354

G auss’s T heorem in G enera l R elativity

By G. Temple

(iCommunicated by Sir Arthur Eddington, F.R.S.— Received November 4,1935)

1 —Introduction

In a recent paperf W hittaker has given a generalization of Gauss’s Theorem on the Newtonian potential which is valid in Einstein’s General Theory of Relativity, and a further extension of this result has been obtained by Ruse.J Both of these extended forms of Gauss’s theorem depend upon the fact that, under the special conditions postulated by the authors, one of the components, G 44, of the Einstein tensor GM„, can be expressed as a divergence in the 3-way, x4 = constant. This can be seen at once if the line-element is expressed in the form

ds2 — V2 (t/x4)2 — ajk dx*dxk, (y, 1, 2, 3),

—as can always be done without any loss of generality. Then

G 41 = - V A2V + V {a ,§

where2VO,fc = dajk/d x \

and A2V is Beltrami’s second differential parameter for the 3-way x4 = constant. W hittaker’s investigations refer to the static field in which x4 is the temporal coordinate and da^/dx* — 0. In the work of Ruse the system of 3-ways, x4 = constant, is chosen so that

9 (a'*Oifc)/3x4 = -In both cases

. G 41 = - V (F %where

F* — ajk 0V / dxk;

and an extension of Gauss’s theorem is obtained by integrating the field equation,

G 44 - iG = - YT 44,

over a portion of the 3-way, x4 = constant.

t ‘ Proc. Roy. Soc.,’ A, vol. 149, p. 384 (1935).X ‘ Proc. Edin. Math. Soc.,’ vol. 4, p. 144 (1935).§ Cf. Campbell, “ Differential Geometry ” (Oxford, 1926), equation (148-4), p. 210.

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Gauss's Theorem in General Relativity 355

T h e re su lts so o b ta in e d h av e c lea rly o n ly a lim ited d eg ree o f g en era lity , a n d i t is a m a tte r o f in te re s t to ex am in e th e w id est p o ss ib le ex ten s io n o f G a u ss ’s th e o re m v a lid in G e n e ra l R e la tiv ity . S u ch a n ex ten s io n w ill necessarily express a n in te g ra l ( th e “ so u rce in te g ra l ” ) in v o lv in g th e s tress ten so r, T M„, a n d ta k e n o v e r a 3-w ay, S 3, as a n in te g ra l ( th e “ flux in te g ra l ” ) in v o lv in g th e firs t d e riv a tio n s o f th e p o te n tia ls , o f th e g ra v ita tio n a l field a n d ta k e n o v e r th e c lo sed 2-w ay, S 2, w h ich b o u n d s th e S 3. W h itta k e r a n d R u se express th e flux in te g ra l in te rm s o f th e “ g ra v ita ­tio n a l fo rce ” exp erien ced b y a p a r tic le w h ich desc rib es a n o r th o g o n a l tra je c to ry o f th e system o f 3-w ays, x 4 — c o n s ta n t, th e g ra v ita tio n a l fo rce be in g defined as th e rev ersed geodesic c u rv a tu re o f th e tra je c to ry . In g en era l th is im plies th a t i f th e so u rce in te g ra l is ta k e n o v er th e 3-w ay, x 4 = c o n s ta n t, th e n th e flux in te g ra l w ill in v o lv e o n ly th e d e riv a tiv es o fth e single p o te n tia l g 44. T h is re s tr ic tio n is u n n ecessa ry a n d , in th is p a p e r , n o such lim ita tio n is p laced o n th e flux in teg ra l.

