an n-dimensional manifold

N, the differential form fields can be

expressed, with respect to local coordinates, as sums of homogeneous forms like UI =

\II dxai ' .. a" where the components \II r! I at' .. a r ' I at' .. al' are real functions and

I

J.

are exterior products

of the coordinate differentials. With respect to the

operations of addition, multiplication with real numbers and exterior mil.!. tiplication

the form fields from an associative algebra. The exterior

r

differentiation operator An r-form

d maps this algebra into itself.

can be contracted, like any covariant tensor, with avec

tor

A; the result is a (r - l}-form cP= A·U/ with components I.D = I I I a 1 · .. a r = Aat \II . A coordinate-independent definition of this operation I at a, ... a~

( t ) See the "references"about mathematical tools" in the bibliography at

the end of these lecture notes.

- 309 -

Ehlers

is contained in the following assertion: For any system of r-1 vectors A 2,···, Ar' we have

For a fixed vector

0/-+ A· f

A the mapping

of the algebra of

forms into itself in an antiderivation, i. e., it is linear and satisfies the product rule

Moreover,

0.

A trivial, but useful consequence of the definitions is the Lemma 1. If

n

is a nonzero n-form at some point

map L-"L·.n. = : W

p

of

is a vector space isomorphism of

N, then the T (N) onto p

the space of (n-l) -forms at p. This lemma immediately leads to Lemma 2.

If.n. is

an n-form at

p,

n f

then the most general (n -1) -form W at

p

0, and

LET (N), p

If

fA)

f

to\) =

0,

f

such that t\){A l' ... , An _1)

whenever (L, A , ... , A ) is linearly independent, is given by 1 n-1 where a f 0. Corollary.

L

a

°

vCt

has the property stated in Lemma 2, then L . "" = 0,

- 310 -

Ehlers

w

and

o whenever

(AI' .... , An_I)

(L, AI' .... ,An_I) is linearly

depe~

dent.

n

Lemma 2 and its corollary should be visualized by considering and

w as volume-functions for n- dimensional and (n-l)- dimensional

parallelotope s, respectively. Another useful fact needed later the proof of which is left as an exercise is Lemma 3. If

n

is an n-form field on

N, L

a vector field and

f a

function, then

df "

(L •

n)

L (f)

n.

(1)

We here recall that a vector is (identified with) a linear differential operator acting on functions:

L(f)

= La f,

a Finally we recall the fundamental theorem (of Stokes):

If

M is an oriented, compact, m- dimensional subma.nifold-with - boun .

dary ~M N

of an n-manifold

defined on

N, and

'f

is a (smooth)

(m-l}-form field of

M, then

(2)

The assumption that

M is compact can be omitted provided

decreases to zero sufficiently strongly at infinity of be allowed to have "corners".

M; also,

if dM

may

- 311 -

Ehlers

3. Volume elements in spacetime Under the assumptions about spacetime stated in section 1 the expression

(3)

y-:g >0,

where g:= det(gab) and

is a nonvcinishing4-form such

that

'? (e 1,e 2,e 3,e 4) = 1 for any orthonormal basis (e j ); it is the volume

element of spacetime. Let

A

be a vector field on

X

and

D

a 4 -dimensional, oriented)

compact submanifold-with-boundary of X - henceforth called a region. Then, according to (2),

Id(AO"1)

=

D

SAO "l

(4)

'() D

The integrand on the left can be rewritten as

d(Ao,)

and that on the right as A

where

0

fJ) (

Aa

6.:'

a'

(5': = a

fT) dx bcd 6 (abed '

-.!.

(5)

- 312 -

Ehlers

~ abcd

'7 [abcd] ,

are the components of

(1234

(6)

"I .

are the components of the (vectorial) hypersurface element a the latter is a vector-valued 3-form.

in

~

X;

With this notation, (4) goes over into

Er, a

(7)

D

the familiar metric-dependent version of Gauss's theorem

in Rieman-

nian space. We shall heneeforth use the term hypersurface for "oriented hypersurface". Since each tangent space

T

q

of X is itself a (flat, oriented)

pseudoriemannian space, it has its own volume element ..,... I&.

g is to be evaluated at p a from

p" p a

:f a.

~X

,r:r;"d 1234 V -g P .

(8)

q with respect to coordinates (x a), and

define an oriented coordinate - system on

Physically important hypersurfaces of T

q

T

the q

are the mass - shells

for

masses m? O. The mass- shell

P (q) con sists of all future directed m q which belong to (proper) mass m;

(4 momentum) vectors p at p 2 " - m 2. An oriented coordinate- system on Take coordinates on

X around

q such that

P (q) is defined as follows.

dm _ --~, v;;)X

1,2, 3, are space-

- 313 -

Ehlers like and

S

X4

is future- directed and timelike at q. Then the restric-

tions of the natural coordinates ordinate- system on

p"

to

P (q), and p 4 ( m

P (q) form an oriented com 0) is determined by

>

cab 2 gab(x )p p = - m .

Po (q)

(9)

is the future light cone of q.

In order to obtain a scalar volume element on the

T -analogue of q

(5),

P (q), m

1 bcd La:= 6'(abciP .

Its restriction to

consider

(10)

P (q) has values proportional to the normal of P (q),

m

hence there exists a 3-form 1T

m

m

such that

(11) since

Pa explicit ely

is a normal of P

m

Setting a

4 in (10) and

(11) gives

(12)

The same volume element is formally obtained from

It

m

= 2

(2 2 H(p) d(p + m ) 1t.

(13)

- 314 -

Ehlers in which H is the Heaviside function of p 4 and

J

is the

Dirac

distribution. For m> 0, m 1t:

P

m

m

is the induced Riemannian volume element of

as a hypersurface of T .

(q)

q

In inertial coordinates at q,

we have the familiar

expression

~3

(14)

E

where

E

4

P

is the energy. Taking polar coordinates in p-space we

obtain

1(m

Vm 2 +p 2'

(15)

or also

m

1'(

m

2'

(16)

The consideration which led to the volume element TC o on the tan gent null cone q

in

P (q) can be generalized to the actual null cone o X. We leave it as an exercise to the reader to verify

Lemma 4.

Let

N~ be the past null cone of q

q, and let u

q

of

be a

of W is oba q tained by drawing null geodesics through q, choosing tangent vectors k future-directed timelike unit vector at q. A normal k

to them such that, at q, k.u

and parallely propagating these a dx a k's along the null geodesics. Also, put v = 0 at q, and put k = dv' = 1,

q

- 315 -

Ehlers obtaining a field of affine distances

v on

J.q

dfl q

ne>note as

the solid angle abtained by proJecting a small bundle of null through and call r E

Jq

q

into that 3-space through

rays

q which is orthogonal to

u, q

D the distance from apparent size of an arbitrary point from q, as measured by an observer at q with 4-velocity u . q

Then

\ 17)

so that

D2d

n q 1\ dv

is a natural scalar volume element on N. q

4. Basic assumptions about a relativistic gas. Geometry of phase space.

(i )

The history of a system of many (classical) particles of negligible size is represented in relativity theory as a complex of timelike or lightlike wordlines. The particles may be thought of as being macroscopic ( stars, galaxies) or microscopic (molecules, atoms, ions, nuclei, photons, ... ), and they may be interacting through long- range and/or short range forces. Without attempting to give a detailed description of the dynamics of such a general system, we lay down a special, simple model for some systems which we call gases. In these systems, the particles are assumed to move like test particles in a mean

( i)

gravitational field

gab

and elec-

The geometric treatment given in this section follows essentially that of Bichteler (1965). See also Chernikov (1963), Lindquist (1966) (Appendix), and Marie (1969).

- 316 -

,Ehlers

tromagnetic field

F ab' except during encounters due to short range in-

teractions which are idealised

as pOint collisions. (I. e. ,the range of

these interactions must be much smaller than the mean free path. ) The mean fields may be external fields - we then speak of a test gas - or may be collectively generated by the gas p8rticles themselves, in which case we have a selfgr2vitating gas

(or a Vlasov plasma).

We proceed to formalize this qualitative picture of a gas. A particle of mass

m (~O) and charge

e

has a worldline

xa(v) which obeys the Lorentz-Einstein equations of motion

a

p,

Dpa a b dv = e F bP ,

if radiation reaction is neglected. The parameter the tangent vector mv

v

(18)

is so chosen that

pa is the (future-directed) 4-momentum. If m

is proper time.

~v

> 0,

denotes, here and in the sequel, the abso-

lute derivative along the world line, Dpa

cis

=

d p aa r c b bc P P .

cis +

(19)

If a particle participates in a collision at x E X, its world line may

have a corner at

x, or the world line may end or begin at

x, if

the

particle is annihilated or created in the collision. In the case of many particles the spacetime figure of a gas is a complicated network of curves, since several trajestories with different direc tions can pass through the same event, and the trajectories through nearby events can have quite dilferent directions,

- 317 -

Ehlers A simplification of the geometrical representation is achieved, as in nonrelativistic kinetic theory, by introducing a phase space. in relativity no preferred space sections

t = const. exist, the

Since relati-

vistic phase space cannot be defined in strict analogy to the ordinary

(1, p) phase space (of one particle), but will correspond to the (;t, t, p, E) ~space.

We define the (relativistic) one particle phase space for parti-

cles of arbitrary mass

M: = {

m

to be the manifold

(x, p): x E X,

P (Tx(X),

P~

0, p future directed.} (20)

This set is indeed a 8-dimensional manifold, if we agree to take

as

local coordinates (x a , pal, where (x a ) is a coordinate-system on

X

and

pa are the corresponding natural vector components.

M is, in fact, a manifold with boundary, the boundary 2 set of states (x, p) having mass zero, p = O. M is a fiber bundle

with base

X. The fiber at

of non-spacelike, future-directed vectors at space at bundle

being the

x is the set

x, i. e., the 4-momentum

x. (If all vectors had been admitted, M T(X)

dM

would be the tangent

over spacetime.)

M is obviously oriented, the (x a, p a)_ systems being oriented coordinate-systems. The equations of motion (18), (19) define on

a~

L =P

d X"

M a vector field

aarabcd

+ (e F bP -, bc P P )

d pI.

(21)

- 318 -

Ehlers

called the Liouville vector

(or operator). The oriented integral curves

(xa(v), pa(v)) form a congruence in

M, the phase flow generated by

L. Physically, the phase flow represents the set of all test particle motions which are possible in the combined gravitational and electromagnetic fields occuring in The rest mass

L. m

as given by equation (9) is a scalar function on

M. It is constant on each phase orbit,

o.

L(m)

Hence the restriction defined by m

=

of L to the hyper surface m const. is tangent to M . We note that m M

m

=

L

Up m xeX

M , with its Liouville vector

m

(22)

L

M of m

(x).

m

(23)

and its phase flow, is the phase

space for particles of fixed mass m; it is seven-dimensional -

corresponds to the Newtonian (1; t,p) -space. (

( 1)

M

t

)

and

It is also a fiber

In classical mechanics, this space is sometimes called "augmented phase space". See e. g. Liboff (1969). p.16.

- 319 -

Ehlers

bandle with base shell at M

by

X, the fiber over

x

now being

P

m

(x), the mass-

x. , being the boundary of

f1~ _ m 2,

t~

oriented submanifold of

given

M

is also orientable. We orient it by choosing a coordina4 a te system (x ,p) on M such that p 4P pEP (x), 0 whenever P

a

<

m

and then take (x a , p'l1 ) as an oriented coordinate-system on

M

m

.

We

then have L = pa m

~

dXa.

+ (e F" pb b

,.,V pbpc) - I bc

'j

<JP"

(24)

We know from ordinary statistical mechanics the usefulness of a mea sure on phase space which is invariant under canonical transformations and, in particular, under the phase now. Let us conseder, therefore, the coordinate-independent 8 form

n : on

M

=

~ "1"

(formed by means of

= _ g dx 1234 A

(3)

and

(25)

(8)) and the 7 - form

...:.Ld

Ip41

on

dp 1234

1234" d 123 x

p

(26)

M . Obviously, at each Doint. m 0,

n

m

I o.

