Annals of Mathematics

A Generalization of the Relativistic Theory of Gravitation, II Author(s): A. Einstein and E. G. Straus Source: Annals of Mathematics, Second Series, Vol. 47, No. 4 (Oct., 1946), pp. 731-741 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969231 . Accessed: 24/05/2013 08:55 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.

http://www.jstor.org

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

ANNALS OF MATHEMATICS

Vol. 47, No. 4, October,1946

A GENERALIZATIONOF THE RELATIVISTIC THEORY OF GRAVITATION,II By A.

AND E. G. STRAUS

EINSTEIN

24,1946) (ReceivedJanuary In a previouspaper (Ann. of Math., Vol. 46, No. 4) one of us developed a generallyrelativistictheory,whichis characterizedas follows: of the four coordinates(xi, (1) Group of real transformations ,X4) (2) As only dependent variable to which everythingis reduced we have the tensorgik, whichis taken thereto be complexand of Hermitiansymmetry. W. Pauli noted,that the theorydeveloped on this basis is such that the limitationto the case of the Hermitiantensoris not needed forthe formalism. (3) It was added in proofthat it seemsnaturalto assumethat the fieldsatisfy the equations (1)

ri = 2(rFa

-

rai) = 0.

It was asserted but not proven,that there exist identitieswhich allow us to overdetermination. adjoin these equations withoutintroducingan impermissible This assertionwas, however,based on an error. The introductionof equation derivationofthe fieldequations fromthe originalone and (1) impliesa different a (slight)deviationofthe latterfromthe fieldequations ofthe firstpaper. The mathematicalformalismof the theoryis preservedhere except for an of tensor densities. alterationrelative to the rules for absolute differentiation Otherwiseknowledgeof that formalismis assumed here. parallel translationof the fundamental ?1. The dependence of the infinitesimal of densities. tensor. Absolute differentiation The connectionbetweenthe gik and the rPkis characteristicforthe theory. It is givenby the equation: (2)

(gik;a

-

=)9ika

gskrta

9i8 gaPk

=

0.

ofthe r fromthe g has the followingproperty: This determination If to the tensorgik correspondsthe translationrFk accordingto (2), then to the tensorPik = gki correspondsflk = rk. PROOF: If one formsthe leftside of (2) forthe Pih and the I** one gets gik.a

-

gk

P,8

-is

tak;

and exchange ifwe introduceherethe g and r, accordingto the above definition, the last two terms we get gki,a Nin -

F

gisi ka fat

~k r8 rai -9ks

This expressionvanishesaccordingto our assumption,since it becomesthe left side of equation (2) if wteinterchangethe freeindicesi and k. 731

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

732

A. EINSTEIN AND E. G. STRAUS

REMARK: The propertyjust establishedhas nothingto do withthe assumption that gik and FLk are Hermitianwith respectto the indices i and k. It is as possibleand as natural to considerthese quantitiesto be real but not symmetric;the numberof independentcomponentsof g and r is then the same as in the case of Hermitiansymmetry. One thus obtains a theorywhich differs fromthe previouslydevelopedone by the signsof certaintermsonly.

of tensor densities Absolute differentiation If we multiplytheleft-handside of (2) by 2g"iwe get (see

.1,(Vg),a -

(2.1)

2I(rI' +

(2.1) by V/-g we get the vectordensity multiplying

(V g),a-

rFe);

+ r:a). ,V-g(rFs. \/sa8 (as

2

,a

(-\/-9

-loc.cit) the vector

This we defineas the absolute derivative(A/-g),a of the scalor density-V/Z. we definethe absolute derivativeof everyscalor densityp Correspondingly P;a -P

(3)

a -

p

(raS +

ra)-

for all tensordensities follow in a well From this the rules of differentiation knownmanner,e.g. ik ik skc ia + gia kaic gik (3.1) giiklr'2 +~ Fsk =s (ra. + r8a). _

It can be easily shownthat the equations goi k ;

=

0; g-i

; 1 =

0; a+;

=

0

are equivalent here too. fortensordensitiescorWhen (2) is satisfied,then the rule of differentiation respondsto the one definedpreviously. For a contravariantvectordensityWPwe get a +

+;a=

2r(Fa

rF-sa

+

r:a)

and forthe divergence (3.2)

2a ;a =

a

1a a +

rarX

also !a

(3.3)

= 2a a

21 ra.

-

Here we see how natural it is to specialize the fieldby equation (1). For on the right-handsides of (3.2) and (3.3) each termhas tensorcharacter,but accordingto (1) therewillbe onlyone term. There are otherformalreasonsforpostulatingequations (1) whichwe should mentionhere. Like in the theoryof symmetricalgik the once contractedcurvaturetensorplays an importantpart. The curvaturetensor Riim

F rIm

-

F1Fm

-

rm,

+ rFmrF

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

OF GRAVITATION

THEORY

RELATIVISTIC

733

has a contractionwithrespectto the indicesi and k whichvanishesidentically in the originaltheory of gravitation. Here we get a rat-

D Ra1m =

_a

am,

whichin generaldoes not vanish even if (2) is satisfied. Namely, if we transformthe right-handside usingthe equation followingfrom(2.1)

(raI+

(2.2)

rla),m -

(ram + rma),I

0

we get RaIm

(rF, m -

=

rm,1)

This will not vanish in general,but, itwill vanish whenthe fieldsatisfiesequation (1). If we contractRkIm accordingto the indicesi and m,we getthetensor Rkl

=

Rkla-rla

-

rb a b

r-

p a

+

r a rpb

This tensoris, in general,not Hermitian,i.e., it is not transformedinto itself ifwe replacethe r by the I and interchangethe indicesk and 1. (In the following we shall use the terminologyHermitianin this sense.) For the anti-Hermitian part we get: 2RkZ =