In c o n s tru c tin g th e w id est p o ssib le g e n e ra liz a tio n o f G a u s s ’s th e o re m th ree d ifficulties a rise in succession . In th e firs t p lace , i f su ch a th e o re m is to be co m p le te ly g enera l, i t m u s t b e v a lid fo r an y S 3 b o u n d e d by a p re sc rib ed c losed S 2. I t is sh o w n in § 2 th a t th is in te rn a l c o n d itio n o f consistency can be satisfied in v ir tu e o f th e v an ish in g o f th e d ivergence o f th e ten so r, G M„ — = — y T M„, p ro v id e d th a t a n a p p ro p r ia tesystem o f te lep ara lle lism is in tro d u c e d in to th e R ie m a n n ia n geo m etry . T h e second a n d th ird d ifficulties a re th e exp lic it d e te rm in a tio n o f th e congruences defin ing th is te lep a ra lle lism a n d th e exp lic it d e te rm in a tio n o f th e flux in teg ra l. I t is sh o w n in §§ 3 a n d 4 th a i b o th d ifficulties c an be rem oved by ex p an d in g a ll th e g ra v ita tio n a l ten so rs as p o w er series in th e c o n s ta n t o f g ra v ita tio n , y , a n d by re ta in in g o n ly th e lead in g te rm s. T h ere th e n resu lts a m o d ified v e rs io n o f G a u ss ’s th e o re m w h ich is sim ple , exact, a n d co m ple te ly general.

T h e u su a l n o ta tio n is em p lo y ed — c o v a r ia n t deriv a tiv es a re d e n o te d by ( a n d th e su m m atio n c o n v en tio n is used .

2— T he C o ndition of Consistency

T h e in te rn a l co n d itio n o f consistency is o b ta in e d by co n sid erin g th e geom etric re la tio n betw een th e reg ions o f in te g ra tio n o f th e flux in teg ra l an d o f th e source in teg ra l. In th is co n n ex io n th e te rm s “ o p en re g io n ” an d “ c losed reg ion ” w ill be used in th e ir to p o lo g ica l sense to describe finite reg ions w hich d o o r d o n o t possess a b o u n d ary .

In th e classical fo rm o f G a u ss ’s th eo rem a n d in W h itta k e r’s ex tension o f th is th eo rem to a s ta tic E in ste in field th e flux in teg ra l is ta k e n over a

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356 G. Temple

closed S2 which lies in the 3-way, x4 = constant. The source integral is then taken over the open S3 which is bounded by the closed S2 and which also lies in the purely spatial 3-way, x4 = constant. Thus the domain of the source integral is uniquely specified by the domain of the flux integral.

In the general reativistic form of Gauss’s theorem the situation is quite different. The space-time manifold is no longer naturally stratified into purely spatial manifolds, and the domain of the flux integral can be any closed S2 in the complete space-time manifold, while the domain of the source integral can be any open S3 bounded by the prescribed S2. Thus there will be infinitely many open S3’s satisfying this condition, and Gauss’s theorem must be valid for them all.

The geometry can be more easily visualized for the analogous problem of the classical form of Stokes’s theorem. In Stokes’s theorem the analogue of the flux integral is the integral of the tangential component of the electromagnetic vector potential taken over a closed Si, around ft. closed curve), and the analogue of the source integral is the integral of the normal component of the magnetic intensity taken over any open S2 bounded by the prescribed Sl9 {i.e.,over any open surface bounded by the closed curve). There are infinitely many open S2’s satisfying this con­dition and Stokes’s theorem must be valid for them all. The condition of consistency is obtained by considering two open 2-ways, S2' and S2 , each bounded by the prescribed Sx. S2' and S2" together form a single closed 2-way S2, and the value of the source integral taken over this closed S2 must be zero. Since this must be true for any closed S2 it follows that the divergence of the magnetic intensity must vanish every­where. This is the condition of consistency for Stokes’s theorem.

In Gauss’s theorem the condition of consistency is obtained by con­sidering any two open 3-ways, S3' and S3", each bounded by the pre­scribed S2. S3' and S3" together form a single closed 3-way S3, and thevalue of the source integral taken over this S3 must be zero. Since this must be true for any closed S3 it follows that the divergence of the inte­grand of the source integral must vanish everywhere. This condition o consistency is expressed analytically as follows.