(27)

- 320 -

Ehlers

nand

n

are related as follows (exercise):

m

!l =

m dm

",{lm .

(28)

To see whether

n is invariant with respect to the phase flow we com-

pute

the Lie derivative of

£L!L ,

the identity (

~

"" : =

d.n. = 0, L'

n

=

= d(L·..n)

tL n

we get

p

with respect to L. Because of

)

iLn. and

n

a

era" Tt

+

L·d.n.

= d W, if we put

1

a

d

+ 6" ~ abcd( eF dP -

ra

d e bcd de P p )dp

The differential of this 7- form vanishes. This is reall.y inertial coordinates at some (arbitrary) event Hence,

( .. ) See, e. g. Hicks

(1965), p. 94.

x.

"1-

(29)

verified by using

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Ehlers

0;

dl.l)

n

i. e.,

The

is invariant under the phase flow t - form W

(30)

(Liouville's theorem).

which arose here rather naturally as a tool will

be seen in the next section to be important in itself; let us note some of its properties. From its definition

(29)

and from

iLW

= d(L' W)+

+ L(dw) we infer: L'

~

o.

0,

(31)

These properties express that '" induces a nonzero quotient manifold

MIL;

~A

(31)

7-

SII '

then

on

with respect to

means that MIL. If

'l, ed in

JE

~

L, i. e., such that

} r-t ~,. f';if""WA('l> , ... ~

M comoving local L =

'eA ,whIch . "." )d;> IS a form

is a tube of phase orbits and

!:

a cross section of

measures, loosely speaking, the "number" of orbits contaiR

W

1

I.U =

the

W can be consedered as a measure on tre

=.:m=.:an=i.::.;fo:..::l..:;d_o"--f,--,, p_h.=cas=..e, -,o-,-r.. .:.b-=..it:..=. ;s. Indeed, if we int roduce on coordinates

7 form on

it is independent of the cross section.

The preceding consliderations can be carried over straightforwardly from

W

m

M, L,

to

M ,L

:=L'n. =pa

m m

m

m

(exercise); one obtains

6"" A1t + a

m

1

IY)

). b rl b c

,.."

21 p,1 P"f"''' (F bP -, bcP P )dp 1\'9

(32)

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Ehlers

(.n. ) =d m

W

m

=

~L

(W m )

m

L'CV m

m

=0

(33)

5. Distribution function, collision density, Liou ville's eguation An individual gas-history - a particular complex of world-lines is too complicated to be useful; we are interested only in the

typi~al,

average properties of gases. Therefore, we imagine a large collection of microscopically different, but macroscopically indistinguishable gas histories, a Gibbs-ensemble of gases. The average properties of such an ensemble are the subject of kinetic theory. (The averaging may have the additional merit that it disposes of certain all-too-classical features of our gas model like sharply defined worldlines and collision events; the average properties may well provide an appro&.imate macroscopic description of a gas whose particles obey quantum laws. ( , ) Consider, then, a gas consisting of particles of different species. Concentrate on one component the particles of which have mpss charge e. A definite microstate, or history, of Ue gas can be

m

and

represen~

ed, as far as the specified component is concerned, as a collection of

( 1 \

Nonrelativistically the Boltzmann equation, e. g., can be "derived" from classical as well as from quantum mechanics; see Kadanoff and Baym (1962): Lowry (1970).

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Ehlers segments of phase orbits in

M , the states occupied by particles m between collisions. (We do not assign phase-orbits to particles during collisions; hence there are no particle orbits in

M

m

transverse

to tre phase flow. ) The distribution of occupied states in terized by the functional

1: the

pact hypersurface

2:--+N m [ 2:]

M can be fully characm which assigns to any co~

number of occupied orbit segments intersec-

ting it. By a hypersurface in

M

m

we mean here and henceforth

oriented, 6 -dimensional submanilbld with boundary of section of an orbit

k

with

I:

an

M . The inter

m

-

is counted positively (negatively) if,

at the event of intersection, the vector basis

(L m , AI' ... ,A 6 ) has the

same (opposite) orientation as the basis of an oriented coordinate system of

Mm' Lm being the tangent to k

and (AI"'" A6 )

an oriented ba

sis tangent to ~ . If

D

is any region in

collisions in

Mm' then

Nm [

d D]

is the number of

D, if creations are counted positively, and annihilations

negatively. For a macrostete, let Since

Nm[X] is

Nm[X]be

the ensemble average of

a kind of flux through

L

N .

m of a fictitious fluid steeaming

in

M with velocity L , we expect it to be expressible as an integral. m m WEI thus need a volume element for hypersurfaces in M . m It is natural to ask whether there exists a 6 - form on M

could serve as such a volume element. From the meaning of

m

which

Nm

it is

clear that this form would have to assign a nonzero volume to any hype.!: su rface - element not tangent to

L , since there could be a flux through m

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Ehlers

it. Using the fact that (see eq. (27))

n

is a non-vanishing 7-form on

m

M

m

and remembering lemma 2, we infer that such

6-form must coincide with

W

a

as defined in (32), except for a non

m

-

vanishing factor. Because of Liouville's theorem, eq. (33), it is advisable to choose this factor to be constant on phase orbits in order that the 6 - form is

) " df /\ c.J "df 1\ (L • m m m L (f) n ~ L (f)" 0 ; we have used (1)). Hence, iN m m -"'In --...,,- m m reoommends itself as an almost unique candidate for the required me-

L

n )" asure.

A hypersurface

m

-invariant

2:

( 0" d(f W

M whose projection into X is a spacem like hypersurface corresponds to a region of an "inttantaneous" ordin~ ry

in

(it,p)-phase space. On such a

X

whose projection into

L:

(and, more generally, on any

is a hypersurface), W

m

I:;

from eq. (32) reduces

to its first part,

p

m

a

If we choose at some point

stem such that

t5"'a A x

is, at

(34)

~ .

with (x, p) E x, normal to

l: an inertial coordinate sy-

r'

(34) gives, at

x,

(35)

which is, except for the (conventional) sign, the ordinary. phase volume element of an observer at

x

with

4 velocity

::x~

. Therefore, Wm

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Ehlers

is the appropriate

6-form we have been looking for ..

We return to our study of

M . According to its physical mea!! m ing, we make the following smoothness assumptions about N , i. e. m about a macrostate of a gas:

l: C

D 1) On any fixed hypersurface

f~

nonnegative density function

Mm there exists a continuous,

Jr'

W

fE

D2 ) Every point (x, p) , every region

D

Mm

cue

I

Nm

l:'c:E

such that for all compact parts

(36)

m

has a neighbourhood

U

such that for

M

[Cl

m

J

oJ ~ A n

(37)

m

D

for seme constant

A

dppending on

U.

Dl asserts that on any fixed hypersurface ~ the measure defined by the expectation value of the number of occupied states contained in parts

~I of

l:

has a continuous derivative, or density function

fE

with respect to the geometrical measure Equation

(35)

shows that

fE

c.J. m (x, p) equals, for any observer who -

se worldline intersects the projection of

!; into

X

orthogonally at

the ordinary density of states in his infinitesimal, ordinary

(1.-P)

x,

phase

space. D2 asserts that the expectation value of the number of collisions in

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Ehlers

D

nm

is at most of the order of the

volume of

D; this assum

ption excludes, e. g., the possibility of having all (or particularly many) collisions occuring on one hypersurface of X. The two assumptions

D1 and D2 imply theexistence of an in-

variant, i. e., hypersurface or observer-independent (one particle) distribution function

(1)

f

m

N

on

M

m

such that for any hyper surface

J

(38)

fm (A)m;

m

E To prove

contained in two hypersurfaces

~

For that, consider a tube dary. Then

l.: f'\ t

17 J ,

out loss of generality gether with the part between

2:1 ('\~]:

\ 1)

A

we have

and

~2'

then

of phase-orbits having

'1.

'S

is

fI:l($~=ft2. (~)

on its boun

~

two cross sections of

.

With~

we assume that these two cross sections to -

and

Nm

2:1

L f1 J are

I\..

table, compact region sect

~EM

(38), we have to show that if a point

of the cylindrical buundary ~ j

~'Ll"\dj form D

of

[dD 1

the boundary 'dD

which lies of an orien-

M . Since no phase orbits can inter m

Nm[~1"'1]

Nm[l:1.,,1].

Because of

The preceding introduction of f is a generalised and "rigorised" version of that given in S!J(ge (1957), p.12 14.

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Ehlers

D1 and with the mean value theorem for integrals this can be written as

N

m

I

= fop ($) w ~1 11:;.":1 m

[dD]

- f~~1 (~)t t.n)m JW ,

where

re-

~.

~

€

L:'. ""g,

But we know from Liouville's theorem that the two integrals on the right-hand side are equal. Hence, using also

~

Jnm (1:",'.}IIlJm )

-1. If one now lets

A

~

J)

bit !Jassing through

S

~

D 2,

I

fE1

(~1)

- f1;1.

f~ (~)

(~2.)

\

shrink towards the or-

, the right-hand side tends to zero since the

numerator is "one order smaller" than the denominator. Also, Consequently,

'I.

f~

=

~f

(s).

~~

in order to emphasize that

~i.--'

We call the common value f

m

f

g,

(S),

is defmed on 1\1 • m m It is easy to verify that our orientation and sign conventions im-

ply

f

m

)

o.

(39)

It is techilically desirable and physically not harmful to require

also D3 ) fm

is continuously differentiable on

Mm.

Having obtained a phase space density

f

which measures the m average density of occupied states, we obtain straightforwardly a M . The average number of collisions in the m region D eM, i. e., the difference between creations and annihim lations of particles of the specified kind in D, is given by N collision density in

= "\

f fm

C1D

J

W ' d(f w ) = mOm m D

Jdfm

[dD]

A

LV = m

DI dfm ,,7Lm ,Wm )

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Ehlers

=I

D

(f

L

)

n. .

m m m We have used equations (38), (2), (33), (32), and (1). Hence a j} f ). b r). b c d f"", (40) Lm (fm) = p ;} x":. + (e F bP - , bcP P ) e)P>" 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.

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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).

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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.

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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)

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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).

,

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Ehlers

(47) in an inertial coordinate system at arbitrary observer at N 4 m -Nm '.

x:

is the number density 4 <_v~m) = N).m ~ d xl = N m /X is the particle flux density,

T 44 = N 4

m

m

<E >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

(51)

<-. .. 'X

pressure tensor. Here

V.

E (= p \

3-momentum, and ( at f

m

and

p

are the 3- velocity, energy, and

),"") denotes the conditional expectation value X

x, evaluated by means of the probability distribution defined by with respect to the chosen inertial system. We also define mean kinetic pressure

p :

= 31 tr -Tm

~3

4 Nm

p

by (tr : = trace)

<"Po"v '\ / x(m)

(52)

and recognize the classical Bernoulli formula. The rest mass density is energy density in

Lemma

p:

=

m

N~.

Writing

for the

(51)3' we formulate

7. For any observer and any distribution f , the inequam

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Ehlers

lities

hold. Most of these inequalities are obvious from (51), (52), and E = m (1 _ Vi ) Vl ' P = E ~; the only nontrivial inequality is the third one, due to A. H. Taub (1 - v

2 t

fi

and

(1948). It follows by considering

2 ! (1 - v ).

as elements of the Hilbert space

"p'I.(P , f dpl2\ and applying Schwartz's inequality to (J.. m m

Equations

(51)

p , then f ' then

If

p« p»

If

m = 0, then

If

and

(52)

imply the well-known relations:

fA' ~ f+ ~

r

r=

~ 3p

3 P

them.

p (nonrel. monatomic gas), (ultrarel. gas),

(54)

(photon or neutrino gas).

In order to obtain balance equations for various macroscopic fluid variables we observe that these latter quantities are moments of the distribution function in 4-momentum space, given by at a\ a. p p ... p fm 11m' The Oth moment is, at least for m

J

> 0,

essentially the trace of the matter tensor. Indeed. (47), (51), and (52) give

r-

3p.

(55)

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Ehlers

The 1st moment is the particle current density

N a, and the m

2nd moment is the matter tensor. We would like to evaluate the divergence of the

Lemma

8.

If

r~th

g

order moment. We first establish C 1 -function on

is a

J

L

m

(g)

To prove this, take an arbitrary region

'S : = { M

M , then m

1lm . D C X, and let

(x, p) : x E. D, P E Pm (x)} be the cylindrical region

lying over

r

rg w = L (g) ;}/' m (~m these integrals into integrals over D:

J"g Wm

dD

(34)

f "'D" Since

=f

G'a {

and

(7))

dD

of

D. Then, as in the derivation of the collision densi-

ty r:bove eq. (40),

(Use

(56)

f

pag 7t m }

Lm (g)!l m =

n m . We transform both

=[~ (J

Ji~JLm(g) 'Tt m } , D

D is arbitrary, (56)

follows.

pa g nm ) ; a

from

J

(26).

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Ehlers

Next, we generalize this to obtain an "equation of transfer".

Lemma

9.

M , then

m

(1 )

(for r

If

f

~

2)

m

is an arbitrary distribution function

L to

, + ~ e

Fa). b

m

on

(f)'TC m m

J

1 ·· b p a ... p ... p a" fm TC m ·

+

(57)

}.: t

( The integrand in the sum is. to be understood such that a).

in the sequence

b

replaces

a 2... a r .)

Proof: Take an arbitrary tensor field v ... which satisfies at ... a .. . a~ ~ v = 0 at somearbltrary event xO' Put p ... p Y .•• a 2: ... a .. ; b at f = g, and apply lemma 8, to obtain at xO: :: a .. m

Evaluate

( i)

L m {. .. )

at

Xo by taking the phase-orbit through Xo

Tauber - Weinberg ( 1961)

and

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Ehlers

differentiate C.. ) with respect to the parameter at L

m

C.. )

xo' getting

~()_ ~(a'1. d ... - v d P ... p a,.) f v a 1... a.. v m

= va..... at' (p at ... Pa" L m (f) m + e where we have used v... gives

F a\ bPb ... par f m + ... ),

(18). Insertion into

(58)

and "dividing" by

(57).

(Generalizations of Lemma 9 have been given by Ph. M. Quan (1966) and C. Marle (1969), but they do not seem to have found applications yet.) Applying

(56)

(with

g = f ) and (57) (with r rn (45), (46), and (47) we obtain

sing the definitions

N a·a =

m'

J

L (f ) 11 m m m

2) and u-

(59)

and

(60)

Equation

(59)

is the balance equation for particles

species considered; since

of the

L (f ) has been shown to be the colm m lision density in phase space, tile right-hand side of (59) is the

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Ehlers

(spacetime) production density of these particles. Equation (60) is the 4-momentum balance equation for the given species; the vectors on the right-hand side represent the electromagnetic and the collisional 4-force densities acting on the col"!!. ponent of the gas with distribution function

f . An example for the m

latter is the force exerted on an electron gas by photons due to Compton scattering. So far, we always concentrated on one component of a gas which may contain other kinds of particles as well; all our equations are valid for any component of a mixture. Let us now first specialize to the case of a monocomponent, or simple gas

consisting of particles all having (proper) mass

m and

charge e. Then, assuming conservation of particles in collisions (59) gives the conservation law

N

a

m ;a

S

L(f)'t(

m m

m

0

a which, of course, implies also charge conservation, J ;a

o.

Assuming also 4-momentum conservation during collisions, (60) re sults in T ab m

;b

o.

(62)

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Ehlers

These equations give on the one hand the macroscopic conserva tion laws basic to fluid mechanics, and they impose restrictions on the evolution of

f ,required by the microscopic conservation laws. m For a simple gas, there are two sensible ways to define the

mean 4-velocity. rent vector N a

m

N a m

One can either use the fact that the particle curis- timelike and put

a a a n uk ' uk u ka = -1, uk

or one can use the normality of the matter tensor the

ua

a

a uDu Da

uk a

a

= -1, u D

is called tre kinematic mean velocity.

ing with

( i)

m

future-directed (64)

(1 )

An observer travell-

is characterized by the property that in his local iner-

tial frames there is no particle flux density (63)

T ab and use

of Lemma 6, i. e., require

0,

uk

(63)

future-directed,

(see

(51) 2);

n

from

is the proper particle number density. a (( ) u D is called the dynamic mean 4-velocity. An observer travellr

The distinction and terminology is due to J. L. Synge'; see Synge (1960)

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Ehlers

ing with it will measure no momentum density, and this characterizes

u Da (see

(51)/ The energy density

f

in

(49)

is the

minimum of the energy densities measured by all possible observers; this property also characterize s The two mean velocities

uk a

u Da

(exercise).

and

aD a

are in general distinct,

their equality characterizes (by definition) adiabatic processes. They are physically characterized by the existence of an observer

ua

who

finds neither a Jll. rticle flux nor an energy flux in his local inertial systems. The necessary and

sl~t'ficient

condition for that is that

[aT b] N c = 0, a very complicated restriction on the distriN m m c m bution function. a If one chooses ...!!!!Lmean 4-velocity u , one can decompose the matter tensor uniquely according to the scheme (Eckart 1940):

T

ab m

r-

(a b) u au b + 2 u q + P h ab + 'T(..ab,

(65)

where ha

projects

T (X) x

b

onto the

$a.b + uaub 3 - space orthogonal to

(66)

a u , and where

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Ehlers

u

o.

a

(67)

U. is the mean energy density, qa the mean energy flux densiJ ab ty, p the mean kinetic pressure and rc; the shear pressure a tensor relative to u a , These quantities change with u a . If u = a 0, Adiabatic processes are characterized by the then q a property that q o for u a = uk a . If, in addition, Tt~ = 0, the

gas behaves, in the process considered, as an ideal gas.

We shall

extend these mechanical considerations later on to the regime of ther modynamics. Consider next a multicomponent gas. We distinguish the particle species by indices charge

A, B, ' , ,; particles of species

m A'

e A' and (if we have microscopic particles) further characte-

ristics like baryon number which we denote by tor

A have mass

bA

etc, Each species has its phase space

MA (instead of

LA; its distribution

and its matter tensor in terms of these, like

function

M ), and its Liouville operamA a a fA' current densities NA' J A'

TA ab, and the quantities which we have defined

u~ , A.

Requiring again 4-momentum conservation, we have instead of (62) T ab = Fa Jb k;b b'

(68)

- 343 -

Ehlers

where T ab

LT~b

k

(69)

A

is the total kinetic stress energy momentum tensor

of the mix-

ture, and

Ja:

=

LJAa

(70)

A

is the total electric 4-current density. Moreover,

(71) The indlVidual particles will in general not be conserved during collisions, but certain combinations of the

NA a

will have vanish-

ing divergence. For example, if we define the baryon current density

(72)

and assume conservation of baryon number during collisions, we obtain

;a

o

(73)

- :344 -

Ehlers

and

(74)

Similarly, we will have

(75)

and (76)

Thus, we obtain macroscopic conservation laws for a mixture and corresponding integral conditions for the distribution functions. Resonable mean 4-velocities for a mixture are the dynamic mean 4-velocity ab by Tk '

ua D

defined as in

(64), with

Tab m

replaced

the barycentric mean velocity, defined by

(77)

and the baryonic mean 4-velocity, defined by

(78)

- 345 -

Ehlers

provided

Ba

is timelike, as it is for "ordinary"matter.

With any choice of mean 4-velocity, one can decompose Tk ab according to

(65), obtaining

r'

p etc. for a mixture. Which 4-

velocity is the most useful one depends on the circumstances; a careful investigation is not known to the author.

7. The selfconsistent Einstein-Maxwell-Liouville equations (t ) Consider a collisionless mixture of particles, so that holds for each component, and consequently

(41)

(71), (74), (76) are

trivially satisfied. Then, we have the macroscopic conservation laws

(73), (75) and the (generalized) Poynting equation (68). It is,

therefore, permissible to assume that

gab' F ab

are the mean fields

produced by the gas, i. e., to require that they satisfy the EinsteinMaxwell field equations: ab

G

F

( 1)

+ /\. gab

[ab, c]

TK ab + T ab M

'

(79)

0,

Compare with Tauber-Weinberg (1961) who apparently first advocated these equations.

- :146 -

Ehlers

where T ab

(80)

M

is the Maxwell stress energy momentum tensor. Indeed, (80) and . ab _ a ,b (79)2 Imply Tl\T 'b - - F b" and if this is combined with (68), there results' (T ab + Tab) = 0, as required by (79). k M;b Hence, the equations (79) together with the Liouville equations

(81)

seem to provide a closed, consistent system of dynamical equations for a gravitating plasma (in the Vlasov approximation). For neutral particles, (79)1

(with

T Mab = 0)

and (81)

a relativistic version of the equations of stellar dynamics

give

(for col

lisionless systems). It is natural to pose the Cauchy initial value problem

system

for the

(79), (81). Formally, there seems to be no obstacle to

sol~

ing it in the usual way by separating initial constraints from evolutiona equations, the former being propagated off the initial

hypersurf~

ce in consequence of the evolution equations, which in turn can

be

- 347 -

Ehlers

solved for the highest derivatives off the initial hypersurface, provided that is not characteristic. A careful elaboration for the present system (79), (81) does

not seem to have been performed, however. (1)

Examples of sulutions to the equations (79)1 (with

T Mab = 0 ),

(81) are known, see E. D. Fackerell (1966), 1968), J. Ehlers, P. Geren

and R. K. Sachs (1868), R. Hakim (1968), R. Berezdivin and R. K. Sachs (1970); see also Misner (1968), Stewart (1969),

( t )

Matzner (1969).

Solutions with electromagnetic fields do not seem to be known at present (Problem).

8. The Boltzmann eguation Consider again a multi component gas with particle species A, B, ... If collisions occur, then the phase space density of all collisions

which particles of type

A

in

participate, LA (fA)' will be a sum (or in-

tegral) of various contributions due to different kinds of collisions, e. g. , elastic and inelastic binary collisions, absorptions and emissions.

( 1)

For a series of papers on the stability theory of static, spherically symmetric solutions of (79), (81) (for Fab = 0), see J. R. Ipser and K. S. Thorne, Ap. J. 154, 251 (1968), and subsequent papers by Ipse r in the same Journal.

( 1)

(Note added in proof) Meanwhile, the problem has been solved by Y. Choquet-Bruhat; see Journ. Math. Phys., 1970, and another forthcoming paper.

- 348 -

Ehlers

Let the symbol (82)

( x;PP A' PB' ... stand for a collisions in which particles of types A, B, . .. respective 4-momenta p A' PB' . . . particles

collide at

x E. X

with

and produce

C,... with PC""; the numbers of incoming and out-

going particles may be arbitrary. (If, e. g., A = B, one has to write pA' PA

instead of pA' PA; this is tacitly assumed here and

in the sequel.) The set of all collisions (82) of a particular type, with PA € PA(x), . ..

is again a bundle over

carries a measure, viz.,

'?

1\ 1tA A

1TB

x

€

X,

X, the collision bundle. It 1\ ...

A 1t C

Augmenting our former smoothness assumption

A ...

D2 concerning the

probability distribution of collisions we make the hypothesis: C 1) In any macrostate of a gas, the average number of collisions (82) in a compact region

J

U

of the collision bundle is

V (x; PA' PB" ..-PC' ... ;)

where

U

S(Ap) ,?AnAA 1'CB· .. A 'fCC' .. (83) i

V is a nonnegative (ordinary, measurable) function. (

to avoid ambiguities in the definition of V, U

) (In order

must be such that 4-momen

tum ranges KA (x), KB(x), ... of indistinguishable incoming or outgOing particles

( i)

Because of the S-factor in (83) the physically important domain of definition of V is (X; p A' PB'" . -PC'" .) : Ap = 0

- 349 -

Ehlers

( A = B =... ) do not overlap. In (83),

Jis the Dirac distribution (on R\

and

(84)

is the 4-momentum difference between "in" and "out" states. The

J -factor in (83) expresses that collisions (82) occur only if they

conserve

4-momentum.