If the parametric equations of the S3 areX U. _ (Tl?

the source integral will have the form

jjf Jw m - ** ** d< (2jjwhere eM1,pir = ± (— g)4 is the fundamental antisymmetric tensor of the

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Gauss's Theorem in General Relativity 357

co m p le te R ie m a n n ia n 4-w ay. T h is tr ip le in te g ra l ta k e n o v e r a n y c lo sed S 3 can be exp ressed as th e q u a d ru p le in te g ra l,

| jjj (M m)m ( — g)* d x 1 d x 2 d x 3 dx*,

ta k e n over th e 4-w ay S 4, b o u n d e d b y th e c lo sed S 3. S ince th e tr ip le in te g ra l v an ishes fo r an y c lo sed S 3, th e in te g ra l o f th e q u a d ru p le in te g ra l m u s t v an ish a t every p o in t in th e S 4. H e n ce th e te n so r M M w h ich o ccu rs in th e so u rce in te g ra l m u s t sa tisfy th e c o n d itio n

(M*% = 0. (2 .2)

W h en th is c o n d itio n is satisfied , th e so u rce in te g ra l (2 .1) ta k e n o v e r an y open S 3 c an b e exp ressed as a d o u b le in te g ra l ta k e n o v e r th e c lo sed S 2 w h ich b o u n d s th e o p e n S 3. I f th e p a ra m e tr ic e q u a tio n s o f th e S 2 a re

X * = X * ( CD1 , CO2) ,

th e n th e d o u b le in te g ra l, i.e ,, th e flux in te g ra l, w ill h av e th e fo rm

( z 3 >

w here F MV is a n a n tisy m m etric te n so r . H en ce , b y th e g en era lized fo rm o f S tokes’s th e o re m ,f

£ 1VT —° /x v p c r J-VX3 F „ ,

dx>dF„pd x a ’

— (F p(X) p + (F ^ p -f- (F vp) O’*

I t w ill be m o re co n v en ien t to express M m in te rm s o f * F M*', th e te n so r w hich is rec ip ro ca l to Fp,„. S ince

*F mp = Fp„ )a n d L (2.4)O F mi, = * F p<r, jit fo llow s th a t

M m = i e ^ p<T sAvpo.

= __ 1_■ —_f ( —p-'yi( - g ) i d x*1 1

= (*F»% . (2.5)

T h e ap p lica tio n o f th e c o n d itio n o f consistency is sufficient to d e term in e im plicitly th e fo rm s o f th e source a n d flux in teg ra ls . S ince th e source

t W eitzenbock, “ Invariantentheorie ” (G roningen, 1923), p. 398.

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358 G. Temple

integral must be a linear function of the stress tensor, TM„, it follows that the tensor MM which occurs in the source integral (2.1) must have the form

M m = pA

where the tensor PA is chosen to satisfy the condition (2.2), viz.,

(PA T /% f= 0.

In virtue of the divergence equations,

(TAM)M = 0, :ithis condition satisfied by PA becomes

(P*)m TA* =*= 0. (2.6)

The flux integral (2.3) is specified by the tensor F M„ or its reciprocal *FM„, which must satisfy the condition (2.5), viz.,

PA T^ = (*F rV

The introduction of the field equations,

G / - | g A* G = — TTA* (2.7)t

now yields the differential equations for *F/41' regarded as a tensor function of the potentials, gM„, and their derivatives. These equations are

PaGa* - £P* G = - y (*F*% (2.8)

The construction of Gauss’s theorem for the gravitational field therefore depends upon the solution of equations (2.6) and (2.8) for the tensors PA and *F'i", and it is accordingly a problem much more difficult than the construction of Gauss’s theorem for the electromagnetic field. In the latter case the condition (2.2) is satisfied by the electric charge and current tensor J>, and the equation for *F^, (2.5), is satisfied by the tensor specifying the electric and magnetic intensities. In the gravitational field there are no simple tensor solutions of these equations immediately suggested by the theory.

3—The System of Teleparallelism

The auxiliary vector PA introduced in the last section can be regarded as determining a system of teleparallelism. It might be thought that in

t The value of the cosmical constant has been taken to be zero.

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G enera l R e la tiv ity th e so u rce in te g ra l sh o u ld b e re p laced by th e fo u r in tegrals

| | j i w T V 3 X - X2 d r 1 d (>. = 1, 2 , 3, 4) (3 .1)

b u t i t is easily seen th a t th is set o f q u a n titie s does n o t fo rm th e c o m ­p o n en ts o f a ten so r, p rec ise ly b ecau se n o in v a r ia n t m e an in g h a s b een a tta ch e d to th e a d d itio n o f ten so rs s itu a te d a t d iffe ren t p o in ts . T o g ive a m ean in g o f th is “ a d d itio n a t a d is tan ce ” re q u ire s th e in tro d u c tio n o f som e system o f te lep ara lle lism . S uch a system is p ro v id e d by th e te n so r P A. T h e d irec tio n s o f P A a t tw o d is ta n t p o in ts , can b e d esc rib ed as “ p a ra lle l ” , a n d th e a c tu a l so u rce in te g ra l (2 • 1), w ith M m = P A T/% can be reg a rd ed as o b ta in e d fro m th e fo u r in teg ra ls (3 1) by reso lv in g th e in teg ra l a t each p o in t in th e d ire c tio n o f P A.

T h e exp lic it d e te rm in a tio n o f P A as a so lu tio n o f th e e q u a tio n s (2 • 6) is clearly a m a tte r o f som e difficulty w h ich w ill in vo lve a k n o w led g e o f th e co m p o n en ts o f th e stress ten so r. A still fu r th e r d ifficulty is p re sen ted by th e so lu tio n o f th e e q u a tio n s (2*8). I t is o b v io u sly d esirab le to express th e flux in teg ra l, an d th e re fo re th e te n so rs F M„ a n d * F M" as ex p lic it fu n c tio n s , free fro m q u a d ra tu re s , o f th e p o te n tia ls , Ml/, a n d o f th e ir first deriva tives. In these c ircum stances th e d ivergence , (* F ^ )„ , w ill n o t invo lve an y te rm s o f th e fo rm

d g p c r

dx* ’ jflll *

Gauss's Theorem in General Relativity 359

(since *F MV is an tisy m m etric , a n d th e re fo re , in th e ex p ress io n fo r th e divergence, th e d u m m y suffix v w ill n o t ta k e th e va lue fx). B u t such term s do o ccur on th e le ft-h an d side o f e q u a tio n (2.8). H en ce it seem s im possib le to express th e flux in teg ra l in a fin ite fo rm .

B o th o f these difficulties c an be rem o v ed by ex p an d in g a ll th e g ra v ita ­tio n a l tensors in pow ers o f th e c o n s ta n t o f g ra v ita tio n , y, a n d by re ta in in g on ly th e lead ing te rm s in these expansions. T h is dev ice reveals a system o f te lepara lle lism w hich is intrinsic in th e geom etry o f th e g rav ita tio n a l field, an d it leads to a sim ple, exp lic it so lu tio n o f th e e q u a tio n s (2.8).

L et the lead ing te rm s in th e expansions o f an d G M„ be

g fJLV&./X.V I I ( y )an d

G m>, = + yR/ni/ H- O (y 2).

T hen K MV m u st be the E inste in ten so r fo r th e “ a-space ” w ith th e line- elem ent

ds2 — dx*

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360 G. Temple

and it follows from the field equation (2.7), by equating the terms inde­pendent of y, that

K — 0,where

K == a'4*' K M„.Hence

K = 0 and K M„ = 0. (3.2)

The equations (3.2) must be taken in the strict sense that vanishes everywhere and is entirely free from singularities. In these circumstances the a-space is necessarily Euclidean, i.e., the Riemann-Christoffel 4-index symbols are all zero. I t follows that, by a suitable transformation of coordinates, the coefficients aM„ can all be made constants, independent of the new coordinates. Henceforward it will be assumed that this transform ation has been performed, and tha t all tensors are referred to the new system of coordinates.