It then follows that the distribution functions

fA' fB' . . .

of

a gas satisfy equations of the form

In this "collision balance" the sum is to be taken over all kinds of collisions in which

A-particles participate, either as incoming or

as outgoing collision partners. The integral goes over the mass shells of all colliding particles except the one whose state occurs on the left-hand side of (85).

r~

is a numerical factor depending

on the type of collision and on whether the state pA

on the left-

hand side of (85) is an "in" or an "out" state; it is defined thus:

r~

>0 «0)

if

A is an "out"

("in") state, and

- 350 -

Ehlers

Ir~ I

n A ( n A ! n B ! ... nC !)

=

-1

,where

nA' n B, . .. are the

numbers of (indistinguishable) particles of types ing or leaving the

V-collision, and

A, B, . .. ente£

nA refer to the number of

particles to which the left-hand state in

(85) belongs.

" (p A' PA I ~ .. I ) (If we h ave a co11lSlon "-7 PA' PB' PB

with

A

f

B

hand side of

(85), then

+ .)

and

PAis the state occuring on the 1eft-

r~

= -2 ( 2! 1! 2! )-1=

This

factor is necessary in order that the various collisions involving identical in

(or out) particles are not counted several times in

the balance (85). The equations (85) are useless as long as the dependences of the functions

V on the state of the gas are not specified. ( ')

It

is clear from nonrelativistic statistical mechanics that in a rigorous many-particle theory

V

ticle distribution functions

fA' fB' ... , but also ( at least) on pair

(1 )

See, e. g., Liboff

(1969).

will depend, not only on the one-par-

- 351 -

Ehlers

correlations

gAB (x, p A;x! pj/ No attempt will be made here to

cope with these difficulties which pose important and interesting problems. Rather, I shall write down a IIreasonable Ansatzll (as people say); then I shall make some remarks about the IIphilosophy II which is used to motivate that Ansatz; then modify it so as to account for the non-classical symmetry character of Bosons and Fermions; and then simply proceed on the basis of the resulting (generalized) Boltzmann equation. Consider the hypothesis

C2 ) V(x;p A' PB' .. ~ PC' ... ) " fA (x, pA) fB(x, PB)" . R(p A' PB' .-:t' PC' .. ) (85)

. In

wh'IC h th e fac t ors

fA' . .. re fer t 0 th e

II· In II

t on l y. st a es

(C 2) is suggested by the assumptions that (a) particles which are about to collide have uncorrelated momenta, (b) the ranges of the coll.isional interactions are small in comparison with the scale on which the

fA'S

change appreciably with x,

(c) collisions take place in spacetime regions so small that the mean differential gravitational field the mean electromagnetic field

Ra bcd

(geodesic deviation etc. !) and

F ab do not affect their frequency

- 352 -

Ehlers

appreciably. (d) the presence of particles not participating directly in a collision does not affect the probability of occurence of that collisioll. These assumptions, which essentially express that the gas is dilute

((d) and, for a gravitating gas, (c)), not too

inhom~

geneous in spacetime (b) ), and in a state of high randomness ( (a) ), indicate the range of validity of the "Boltzmann collision hypothesis"

C 2; each of them poses a problem of justification and

indicates desirable generalizations. If, e. g. , (c) were not true, then

R

might be expected

the principal directions and eigenvalues of In order to support the assumption

(t )

R abcd

to depend and

with two incident and

q

Fab.

C 2 further and to

it to scattering theory, let us consider a collision (p A'

on

p~

relate PC' ... )

emerging particles, and let us consider

Compare Marle (1969), p. 88.

- 353 -

Ehlers

those collisions for which the momenta are contained in small ranges KA CPA (x) etc. According to

(83)

and

(86), the number of those collisions

per unit spacetime volume is

(87)

Regarding the

KA -particles as a beam which hits the

KB -parti-

cles forming the target, we recognize that the number densities of projectiles and target particles, relative to any inertial frame with 4-velocity

u, are given by

(see (44) )

whereas the relative velocity of these particles is

I

(u.p A) PB - (u'PB) (u'p A) (u'PB)

PAl

(88)

- 354 -

Ehlers

Hence, we can rewrite

(87) as

(89)

where (90)

Equation

(89)

is recognized as the standard definition of the dif-

ferential scattering cross section

dQu

for scattering of

pA' PB-

particles into the ranges quation

(90)

K, .. , relative to the u-frame, and ec is indeed the correct expression far that cross sect-

ion which can be derived in the non relativistic limit either from classical or from quantum

0(.)

mechanics, and

!)

( £)

in the relativistic domain from quantum scattering theory.

( t)

See, e. g., Brenig and Haag

(1959)

- 355 -

Ehlers

In this case R is simply related to the

S operator

(1 )

.)

( In the relativistic case, a classical derivation is not available, since there is no well developed theory of interacting particles. ) In a certain sense, we have now justified assumtions stated above the

A- and

(85), since under the

B-particles in the gas should

behave as if they were members of beams in a collision experiment. One correction, or generalization, of (85) shall now be made. If the particles are atomic or sUb-atomic, then assumption (d) is definitely wrong. In the case of Fermions, the presence of particles in the final states decreases, because of the Pauli principle, the collision probability, whereas for Bosons it enhances that probability (stimulated emission and scattering). This is incorporated by writing, instead of

(86),

(91)

( 1 )

See,e. g., Brenig and Haag

(1959)

- 356 -

Ehlers

Here and in the sequel the upper sign refers to Bosons, the lower one to Fermions.

is the volume of a phase-cell which corresponds asymptotically to a non-degenerate p-eigenstate of a free (quantum) particle

( 1 )

degeneracy rC' Hence, sCfC(x, PC) equals approximately the

of spin average

occupation number of simple one-particle pc-eigenstates localised near x. (In the "classical limit"

fC«s~l,

(91)

reduces,

of course, to

(86). A "pseudoproof" of

(91) can be given within the Fock-space

malism, but that will not be reproduced here

( t )

One simplification is possible and useful in If

u =

ApA+ fPB -

for-

.

(90).

and these frames include the center - of - mass

frame of the collision as well as the rest frames of the incoming particles-

( 1 )

( \)

Weyl

(1911), Peierls

See Bichteler

(1936)

(1965), Ehlers and Sachs

(1968), Ehlers (1969).

- :357 -

Ehlers

of

u, whence the corresponding expression

dQ

(92)

is often called "the" (relativistic) cross section. We have now suppressed the arguments henceforth as a(3q - 4) - form, where tes and we imagine that differentials dQ =

T(C

~(-PA'PB)

~ ' .. )

KC"'" q

dQ

is the number of final sta-

has been "absorbed" into four of the

1\ ... : (If q = 1, dQ is a

J(l2

considering

~- "function":

2 2 2 [mA+mB-mC ] - (Pi PB))

G"a

is the ab-

sorption cross section.) Inserting

(91) into

(85) we obtain the generalized Boltzmann

equation

where we have simplified the notation in an obvious way. In particular, in

(93)

and henceforth,

(94)

- :l5H -

E:hlers

Sc r~.)

the factors former

have been absorbed into

The equation

(93)

E. ..

(r

R l\

equals the

has first been formulated in special relati-

vity for a classical (i. e., Boltzmannian) gas with elastic binary interactions by Lichnerowicz and Marrot (1940); for other treatments and generalisations see Tauber-Weinberg (19CiO, Israel (1963), Bichteler: (965) and the papers mentioned in the introduction. Henceforth we shall require the Boltzmann equation (93)

to

hold for the distribution functions of any gas. (Other "reasonable" alternati ves for

V

which lead to different kinetic equations are

po~.

sible, but will not be discussed in these lectures.) One important symmetry needs to be mentioned. If the pic collision law

(S-matrix) is invariant with respect to the total

flection, PT, then the collision "matrix" R:: is invariant pect to an interchange of incoming and outgoing states

(" )

microsco-

See Brenig and Haag

(1959).

(

, ):

with

reres-

- 359 -

Ehlers

RAB .. . C.. .

R C .. .

AB .. .

(95)

We also add that, for Fermions, it is necessary that

(96)

due to the exclusion principle. The conservation law

~ (~p)

(71)

is satisfied by

- factor. Other conservation laws like

(93), because of thEl (74)

can and have

to be incorporated by similar restrictions on the R-functions; this will be assumed in the sequel. It is now clear that we can generalize the selfconsistent field

equations of section

7 so as to take into account collisions; we just

have to replace equation

by

(81)

(93). The remarks about the Cauchy

problem made in section 7 still hold; a rigorous analysis for the system (79), (93)

has not been performed, however.

In a given spacetime a test gas, Bichteler for

( t )

(with a metric of class

X

C 2 ), i. e., for

has solved the (local) Cauchy problem

(1967)

(93). Besides existence and uniqueness of exponentially bounded

on the mass-shell, i. e.,

~(x)pa. fA (x, p) I~b(x) e J& ,with band

I

depending continuously on

x.

(t )

,

Ja

- 360 -

Ehlers

nonnegative, continuous and a. e. differentiable solutions for given initial distributions of the same type, Bichteler has established the continuous dependence of the solution on the initial distributions,

the metric, and the cross section (i, e., R::). He assumes

throughout that the total cross sections (all final states)

f dQ

are

bounded. (This last aSBumption, though perhaps valid for strong interactions. ( , ), does not seem to hold, e. g., in the case of weak interactions. ( t) Bichteler obtained his results by applying Banach's fixed point theorem to an operator given naturally by means of (93). defined on a suitably chosen complete metric space of exponentially bounded distribution functions. As Bichteler pointed out, his results lend some credibility to the

(formal) Chapman-Enskog approxima-

tion which will briefly be discussed later.

( i)

See Eden (1966)

( t.)

See Bahcall (1964)

- 361 -

Ehlers

9. The second law of thermodynamics (H-theorem). We define the entropy current density

of a gas to be

the

4-vector field

(97)

with

sA

clefined as before (below (91) ).

The expression

(97) can, in a sense, be derived from an in-

formation-theoretic point of view as indicated in Ehlers (1969). In the classical limit

sa =

sA fA.-.. 0

L {f

it reduces to

pafAlog (s AfA) TCA -

N~ },

(98)

A and one recognizes in the first term of

S4 the Boltzmann entropy

density. Generally, _Sau

is to be interpreted as the entropy densia ty relative to an observer with 4 - velocity ua. Using Lemma 8 one obtains

~

sa; a =

f

LA (fA) log( (sAfA)-b)

rcA'

(99)

i\

inserting LA (fA) from the Boltzmann equation

(93) and assuming the

- 362 -

Ehlers

PT - symmetry

(95)

for all collisions involvee one gets

a

sum of terms; one from each kind of collision and its inverse, of the form

where we have again used the notation

(94). Each such integral

is nonnegative, since its integrahd has the form ( log

a b)

(a - b). Hence,

;a

~o.

(101)

This is the relativistic form of Boltzmann's H-theorem ( TauberWeinberg (1961), Ehlers (1961), Chernikov (1963), which expresses locally the content of the second law of thermodynamics framework (101) surface in

in the

of kinetic theory. implies that the flux of X

Sa

through any closed hyper-

is nonnegative. Hence, for an adiabatically enclosed

- :l(j:l -

Ehlers

or isolated gaseous body the total entropy

S[~] :

(102)

evaluated on a spacelike cross section of the world tube of the body, never decreases towards the future. (Notice that in the classical limit

(93),

srI]

consists of the total number of particles

Boltzmann contribution. If the total particle the Boltzmann Notice that due to If

and the

number is not constant,

S-term alone does not necessarily increase. ) (101)

does not follow from

(93)

if the collisions are

PT - violating interactions. (95) does hold, and if collisions occur frequently in a gas, then

the competition between collisions of a certain kind and their inverses suggests the tendency of the gas to evolve in such a way that the difference in the integrand of

( 100)

the entropy production density tions

te'lds towards zero, so that ultimately Sa

;a

vanishes and the Liouville equa-

(41) holds, orovided there are no disturbing external influences.