The new system of coordinates is uniquely defined apart from arbitrary linear transformations with constant coefficients, and it determines a definite system of teleparallelism in which two vectors a t distant points are parallel and e q u a li f they have the same set o f components. More­over this system of teleparallelism is clearly intrinsic to the Riemannian geometry o f the gravitational field. This property is all the more notable inasmuch as the absence o f any teleparallelism is the most characteristic feature of an abstract Riem ann space. The apparent contradiction arises from the fact that a Riem ann space which represents a gravitational field involves a param eter y t such that, as y -> 0, the Riemann space tends to the associated Euclidean a-space which determines the tele- parallelism.

Henceforward it will be convenient to m ake aM„ the fundamental tensor and to raise and lower suffixes with respect to a according to the formulae

a*MaA„ = 0 if |i ^ v, or 1 if [x = v,A / = a - A M<r, A m„ = v.avA s , etc.

Then, neglecting terms of order y2,

G / == y R / and G = y R / = yR, say,t This is true not only for continuous distributions o f matter but also for discrete

particles. In the latter case the field equations are KM„ = 0, except along the lines o f the particles. Along these lines the potentials become infinite and to deter­mine the nature o f the corresponding singularities in it is necessary to treat t e world lines as limiting cases o f world-tubes filled with a continuous distribution o matter. The specification o f the singularities will then involve the parameter Y-

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Gauss's Theorem in General Relativity 361

so th a t th e field e q u a tio n s (2 .7 ) b e co m e

H en ce , i fRam — = — TV\

px qa a n d * F M,/ -#» *E M,/ as T 0,

e q u a tio n s (2.8) a n d (2 .6 ) b e co m e

Q A R A" — i-Q ^R = — (3 .3 )a n d

TV* dQ ^/dx* = 0 , (3 .4 )

th e c o v a r ia n t d e riv a tiv es b e in g re p la c e d b y o rd in a ry d e riv a tiv es , a s a ll th e 3 -index sy m b o ls a re o f o rd e r y.

I t fo llow s f ro m (3 .3) th a t Q% th e le a d in g te rm in th e e x p an s io n o f P A, is sufficient to d e te rm in e *E M% th e le a d in g te rm in th e e x p an s io n o f * F Ml/. M o re o v e r f ro m (3 .4) Q A c a n b e ta k e n to b e a n y v e c to r w ith c o n s ta n t c o m p o n e n ts so th a t th e sy stem o f te le p a ra lle lism d e te rm in e d b y Q A is th e sam e system as th a t d e te rm in e d b y th e a -space .

4— T he M odified F orm of G a u ss’s T heorem

T o c o n s tru c t th e m o d ified fo rm o f G a u s s ’s T h e o re m it o n ly re m a in s to so lve e q u a tio n (3.3) fo r C o v a r ia n t d e riv a tiv es w ill n o t o c cu r inth e fo llo w in g an a ly sis so th a t th e sy m b o ls

(/)* » (/)»*> ( / > a n d ( / Vc a n b e u sed to d e n o te

d f/dx* , d2f l d x lx- d x v, a av ( / ) „ , a n d a Mtr (/)<„,.T h e exp lic it ex p an sio n fo r R A/X is

R am = i aP<7 { ( P a/O po- + ( P p<t) am ---- O a<t) pM ---- ( P pm) a<t} ,w hence

R am = i { ( P am) pp + ( P ) am — ( P ap) p* — ( ( V ) ap>,w here

P = P /.By su itab ly g ro u p in g th e te rm s in (4.1) i t is co m p ara tiv e ly sim ple to

express Q AR AM — ^ Q PR in th e re q u ire d fo rm as th e d ivergence o f an an tisy m m etric ten so r. L e t

A amp = ( P a* )' — ( P ap) m.

H ence R am = i { (A aw0 p — ( V ) a}a n d R — i{(A MMp)p - ( V ) m)

— i { (A mmp)p + (A ppa)M}, since A amp is an tisy m m etric in (x a n d p.