Unfortunately, precise theorems supporting this physical expectation are so far missing in relativistic kinetic theory; even at the nonrelati vistic level little is known. (For a brief discussion see, e. g., Uhlenbeek and Ford

(1963), p. 31.) Any result in this direction would be of interest. It

would also be of some interest to know whether in situations of gravitational collapse

S

may increase toward>: 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

<;rab,

-e-"" ,

We have shown earlier that the "fluxes" vanish at an event there is local equilibrium at

x if

x, and that the IIforces" vanish in

a region if there is (global) equilibrium in that region, Hence, one is driven to conjecture that, in a near-equilibrium process, the fluxes

(which are "caused" by the forces) depend linearly and ho-

mogeneously on the forces, with coefficients depending on the thermostatic variables sumption

s, n A' This assertion is indeed used as an as-

in phenomenological approaches, and leads to (more or

less) well-known relativistic linear transport and reaction eguations for the shear viscosity flow

wa :

~

= qa

reaction rates

because of

i ~ , the diffusion currents i Aa, and the

v, The corresponding matrix which transforms p

( 0ab'.{t",,) into te

fA

1'Ca b' the volume viscosity 1t, the heat

----

(1'tab , TC",,)

must be positive-semi-defini-

(147), If one requires, as is natural for a fluid,

that this matrix is invariant with respect to rotations

( in the 3-

- 382 -

Ehlers tangent-space orthogonal to

u a ), the matrix reduces; and one

obtains a further simplification by assuming Onsager-Casimir symmetry. All this follows strictly the standard theory. However, we should not make these assumptions, but deri ve them from kinetic theory. This has not yet been done in the generality maintained here, but it will undoubtedly be done soon ( 1). Such a derivation will supposedly) give not only the form of the transport and reaction equations, but will also provide formulae for the transport and reaction coefficients in terms of thermostatic variables and cross sections. Two classical methods for doing this offer themselves; the Chapman Enskog method, and the Grad method of moments. Both these methods have, in fact, been adapted to relativity; the former by Israel (1963) plete form, by

and, in a mathematically more com-

MarIe (1969). (Israel, however, gives more de-

tailed results, particularly for a special type of "Maxwellian" gas.) The method of moments has been taken over into relativity by Chernikov (1964)

and in a more geometrical (and also

analytically more powerful) manner by MarIe (1969) ) and, independently,

(see Marle

by Anderson and Stewart

(1969), Anderson (1970). Mathematically,

(1966),

(see Stewart

MarIe's tre.atment is the

most complete one as regards the discussion of the "relativistic Hermite-Grad polynomials",

(')

whereas Anderson and Stewart have

(Note added in proof) See a forthcoming paper by J. M. Stewart, to appear in Lecture Notes in Physics, Springer- Verlag.

- 383 -

Ehlers

gone further towards physical applications (transport coefficients from cross sections.) In all of this work, the gas is a simple Boltzmannian one; in that case, both methods give the transport equations for

b' 1C and q expected on the basis of (147). In a a particular. Israel (1963) and Anderson and Stewart (1969,1970) T(

both emphasized that a relativistic gas has (in general) a positive bulk viscosity, in contrast to a non-relativistic gas. The bulk viscosity vanishes both in the nonrelati vistic ultrarelativistic ce between m

=

( T - 0 ) and in the

(T-OOI limit. This result "explains" the differe~ D)-gases and (m> 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 <3"a

( 143)

A where./\. is that part of the boundary of the world tube which lies between ~ , and ~1

L . This ass;rtion

L: t ,and

where'" is assumed to be later

than

~2

inequality is a precise version of the somewhat vague

~S ~ J~

which has first been postulated

in general

relativistic thermodynamics by Tolman (1934). In a similar

fashion

one can derive other "global" thermodynamical laws for moving, finite systems enclosed in containers (timelike cylinders in

X) from

the basic differential relations discussed here; again, there ambiguity. (For another example of such a derivation, see

is

no

Starusz

kiewicz (1966).) Last - but not least - I would like to mention that the

long-discu~

sed paradox concerning the acausal nature of temperature propagatlOn (mathematically: the parabolic character of the corresponding system of equations) has been resolved by the observation that the general, "anormal" solutions resulting from the method of moments obey hy perbolic equations with non- spacelike characteristics (Stewart

1969),

and only the special, so-called "normal" solutions give rise to the p~ radox, which is, therefore, due to an inadequate approximation.

- 385 -

Ehlers REFERENCES A. References about mathematical tools

R. ABRAHAM and J. E. MARSDEN, Foundations of Mechanics, Benjamin 1965 Y. CHOQUET-BRUHAT, Geometrie differentielle et systeme exterieurs, Dunod 1968 H. FLANDERS, Differential forms with applications to the phX sical sciences, Acad. Press 1962 N. J. HICKS, Notes on differential geometry, van Norstrand 1965 A. LICHNEROWICZ, Theories relativistes de la gravitation et de 1'electromagnetism, Masson 1955 A. LICHNEROWICZ, Linear algebra and analysis, Holden-Day 1967 M. SPIVAK, Calculus on manifolds, Benjamin 1965 B. References on or related to relativistic kinetic theory

( The following list is not meant to be complete, but includes only papers which are closely related to the preceding lectures.) J. L. ANDERSON, in: Relativity; M. Carmeli, S. J. Fickler,

L. Witten (Ed.), Plenum Pro 1970, p. 109 W. L. AMES and K. S. THORNE, Ap. J. 151,659 (1968) J. N. BAHCALL, Phys. Rev. 136, B 1164

R. BEREZDIVIN and R. K. SACHS, in: Relativity; M. Carmeli, S. J. FICKLER and L. WITTEN (Ed.), Plenum Press 1970, p. 125

- 386 -

Ehlers

K. BICHTELER, Beitrage zur relativistischen kinetischen Gastheorie,_ Dissertation, Hamburg 1965

K. BICHTELER, ·Z. Physik 182 , 521 (1965)

"

, Commun. math. phys.

R. H. BOYER, Amer. Journ. of Physics

!, ~,

352 (1967) 910 (1965)

W. BRENIG and R. HAAG, General Quantum Theory of Collision Processes, in: Quantum Scattering Theory, M. Ross (Ed.), Bloomington

1963

N. A. CHERNIKOV, Acta phys. Polon. 23, .629 (1963) ""

" " " ~, 1069 (1964)

""

" " " ~, 465 (1964)

""

Physics Letters ~, nO.,2, 115 (1963)

C. ECKART, Phys. Rev. 58 919 (1940) R. J. EDEN, Lectures on High Energy Physics, University of Maryland Technical Report, 1966 .J. EHLERS, Akad. Mainz Abh., Math. Natur. Kl., Jahrg 1961,

791

"

"

, General Relativity and kinetic theory, in Rend. d. Scuola Internaz. di Fisica "Enrico Fermi", Corso 47 (1969)

J. EHLERS, P. GEREN and R. K. SACHS, J. Math. Phys .

.!!.'

1344

(1968) J. EHLERS and R. K. SACHS, Kinetic theory and Cosmology, Brandeis Summer Institute in Theoretical Physics, 1968 E. D. FACKERELL, Ph. D. thesis, Univ. of Sidney 1966,

- 387 -

Ehlers

E. D. FACKERELL, Ap. J.

"

s. R.

~,

643 (1968)

1.,

, Proc. Astron. Soc. Australia

"

86 (1968)

de GROOT and P. MAZUR, Non-equilibrium thermodyna-

mics, North-Holland, 1962 R. HAKIM, Phys. Rev.,

173, 1235 (1968)

J. IPSER and K. S. THORNE, Ap . .J. ~, 251 (1968) W. ISRAEL, Proc. Roy. Soc. 259, 129 (1960)

"

"

, Journ. Math. Phys.

i,

1163 (1963)

L. P. KADANOFF and S. BA YM, Quantum Statistical Mechanics, Benjamin 1962 G. A. KLUITENBERG, S. R. de GROOT and P. MAZUR,

Physica~,

689 (1953)

"

"

1079 (1953)

"

"

, Physica

~,

G. A. KLUITENBERG and S. R. de GROOT, Physica 20, 199 (1954) ""

"

"

21, 148 (1955)

"

"

"

~1,

"

169 (1955)

P. T. LANDSBERG and J. DUNNING-DAVIES, in: Statistical Mechanics of Equilibrium and Non-Equilibrium, J. MEIXNER (Ed.), North-Holland 1965, p. 36 R. L. LIBOFF, Introduction to the theory of Kinetic Equations, Wiley 1969 A. LICHNEROWICZ and R. MARROT, C. R. Acd. Sc. Paris 759 (1940) R. W. LINDQUIST, Ann. of Physics

~,487

(1966)

J. LOWRY, Ph. D. Thesis, Univ. of Texas at Austin, 1970

210,

- 38t! -

Ehlers

C. MARLE, Ann. Inst. Henri

" 11

c. W.

""

A .!Q, 67

Poincare

"

"

(1969)

A.!Q, 127 (1969)

,C. R. Acad. Se. Paris 263, A, 485 (1966)

MISNER, Ap. J.

~

431

R. PEIERLS, M. N. 96, 780

(1968)

(1936)

PHAM MAU QUAN, C. R. Acad. Paris 263, A, 106 R. K. SACHS and A. M. WOLFE, Ap. J. 147,73 A. STARUSZKIEWICZ, Nuovo Cimento

~,ser.

(1966) (1967)

10, p. 684

(1966)

J. M. STEWART, Ph. -D. thesis, Cambridge (Engl.), 1969

E. C. G. STUECKELBERG and G. WANDERS, Helv. Phys. Acta 26, 307

(1953)

J. L. SYNGE, Relativity: the special theory, North.- Holland 1956 "

" , The relativistic gas,

"

" , Relativity: the general theory,"

A. H. TAUB, Phys. Rev. 74, 328

""

1957

"1960

(1948)

G. E. TAUBER and J. W. WEINBERG, Phys. Rev.

~,1342

(1961)

G. E. UHLENBECK and G. V. FORD, Lectures in Statistical Mechanics, Ann. Math. Soc., Providence A. G. WALKER, Proc. Edinb. Math. Soc. H. WEYL, Math. Ann.

!!.'

441

i,

(1911)

R. A. MATZNER, Ap. J. 157, 1085

(1969)

238

1963

(1936)

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. 1. M. E. )

ABSTRACT MINKOWSKI SPACES AS FIBRE BUNDLES

K. B

Corso tenuto a

MARATHE

Bressanone dal

7 al

16 Giugno

1970

ABSTRACT MINKOWSKI SPACES AS FIBRE BUNDLES

K. B. Marathe University of Rochester

A new definition is proposed for representation spaces of Special Relativistic physical events. These spaces - called here Minkowski Spaces - are given a structure of a fibre bundle over a four dimensional manifold. The Lorentz group appears as the structure group of this fibre bundle. It is shown that there is a unique torsion- free connection

in this bundle which induces the usual pseudo-Riemannian structure on the base manifold.

- 392 -

Marathe

MINKOWSKI SPACES AS FIBRE BUNDLES

K. B. Marathe University of Rochester

1.