2 BVOL. c liv .— A.

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362 G. Temple

Hence _ , .R == ( A / %

and equation (3.3) becomes

— 2 (*E-% = Qx (A a )p — QA (Apmp)a — QM (A^Op = QA ( A r X - Q" + Q* (A /%

on changing the dummy suffixes and using the antisymmetry of A/*. It is now seen that this equation possesses an explicit solution in the form

*EM" = — i {QAA / “ — Q *A /A + QMA / A). (4.2)

The structure of the tensor *E#H' is seen more clearly if the coordinate system is chosen so that

Then

*EM*' = — i A / v while

QA = 0 if= 1 if X = k .

\ (ft/*)’' + 1 ( P R if t* K> and v ^

*E*K = 0, (all fx)

Gauss’s Theorem can now be written in its modified form as

. (4.3)

(4-4)

where is the fundamental antisymmetric tensor of the a-space.This equation (4.4) is equivalent to four equations of which the first,

given below, is obtained by writing

Q1 = 1 , Q2j Q3 = Q4 == o ,

viz.,

HU1* d(X 2, X3, X4) rp ,

0 ( t \ t 2, t 3) 12 9 (X 3,

0 ( T X,

X4,

T2,x2 + T i3T3)

0 (X 4, X1, X2) 0 ( t 1, T2. T3)

— i-p 4 ^ ( x 4,

1 a R

X2, X3) ) / - -----T2, T2) / V “

c/ t 1 c/ t2

= “ * f | {[(pi3)4 - (p,4)3] 0 (X 1, 0 (CO1,

X2)CO2)

+ [ ( P i4)2 •/ g 2 \4 1 a ( x 1, C P i ; J 0 ( c o \

X3)CO2)

+ K (V)3 - < W>)2] 3 ’ % d o t , (45>

where a is the determinant of the a ’s.

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Gauss's Theorem in General Relativity 363T h e th e o re m in th is fo rm is v a lid fo r a n y g ra v ita tio n a l field , th e flux

in te g ra l m ay b e ta k e n o v e r a n y c lo sed S 2 a n d th e so u rc e in te g ra l m a y b e ta k e n o v e r a n y o p e n S 3 b o u n d e d b y th e c h o se n S 2. T h e fo u r e q u a tio n s (4.5) w h ich c o n s ti tu te th e th e o re m fo rm a s ing le te n so r e q u a tio n in th e a -space . By a su ita b le ch o ice o f th e S 3 th e in te g ra l o f th e so u rc e in te g ra l c an b e m a d e to b e a n y ch o sen c o m p o n e n t o f th e s tre ss -te n so r in th e a - space . T h u s , i f S 3 lies co m p le te ly in th e 3-w ay, x 2 c o n s ta n t, th e th e o re m b ecom es

A co m p le te so lu tio n is g iven o f th e p ro b le m o f c o n s tru c tin g th e w id est ex ten sio n o f G a u ss ’s T h e o re m o n th e N e w to n ia n p o te n tia l w h ich is v a lid in G e n era l R e la tiv ity . T h e th e o re m is ex p ressed in th e fo rm — “ so u rce in teg ra l ” eq u a ls “ flux in te g ra l ” — w h ere th e flux in te g ra l invo lves th e p o ten tia ls , a n d th e ir f irs t d e riv a tiv es, a n d is ta k e n o v e r a c lo sed 2-w ay, S 2 ; a n d th e so u rce in te g ra l invo lves th e s tress te n so r , T M„, a n d is ta k e n over an y 3-w ay, S 3, b o u n d e d b y th e p re sc r ib e d S 2.

T h e g ra v ita tio n a l te n so rs a re ex p an d e d as p o w e r series in th e c o n s ta n t g rav ita tio n , y , a n d a tte n tio n is c o n c e n tra te d o n th e le ad in g te rm s. T h is device reveals a system o f te le p a ra lle lism w h ich is in tr in s ic in th e R ie - m a n n ia n geom etry o f th e g ra v ita tio n a l field a n d en ab les G a u ss ’s th eo rem to be c o n stru c te d in a n exp lic it fo rm .

+ [(IV ) 3 - | a/ — a t /w 1 d<x>2.

5 — Sum m ary

2 b 2

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