Introduction

The study of Minkowski spaces arose early in this century in connection with the special theory of Relativity. The

concept

of four- dimensional space as a setting for physical phenomena was introduced by Hermann Minkowski. However, we do not find many investigations related to the discussion of the structure

of

these spaces from the mathematicians viewpoint. The questions of topology and differential structure if regarded asaa four-dimen sional manifold have not received due consideration. The Lorentz group plays fundamental role in all considerations related to the special Theory of Relativity. It is then natural to ex pect that this group should arise in a characteristic manner in the consideration of the appropriate mathematical structure taken as a setting for the physical phenomena in the domaine of the

special

Theory of Relativity. However, it is known that regarding a physical event as a point in

IR 4

or in a suitable four- dimensional ma-

- 393 -

Marathe

nifold does not lead to the Lorentz group (Ref. 1). To overcome this difficulty alternative topologies have been proposed for Minkowski Space by Zeeman (Ref. 1) and Celnik (Ref. 2). The topology

fR 4

MF

proposed by Zeeman for

has among other properties the property that the

Lorentz group is the group of homeomorphisms of MF. Though this space has some interesting properties it is topologically quite complicated. In particular it is not locally compact and therefore standard methods of calculus cannot be employed in it. In this paper we propose for the space of physical events (Special Relativistic) the structure of a fibre bundle over a four dimensional manifold

M. The new definition is equivalent to sal

ing that a physical event is characterized not just by a point in the manifold but rather by the point te chart at

P

int

P

and a fixed coordina·

P. This coordinate chart corresponds to a unique

near frame in the bundle of linear frames

P

Lover

~

M. The po-

together with this frame give the point in the bundle of

linear frames over

M, which we shall call a representation of

the physical event under consideration. Which points of the bundle

L

are to be regarded as representing physical events? The

answer is provided by the following theorem which is the main -

- 394 -

lVIarathe

stay of the proposed definition. Theorem 1: Let fold,

L

lVI

be an arbitrary four-dimensional mani-

the bundle of linear frames over

lVI, G the four-

dimensional general linear group, H the Lorentz group and (5'"

a cross- section of the associated bundle

E (lVI, G/H, G, L) then 1. There exists a unique (depending on tr) reduced subbundle

P (lVI, H) of

L w::'th structure group H,

2. There exists a unique torsion-free linear connection in

which makes

P

lVI into a pseudo-Riemannian space of signatu-

re ---t, 3. The holonomy group of this connection is a subgroup of and there exists a connection whose holonomy group is

H,

H.

We call P (lVI, H) a representation space for physical events or in the standard terminology a lVIinkowski Space. With this defini tion two representatives correspond to the same physical event if and only if they belong to the same fibre of

points of

P

P. These two

are then related by a Lorentz transformation.

2. Geometrical

Preliminaries

By a differentiable four- dimensional manifold

lVI

of class

- 395 -

l\1arathe

c"

we mean a Hausdorff, connected, locally Euclidean topo-

logical space with a fixed four-dimensional C r atlas. r need not be infinite, but in what follows we asSl me it to be large enough to ensure the smoothness of the operations involved. We denote the tangent space at frame

I

at

x

x E M by T (M). A linear

x

xEM

tangent vectors in the tangent space T (M). We define the bun

x

dIe of linear frames

L(M, G, 1'(), written as L, as follows. As

a set it contains all the linear frames I at all pOints of M. The group

G

acts freely on

g E G and I E L then x m

Y = gl' i

gl

x

L

by the following action.

is the frame mEL x

If

given by

X

m where m

mx = (Y I , Y2, Y3, Y4 ), and g = (gi ), the summation convention being used. The action of

G defines an equivalence relation on

a natural manner. Denoting the relation by there exists g(G only if L/G

much that m = gl. Clearly

we have

I....,m if

1- m

if and

x

y

x = y. Under this equi valence relation the factor space

can be identified with Let

'V

L in

rc:

L---" M

M.

be the natural projection given by

- 396 -

Marathe

n(l)=x x

A differentiable structure on

,(1 (U)

L

diffeomorphic with

is introduced by making

U XG

where x E U eM

( U, d ) is a coordinate chart at x. Let x coordmate functions. Then for

In

T (M). If 1 = (X.) x

L

x

k

1

k

then (x , x.) 1

We regard The bundle

E

k = 1, 2, 3, 4 be

k = 1, 2, 3,4 form a basis

~x

then with respect to this basis

can be taken as local coordinates.

with fibre

on

L

G/H (considered as a right coset is constructed as follows: Consider

LX G/H

a.oJ) = (a.l,f.a-1)( LXG/H The quotient space 01 denoted by

and

H, the Lorentz group, as a subgroup of G.

space) associated with the action of G

-~

k

LX G/H

as a given by

for

aEG

and

(l,S) € LXG/H.

under this action of G

E (M, G/H, G, L), written as

E. The map

is

- 397 -

Marathe

and a differential structure is introduced in manner by using

in a natural

E

1'tE . (Ref. 3)

(l,~ )-+~'l

The surjective map

of LXG/H ---+ L/H fac-

tors through

E

sequently

can be regarded as a fibre bundle over

L

and allows us to identify

E

H. Let 1:': L---+ E = L/H

structure group

with

L/H. Co!!. E

with

be the natural

pro jection.

3. Proof of Theorem 1 The proof is divided into two lemmas. Lemma 1: There exists a unique reduced subblIDdle with the Lorentz group Proof: Define

Pc: L

H

tion,

1:'-1

there exists

'to) hE H

=

and Since

such that

=

---(0)

-r

is onto by defini-

TO)

= e"(x).

~

M

Now if

hI and conversely. Thus the set of all H-re

lated frames in a fibre of P

(crM)

then 1, mare H-related i-e. There exists

such that m

cation that

't- 1

is non-empty. Therefore, for every x

1 E peL

"((m),

L

by

G'(x) E E = L/H

[G' (x)]

of

as its structure group.

p =

For x E M,

P

L

form a fibre of

is a principle fibre subbundle of

P. The verifi L

is now strai-

- 398 -

lVIarathe

ght forward. Lemma 2: There exists a unique torsion-free connection in the bundle

P

which makes

lVI

into a pseudo-Riemannian

space with fundamental tensor of signature - - -+ ..

1R. 4

Proof: We first define a pseudo-inner product in equipped with the fixed basis e 1,e 2,e 3,e 4

where

e 1 = (1, 0, 0, 0), e 2 = (0,1, 0, 0), e 3 = (0, 0,1, 0), e 4 = (0, 0, 0,1) by (x,y)

=-xIYl-x2Y2-x3Y3+x4Y4

The product defined by (1) is invariant under i. e. h EH

implies

Now regarding each

1R4

1 E Pc L x

onto

hy)

T (lVI) x

(1)

H.

= (x, y)

---(2)

as a linear isomorphism of

we define a bilinear form on

T (lVI) x

by

g (X, Y) where

---(3)

X, YET (M).

Definition of

x

P

dent of choice of

and (2) show that definition (3)

is indepen-

1 E P. (1) shows that the form x

g is sym-

metric. It then can be identified with a symmetric covariant

- 399 -

Marathe

tensor of order 2. If

g.. are the components of g with IJ k respect to the coordinate system (x ) then the unique torsion-

free connection

r in

is defined by the usual relations

P

J gjk ') xh ) - (4) where

r

ij g

are the components of the dual tensor

g'" to g.

is the unique torsion-free metric connection for the pseudo-

Riemannian structure on gnature -.-.-.+

of

g

M

defined by the tensor

follows from its definition and

g. The si(1).

The theorem of Hano and Ozeki (Ref. 4) states that the struc ture group

G

of

L

can be reduced to a subgroup

only if there exists a connection in is

L

H if and

whose holonomy group

H. This last result and lemma 1 and lemma 2 constitute the

proof of theorem 1.

4. Conclusion In view of the results of theorem 1, it seems reasonable to define

P

as a representation space for Special Relativistic events.

Each event is assigned an element of unique as is to be expected.

P. The assignment is not

- 400 -

Marathe

Two points (x, 1) x

and

(y, m)

same event if and only if

in

y

x =y

P

correspond

and there exists h

to the E

H the

M = hI, Thus the Lorentz in va x x riance of Special Relativistic Laws is built into the definition Lorentz group such that

of

p,

The result proved here applies to manifolds which are not necessarily flat. It may thus serve as a spring-board for a corresponding definition in the case of General Relativistic events, However, no definite results in this direction have yet been obtained, Taking

M =

IR. 4

we can precisely characterize the distinct

representation spaces; they are in elements of the map

1-1 correspondence with the

G/H, This follows from the fact that for a E G/H -va

defined by

G": x - - - . (x, a) a

of M ---. E(M, G/H, G, L) is

a cross- section and 6" f G' if and only if a f b a, b E G/lL a b The bundle obtained by using in definition (0) is denoted by a P 6" a' This may be interpreted as saying that within the same P

rs

the representatives of the same physical event are connected by transformations belonging to the Lorentz group, but an non- singular transformation

a E G

arbitrary

will take representatives in P e"e

- 401 -

Marathe

into

P 5 a'

e being the identity of

frame as a cross-section of P.

1

L

G. If we define inertial

then we have a unique bundle

whose fibres are inertial frames. For example a frame

e = (e 1,e 2,e 3,e 4 )

satisfying

and

g(e.,e.)=0 1 J

may be defined as an inertial frame at each pOint of For this last case a different approach has also been proposed by Eberlein (Ref. 5)

- 402 -

.i\larathe

ACKNOWLEDGEMENT

My thanks are due to Professors Eberlein (Rochester), Ste.n (Rochester), Ehlers (Texas) and Lichnerowicz (Col lege de France) for useful discussions.

- 403 -

Marathe

REFERENCES

1. E. C. Zeeman, Topology, Vol. 6, pp. 161-170. 2. F. A. Celnik, Soviet Math. Dokl. Vol. 9 (1968) No.5, pp.1151-52. 3. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. 1, 1963, Interscience. 4. ,J. Hans, H.Ozeki, Nogoya. Math. J. 10 (1956), pp.97-100. 5. W. F. Eberlein, Bull. Am. Math. Soc. Vol. 71;No. 5, pp.731-736.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. 1. M. E. )

SUR LA PROPAGATION DE LA CHALEUR EN I

RELATIVITE

G. BOILLAT

Corso tenuto a

Bressanone dal

7 al

16 Giugno

1970

/

SUR LA PROPAGATION DE LA CHALEUR EN RELATIVITE G. Boillat

1.- Preliminaires. Supposons qu'un champ ~(x~) veri fie un systeme d'equations aux derivees partielles quasi lineaire du premier ordre (ou ramene

ou les A'" x

(i

de

(2)

ordre),

A"((~)~oI. "f(~,xf», ~,,0,1,2,3,

(1) i

a cet

sont des matrices carrees NX N, xo une variable de temps,

= 1,2,3) =

des variables d'espace. A la traversee de la surface d'on-

° Ie gradient du champ est discontinu et on a,

~(x~)

[~1= 0, [~J.l'" ~ctb~, ~.t'do(l(, A<J,
d'ou, en introduisant la vitesse normale

A

= 0,

de propagation dans la

direction Ii,

~ = - Cfo/\VI(\

4

n

VI{ /\V~\ i

An = A n.1

0,

Si AO est reguliere (det AO J 0) on peut supposer que AO = I. On dira alors que Ie systeme (1) est hyperboligue (au sellS large) si les valeurs propres de la matrice A sont reelles (sans etre necessairement n

distinctes) et s'iI existe un systeme complet de vectellrs propres, c'est-a-dire N vecteurs propres lineairement independants (ce qui est toujours vrai pour N valeurs propres distinctes). Dans ces conditions une perturbation initiale arbitraire peut toujollrs se decoQPoser en une somme de perturbations (~~) sur la base des vecteurs propres, perturbations qui se propagent ensuite de fagons differentes. Lorsque Ie systeme n'est pas hyperbolique, distinguons trois cas. a) D'apres (4) il est clair que la somme Sk des produits k a k des racines du polynome caracteristique est proportionnelle a l'inverse du determinant de AO. En particulier,

- 408 -

Boillat

°

, Sl'

IL A 1=

1coer • II det A I -1

, SN 1

m ~ 1=

Idet A 1 Idet AOI- l n

11 en resul te que si, dans certaines conditions, det AO ~ 0, l'une

au moins des sommes Sk tendra vers l'infini et il en sera par consequent de

m~me

de l'une au moins des vitesses de propagation, ce qui

n'est pas acceptable du point de vue physique. Ainsi on parle d'action instantanee

a distance

quand on considere I' equation de Laplace c-2Utt -

comme limite de celIe de d'Alembert

A ~

u .. 0 lorsque c

1 u=

0

~ ~

•

b) Si une valeur propre est complexe il appara!t un manque de stabilite par rapport aux donnees initiales

(1).

c) Si Ie nombre de vecteurs propres est inferieur

a N,

il est

possible de produire une perturbation ~B. telle qu'un choc (discontinui te du champ lui-m!!me : [B.

J ~ 0)

se produise immediatement (2).

Nous verrons diverses circonstances se manifester suivant Ie choix de l'equation de la chaleur. Bien entendu, nous chercherons

a obtenir

un systeme hyperbolique.

2.- Tenseur d'impulsion-energie. Pour un fluide parfait quand on neglige la conduction de la chaleur ce tenseur s'ecrit (3),

(5)

1;o(r = (~+ p)uo(u~

- pg"-f'> = 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<..)V""uf> - '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
+

ul h/r

Nous introduirons encore en un point Ie rep ere profTe oartesien du fluide, (16)

g

a
.. diag(l,

-1, -1, -1), un .. 1, u i

Dans ce repere, V°d. ,

= 0,

.. _ ~ij,

qO

=

°

et d'apres (}) nouE aurons la correspondance, (18 )

\

~-+

U ~ - c\ , Q --')0 q. n ~ qn ' C ~ - 1.

= o.

- 410 -

Boillat

,.- Equation de Fourier. On peut transposer directement l'equation classique de Fourier en relativite, (19)

c(.

q

=-

vo(~'\

K0

vrQ.

Cependant on voit immediatement (Eq. 17) que dans Ie repere pro pre les equations (19) ne contiennent auuune derivee par rapport au temps. On se trouve donc dans Ie cas (a) du §1. Eerivons alors avec Eckart (20)

qo(, = _

K'loif'(UrQ _ QU"\,u(J

Nous evitons ainsi l'inconvenient (21)

QU ~uo(,

=

= -

(5),

Ktf>\(Q'I~). et nous avons,

pree~dent

'f'f'~f'd Q.

Contra.ctant eette equation avec ~ on obtient en utilisant (13), (22)

U2(Er/r) + C(dQ/ Q)

a

O.

II nous suffit ici de calculer les vitesses de prlpagation dans une direct ion orthogonale au vecteur

q" .

Hous ferons done Q. = 0 e t en con.

traetar,t (14) avec ~r' en supposant U I 0, en se servant de (13),(15) et en remarquant que (10),(21) entrainent,

u ~qo(. rA

q iuc<.

= _

J..

= 0,

nous aurons.

Avec (22), on en deduira, 2

9 SQU

4

2

2

+ U C(QVPQ - rQSr - f) - C Pr

= O.

Sous les hypotheses usuelles (6), p

r

')

O.

Ie produit des racines de cette equation bicarree est negatif; il existera des racines imaginaires (§l,b). Cependant il est interessant de calculer ces rarinec. Avec les tions deG gaz polytropiques(avec Po ~ 4 _ ('(_ 2) ~2 _ ((_ 1)

=

I

0 et G = 0; voir § 13) or. trouve

O.

En dehorc de Ie solution imaginBire ~2

~2

y_

1,

equa~

- 1, on a,

- 411 -

Boilln

une valeur que nous retrouverons plus loin (§ll).

4.- Equation de la chaleur. En 1948, C. Cattaneo (7) proposa un nouveau modele d'equation de la

}haleur qui fut redecouvert dix ans plus tard par P. Vernotte (8) et mis recemment sous forme covariante par M. Krany~

(9).

C'est ce modele

que nous etudierons apros y avoir fait quelques changements. Nous chercherons une modification simple des equations

(19),(20)

sans

choisir a priori l'une ou l'autre,

(23)

qd...

=_'{rJ.f'(j)~,.,g+

rl\,u,..).

(C'est l'expression classique dans Ie repere propre s'il est repere d'inertie.)

T'"

Ajou tons un terme au premier membre. Comme il n' y a aucune raison pour que ce terme soit orthogonal a u( nous devonn recrire (23),

(24)

Q.rJ.. +

avec (25)

Q~

T""

=

= qu~

second membre,

+ q~ •

- Pour que Ie Nous avons introduit une nouvelle variable de champ q. champ soit hyperbolique il est necessaire que sa derivee par rapport au temps figure dans les equations du champ, c'est-a-dire dans T~ puisqu'elle n'est pas ail1eurs. On volt tout de suite que l'on n'obtient rien d'interessant en prenant

U1Vy(qU~).

T'"

proportionnel

a g
ou

On est ainsi conduit a faire apparaitre la deCivee temporelle

de Q.~ c'est-a-dire a prendre,

T" =~lVrQ.(

lv./ 1 Qri..)

'"

(s1 i- = cte). Nous ecrirons

(26)

(10)(11)

Q~ + lVf"(hQ~) = - '(rA("(V 'd,..g +

rlvyu t )·

On trouvera cette equation dans (9) a cela pres que nous ne supposons pas, dorenavant, h et

V

constants.

Nous definirons,

(27)

Flh .,' ~/h v , 'j1 rf + t"" + q =e. + p + r + q A 'j1 U adjoindrons a (26) l'equation d'etat, h = log c

et nous

h,

Q,

- 412 -

(28)

F(Q,r,q,h)

Boillat

= O.

5.- Perturbations. De (26) on deduit,

(29)

- UOqoC. .. Ui'(qbh

+~q)

+ Uqo(Dh +v'toll'f,Jg + U(q +r»)ucl.,

que lion porte dans (14) pour obtenir,

(30)

Aou~

- -r~\fo(.(~p -ll'SQ) + Uqt'(fh + ~r/r) .. O.

Nous y ajoutons (13),(15) ainsi que les equations qui derivent des equations d'etat,

= O..

(31)

rll ~ug( + UOr

(32)

rQU ~S = (~rJ. + UUol.)~qo( ,

(33)

u ~uo( rA

(34)

(5)

1,(

= 0' I A q DUo(

+ U 'bqo{ rA

.. 0,

op = Po dO + Pr ~ r, 'b S = Sg bQ + Sr ar, F a + Fr ~ r + Fq ~ q + lo'h ~ h = O. Q

Q

6.- La surface d'onde U .. O. Cette surface se deplace

a.

la vitesse du fluide. 11 vient immediate-

ment, et (si Q. ~ 0),

d ut'

" 0, r, ~qGC. = u b'q.( = 0, iGC.

g(

F- dq + Fh ~ h " O. q

II en resulte qu'il correspond trois vecteurs propres

a cette

vitesse;

Oq.( a deux degres de liberte et Oq (ou Oh si Fh etait nul) peut @tre choisi arbitrairement.

1.- Determinant ce.racteristigue. Nous supposons U I O. Contractons (30) avec ~r et utilisons (31),

(36)

U(A - Q.)~r/r + C(dp -I>~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<. ,( Nous devons done trouver N " 9 vecteurs propres pour que Ie systeme 80it hyperbolique. On pressent tout de suite que (40) est un pOlyn6me de degre 5 (au plus) en ~ : les deuxieme et quatrieme lignes du determinant sont de degre 2, la deuxieme de degre 1. On verifie que le degre est bien egal a 5 (nous ferons le calcul plus loin). 1") 6i A • 0 ne verifie pas (40), a toute solution ~W ;. 0 correspondra un vecteur 0 uo( donne par (0) et un vecteur ~ q0( donne par (29). Nous aurons ainsi (certainement si toutes les racines sont distinctes et non nulles - differentes de U • 0) 5 vecteurs propres. D'autre part si ~ n'est pas une racine de (40) }, = 0 et ouo( • • q~ • 0 a moins que A • O. Dans ce cas (cf. 29,31,33.38).

(41)

-

.l .. (q + ....I )~uG(,

u ~uo( 0(

= (811\. }uo(

• q ~ uO( ri..

.. O.

- 414 -

Une des composantes de

Boillat

auP<. est arbitraire. A la solution A = 0 cor-

respond un vecteur propre. Le nombre total des vecteurs propres est donc 3(U

= 0)

+ 5(aol. de 40) + l(A

= 0) = 9.

Le systeme est hyperbolique. (En supposant, bien entendu, que les valeurs proprea sont reellea,

ce

qui.reste

a

etablir.)

2°) Supposons maintenant que (40) soit Ie produit par A d'un polynome P4

du quatrieme degre. Aux racines de ce polynome correspondront

4 vecteurs propres de la fagon qui vient d'@tre dite. Pour A = 0 on pourra choisir arbitrairement une des composantes de O'W, les autres seront determinees de fagon a satisfaire les equations (30),(35),(37) (nous verrons que cela est possible). Le vecteur ~ uo( sera seulement soumis aux trois conditions (31),(33),(38) (42)

u~ ~u,( = 0, qQ(~/'

= -

I

U~r/r,

qo( ~uo(=

qh

+ ~q

;

une de ses composantes pourra encore 3tre prise arbitrairement. Ce qui revient a dire qu'a la solution A = 0 correspondront 2 vecteurs propres dont Ie nombre total sera 3(U

= 0)

I

+ 4(P4 = 0) + 2(A

= 0)

= 9.

(On a suppose, cela va sans dire, que P4 ne s'annule ni avec U ni avec A.) Le systeme sera encore hyperbolique. d'ex~ence de la solution double A = O. Nous allons montrer que par un choix convenable de F il est pasible

9.- Conditions

d'obtenir une solution double A c 0 ce qui permettra comme on l'a vu oi-dessus de simplifier Ie determinant caracteristique (40). On tire de (30) avec A = 0, (43)

1p

=

).lbo, "h + Ur/r = 0,

soit, avec (35), (44)

(PO -].I )~o

(45)

FO ~O + (rFr - Fh )(3r/r) +

+ rPr '$r/r

=

0,

Fq bq

=

o.

Supposons F~ f o. On peut alors (comme Pr f 0) a l'aide des equations ci-dessus exp~imer ) h, ~r, dq proportionnellement a. ~ O. Le premier membre de (37) devra 3tre identiquement nul si l'on y porte ces expres-

- 415 _

Boillat

sions, compte tenu de 11ega1ite Q = son de 1a presence du terme en

F-q

ce qui est impossible en rai-

~U,

iJ C. On do it donc avo ir

= 0.

A10rs 07) donne 'Oq et (44),(45) 11equation aux derivees partielles que doit satisfaire F(Q,r,h) a 0, a(F)

(47)

= 0,

II est commode de prendre F sous 1a forme suivante reso1ue par rapport

a.

(48)

h,

F • - reh +

h = eh = p/r.

~(Q,r),

(47) devient,

(49)

(v-

PQ)~r + P~Q

a

0,

clest-a.-dire, si lIon ecarte 1a solution

p

D

cte (qui n'est d'ai11eurs

pas l'hy,othese qui conduit aux resu1tats les plus simples; voir §11),

(50)

~)p = V •

10.- Llouue excentionnelle A .. 0. A= (51)

° clest

~ U- Q

=

° soit,dans Ie repere propre (cf. 18),

~ = - qn/l1

En ecrivant que ~ 2 ~ 1, ou ce qui revient au m~me ici que 1a vitesse radiale ~ u~ - q~ est du genre temps (ou isotrope) on a 1 l inegalite, (52)

q2 ~ 'rI2 •

Cherchons maintenant sli! est possible que llonde soit exceptionne11e (12)(13) crest-A-dire ne produise pas de chocs sur Ie front dlonde. Pour qu 101 1 en soit ainsi on deVl~. avoir (14),

soit,

U~" + 11

q.(6uO<

_ ~.( ~qo(.

•

O.

De (32),(42) on tire,

~.(Oq'" = U(rQ dS -

Uv(

~qO<)

=

U(rQ

bS

+

qh

de sorte que 11equation ci-dessus slecrit avec (31),

+

~q),

- 416 -

6(rf + t'"+

q) -

'i1'br/r - (r01S +

Boillat

q1h

+ ~q) = O.

Mais, d'apr~s (7),

r~f-rOh= ~p, si bien que,

- r)r/r = 0, puisque = I- e -h et compte tenu de hh +br = 0, jjbO +br ,,0, ~p + Sf'"

ou,

r

( 43, )

c'est-a-dire, en vertu de,

(53)

~p=O

- qui derive de (48) et ~-

~)p " - h, Ces equations (qui se deduisent l'une de l'autre avec (50)) ne sont pas satisfaites par ~ = O. II en resulte que pour un tel choix l'onde A = 0 produira des chocs. (Les discontinuites des derivees mieres de

variables de champ, discontinuites qui se propagent

pre~

a la

vitesse radiale ~ u~ - q~ , deviendront infinies au bout d'un temps fini si l'onde est initialement acceleree (13).) En revanche, supposons

(55)

V

= K(~),

une certaine fonction de

r

p,

alors (54) donne,

= -K(p)O + funct(p).

Ainsi l'onde A = 0 ne produira pas de chocs (semblable en cela aux ondes d'Alfven de la magnetohydrodynamique). On remarque que si l'on prend simplement,

(56)

P.

= - K(p)O,

Ie second membre de l'equation de la chaleur est celui d'Eckart. Nous recrirons done (26), (57)

Q..i. +

iVr(pl/r)

= -

K(p)fl.rV~ (O'tt),

- 417 -

Boillat

avec la condition qui decouIe de (50), (55),

11.- Expression expljcite de I'eguation caracteristigue. Revenons au determinant (40) et cherchons l'expression de l'equation du quatrieme degre apres mise en facteur de la solution A = O. Avant de considerer Ie cas general notons une approximation particulierement simple (11). Elle consiste a prendre h ( .. ¢/r) = ~ = cte. On obtient alors facilement, pour un gaz polytropique (pY

(59)

Q< +

rlVfQr{

= -

~Rryo(rV~(or(.),

et dans Ie repere pro pre (outre les vi tesses

(60)

= p/r = RO),

A.. 0,

~ = - q/

II ),

+ J.

=+(Y_l)2,

(61)

ou )' est la constante des gaz polytropiques, egale au rapport des chaleurs specifiques :)' • Cp/C y • On remarque que (61) atteint la valeur limi te pour t . 2 (vo ir §14). Une combinaison lineaire de oolonnes (indiquee par les t@tes de colonnes) puis I'addition de la deuxieme ligne sous la forme,

(62)

a la

derniere mettent (40)

FO

a(F) + q(V- PO)Fq

F-q

Fh - qF-q

(PO - V)C

( I> - PO)UA

0

- UQ .. 0,

(PO - t> )Q

0

-A

q2u

2 rOSOU + POC

(~ - PO)B

2U 2

0

avec, B ..

r~)p u2 -

2UQ +

r~)p

C.

En realite, on n'obtient pas directement cette expression mais plutOt (a la place du terme (V - PO)B), ru 2 { rOPrSO

+ (V - PO)(rOSr + f»)- 2(V - PO)UQ + rprv

r..

- 418 -

Boillat

On met ~ - Pg (suppose maintenant non nUl) en facteur et on tient compte de (49),

J> - 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 <.

X { 1,

Non seulement la surface dlonde doit @tre du genre espace mais la vitesse radiale doit encore @tre du genre temps. Ced implique,

- 419 -

Boillat

soi t, 4(X - l)(XPX + YPy)P X + (q2X - y 2jX)Pi

(67)

~ 0,

pour toute solution de (65). 12.- Fluide incompressible. Lorsque la conduction de la chaleur est negligee on sait que la vitesse du son dans Ie repere propre est egale a. la racine carree de (dp!de)s On est done amene (15) a. definir Ie fluide incompressible comme correspondant au cas limite,

~)s

(68)

= 1

==.a,. ~ -

p ..

Cf(S).

lei nous procederons de maniere analogue. Nous chercherons une equation d'etat reliant la pression p a. deux variables thermodynamiques. Toutefois la vitesse normale ~(qn) depend maintenant de la direction de propagation ~ par l'intermedi~e de la quantite q . Nous supposerons n

done que la vitesse limite est atteinte a. la frontiere de l'intervalle de variation de qn:

p( 1, £,q)

=

~2(~ q) 0, ~

2

=

a

1,

1. Nous ecrirons, par consequent, 2 t q,. (q) ,

soit,

a'est-a.-dire, en posant,

(69)

e- =«', p

~ ~)r

+

r~)Q~)~

L'equation d'etat ne doit pas dependre, bien entendu, de q ni de

-q qui

figure dans ~ • II faut done, (71) Afin de satisfaire la derniere equation on peut faire deux hypotheses:

r

a) ou = cte, b) ou bien supposer que la vitesse de la lumiere est atteinte quand l'onde A = 0 attaint aussi cette vitesse ce qui a lieu quand q2 ,.~2 (cf. 52). Dans ces conditions

'if

+£q = 0 est verifiee pour l'une des

- 420 -

£

deux valeurs de Venons-en

a.

Boillat

et il suffi t de supposer

£f..

'f{~).

1-

une autre possibilite : A2(qn) = 1, qn€

Puisque cet te valeur de ~ doit 3tre extremale : d ~ eliminant Y entre P(l,Y) = 0 et Py(l,y)

4c2 (a 2

+ a l + ao )

= (2b 2

= 0,

I dqn

q, + q [ =

O. En

on trouve,

2

+ bl ) ,

qui ne fournit aucune equation d'etat et renforce notre premiere fagon de voir.

13.- Fluides polytropigues. La comparaison de (68) et (71) nous invite a etudier Ie cas ou

~

ne

dependrait que de l'entropie,

(72)

~

= ~(S).

En utilisant l'identite thermodynamique,

(58),

nous pouvons recrire

(l/v) ()vlOs) p

= ~(s)/K(s),

soit, V = pi (p)/(S)

(74) Avec

(73)

(75)

et

(7),

o=~

, F(S) ..

f~ dS.

on obtient,

p/

f .. 1 + i .. Pe F + G,

+ G' ( s) ,

ou P et G sont des fonctions de p et S respectivement et ou la derivation est notee par un accent. D'apres (27),(48),(56),

(76)

~ ..

r(f -

7

0) +

q = r(G

-

1

GI) +

q.

En raison de l'arbitraire sur P on en deduit que

';1 ,. q ~

G = O.

q

2

~)f

2

Alors, (78)

e+p

= f/v • p/p' ~ ~

= e(p);

f ..

70 .

Si nous suposons G' • 0 (G .. ete) nous avons, pour les chaleurs specifiques a pression et volume constants,

- 421 -

1. ~Q)

1

Cp

Q

'OS p

P (!) K

= K

I

+

Boillat

!

K'

(80) Par integration de

f~;

EO

Log

(79),

f

+ F(S)

et Ie coefficient de la derivee au fil du courant du vecteur Q~ dans llequation de la chaleur (57) slecrit encore, (81)

~

K(S)P' (p)exp

=

J~;

Si nous supposons de plus que Cv nlest fonction que de S, (82)

~;~

-

.. cte ..

t-1 ~

P '" a(p + po) (Y-l)/Y ,

1. _ 1. .. (¥_ I)! . Cv Cp K En particulier, on obtient les equations des fluides polytropiques en prenant, iC.§l 1 = cte.. C

KTSY

Pour les gaz, p

o

p

1(,«2 et si lIon prend,

= 0, G .. 1,

on aura, i=CQ, p

cp ..

rR/(l-l),

pV=RQ.

14.- Le fluide incompressible comme fluide polytropigue. rf '"

e+ p .. f/V.

Dans llhypothese la moins restrictive (b,§12),

t -p ce qui entraine,

P G -F = pI - 2p + pIe

.. lY() 1 S ,

- 422 -

(86)

P

= a(p

+p

o

)t,

Boillat

G = 0.

Ainsi Ie fluide incompressible correspond

(87)

~

- p = 2po = cte,

~

- ;p '>

0....,

a '( =

2. Comme (16),

p

o On se retrouve done dans Ie cas (a) du §12; la vitesse de la lumiere est atteinte dans une direction parallele au vecteur courant de chaleur.

15.- Vitesses de propagation. Avee (84), (65) donne, dans une direction orthogonale au veeteur

Plagons-nous dans Ie cas des gaz (p = 0, G = 1) et supposons que lion o puiase negliger Ie terme en ~I (en admettant en particulier que ~ varie peu). Alors,

(89)

~4

_

(y _ 1

+ rf) ~ 2

If

+

'U .. 'i1

0,

Si Ie dernier terme est(generalement)petit par rapport au carre du coefficient de ~ 2,

(90)

~2 ~'(-l+!!.

/1

1

IV

=

~ j (r -

1 +

r;).

On voit que (91) lorsque

\(\12 --'-' - , v_I, ,

q~

()O

qui est equivalent

\ 2 ----r ~ 0, ("2

(cr. (61). D'autre part quand

aI

q ~ 0,

rf/11

=f

dans 1 I approximation non relativiste; par con-

sequent,

La deuxieme valeur coincide sensiblement avec (60) (qn C'est, toujours

a la

meme approximation,

v 2jt s

ou

= 0,

e

~

r).

vest la vis

tesse duson quand on neglige la conduction de In chaleur,

- 423 -

Boillat

Passons au fluide incompressible. En negligeant encore ~, on cherche les solutions de p*(X,Y)

= 0 obtenu en remplaQant les coefficients de

(65) par, lim

(9})

~, ~O

~'a2 ' etc •••

On s'apergoit alors que la vitesse limite (~= i 1) est atteinte dans toutes les directions: P*(l,y) = O. En consequence, on obtient aisement les deux autres vitesses (en exprimant par exemple la somme et Ie produit des racines en fonctions des coefficients),

soit,

q ~2

+}q ~ _ C rO

n

p

On notera que la condition ~2

(95)

q > }q

=

0,

CprO

<1 dans

=

2(p + P0 ).

toutes les directions entraine,

+ CprOo

On sait que l'eau peut @tre consideree comme un flu ide polytropique

avec Y = 7, p > O. Si jamais (88) peut @tre utilisee on se rend compte o en tout cas qu'on ne peut plus negliger Ie terme en~' (pour que

0<~2 (1). Remerciements Nous exprimons notre gratitude Nos remerciements vont aussi

a MM.

a Mme

C. Cattaneo et A. Lichnerowicz.

Y. Choquet-Bruhat et M. J. Ehlers.

References (1) G. BOILLAT, C. R. Acad. Sc. Paris, ~ A (1970), 217. (2) G. BOILLAT, Ibid., 11}4. ()) A. LICHNEROWICZ, dans ce livre. (4) PllAM MAU QUAl!, Ibid.

(5) C. ECKART, Phys. Rev., ~ (1940), 919. (6) D. ter HAAR & H. WERGELAND, Elements of Thermodynamics, AddisonWesley (Reading, 1966).

- 424 Boillat

(7) C. CAT'l'ANEO, Atti del Seminario matematico e fisico della Universita di Modena, ~ (1948), 1; c. R. Acad. Sc. Paris, £iI (1958), 431. (8) P. VERNOTTE, C. R. Acad. Sc. Paris, ~ (1958), 3154. (9) M. KRANY~, Nuovo Cimento, ~ B (1966), 51; ~B (1967), 48. (lO)G. BOILLAT, Lett. Nuovo Cimento, l (1970), 521. (ll)G. BOILLAT, Velocities of Heat propagation in Relativistic polytropic Fluids, Ibid. (a paraitre en 1970). (12)P. D. WL~, Ann. Math. Studies (Princeton), 2i (1954), 211; Comm. Pure & Appl. Math., 1Q (1957), 537. (1 3)G. BOILLAT, La propagation des ondes, Gauthier-Villars (Paris, 1965). (14)G. BOILLAT, Journ. Math. Phys., 10 (1969), 452. Cf. Y. CHOQUETBRUI~T, Journ. Math. Pures & Appl., ~ (1969), 117. (1 5)A. LICmmRO"IIICZ, Theories re1ativistes de la gravitation et de l'electromagnetisme, Masson (Paris, 1955). (16)c. CATTANEO, Lincei-Rend. Sc. fis. mat. e nat., ~ (1969), 698, a propose de remplacer t? par ~ + p de fa90n a obtenir I' equa tion d'etat ~ = cte au lieu de (8 7 ).

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