+ 2rkPF4;

raPk

+

-ra.d

considering(2.2) thisbecomes k14

=

-

(rk;

I+

rL;k),

hence the anti-Hermitianpart of Rkz vanisheswhen (1) and (2) are satisfied. It would be easy to give furtherargumentsto show that equation (1) is suitable for the space structureused. However, the above should suffice. It is now ourtask to findcompatiblefieldequations (on the basis ofa variational principle)so that equations (1) and (2) are part of the fieldequations. First we want to make anotherformalremark,which servesto preparethe derivationofthe fieldequations. If in (3.1) we contractto form0tt'+;aand 9+ ;, thenby subtractionwe get (3.4)

2

a-

g;a)

,_

-ia-

ra

part of gAi. Hence, if (2) where gia is the symmetric,gi the anti-symmetric is satisfiedwe have identically (3.5)

0

(-ra).i

a scalar identityas a resultof (2). The equations (1) satisfy,therefore, equation (3.4) we see that equations (1) and (2) imply (3.6)

=via

0.

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

From

734

A. EINSTEIN

AND E. G. STRAUS

Field equations

?2. Hamiltona. We now choose the Hamiltonian gikPk + Pik

?bigiV a.

giri

is the Hermitianizedcurvaturetensor

Pik=

-

rzka

+

2(Ptak

-

rak,i)

rib

Pbk +

rk

Pab.

The variationis performed accordingto the variablesg ri,Pk, 21, bi whichplay the role of independentfieldvariables,wherethe lattertwo (purelyimaginary) quantities play the role of Lagrange multipliers. (Neither (1) nor (2) are assumed satisfieda priori.) The variationaccordingto the WI and bi yieldsthe equations (4)

ri

(5)

= 0

gvia = 0.

For the variation accordingto the r we use the method which has been establishedby Palatini forthe case of symmetricg and r. It is easy to verify that 3Pik =

(sr

.b) ;a

-

;k -

I, (ar)

I(6Fa)

;i

consideringthis the variationof the &~-integralaccordingto vanish at the boundariesof integration). 0 = -g+-;a +

(6)

+

1gi8= 2 1g

L

ak

r a

_Y1O -

+2215k

(for sr which

+ 2g+A;8aa

2?+;;8sa

k -_ I

r

r

i21&.

The secondlineof (6) vanishesbecause of (4). If we contract(6) firstaccording to k and a, thenaccordingto i and a we getthe two equations (6.1)

{g+-;. ++ 2g+;8 + 4d Si

Ig+-;s-ma

=

0 0.

Adding these two equations we get

+

(6.2)

+

=

0..

Equation (3.4) which was based on the definitionof absolute differentiation yieldsconsidering(4) and (5) (3.7)

-

0.

i

Hence g+-;. and g'+-;8vanishand therefore(6.1) impliesthat W2vanishes. Equation (6) reduces thereforeto (6.3)1

9+-;a

=

0.

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

RELATIVISTIC

THEORY

OF GRAVITATION

735

Equation (5) is impliedby equations (4) and (6.3) accordingto (3.4). The variationofthe &S-integral accordingto the gik yields (7)

Pik -

(bik

-

bk,i) =

0

or separatingaccordingto symmetry Pik = 0

(7-1)

(7.2)

Pik V

2(bik -

=

bki)

0

or, aftereliminationof the auxiliaryvariables b (7.3)

Pik,1 + Pk1,i + Pli,k = 0. V

V

V

Compilingthe resultsofthe variation,we get the fieldequations (whichdeviate slightlyfrom(15b) of the firstpaper) (8.1)

9+-;a = 0

(8.2)

ri = 0

(8.3)

Pik =

(8.4)

Pie, + Pkl,i + V

0

V

Pli,k = 0 V

The derivationof these equations froma variationalprinciple(with real ) guaranteestheir compatibilitysufficiently. If we comparethe systemof equations with that of the previouspaper, we realizethat equation (8.2) is introducedat the cost of weakeningthe equations whichare derived fromthe curvature. Ofthe equations (8.4) only three are independent,while in the originalformulationof the theoryit corresponded to six equations; in addition the order of differentiation of the last equation has been raised by one. The introductionof the last termin the Hamiltonian, whichcaused this raise of the degree of differentiation, is necessaryin order that (8.1) will hold, whichis obviouslythe only reasonable determinationof the r fromthe g. Consideringequations (8) the question arises, whether(8.3) and (8.4) could not be replacedby the strongerequation. (9)

Pik = 0.

The question of the justificationof such an equation caused us considerabletrouble. This equationwouldobviouslybe justifiediftheequations(9), (8.1) and (8.2) wouldsatisfy 3 independentadditionalidentities. The assumptionof the existenceofsuch identitiesis weak fieldssuch additionalidentitiesdo indeed strengthened by thefactthatforinfinitely exist. Namely,if we put (neglectingthe special characterof time) gik = 5ik + -Yik

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

736

A. EINSEEIN

AND E. G. STRAUS

and neglectthe square of y as comparedto 1, then we may replace equations (8.1), (8.2) and (9) by the linearizedones

-rak=o?

ysk ,a -ra k.-

r6.,. -

rag)

=

-

rxki

-k

0 =

O.

From the firstequation we solve forr rid =

?

i(-Ytb

+

7iaik

7Yak,i)

thesecondequationthengives (MM5

0

=

)Yia.a V

and the third,consideringthis (GiL- )

-

+

7ki.aG

+

Yia.ala

Yakc.ai

-

=

'Yaa.ik

0-

The latterantisymmetrized can be replacedconsideringG, = 0 by (Uik

V

=)

'Yik.aa V

=

We now have the identity UikkV

-

Gik

-

0.

weak fields,therewould correspond If to this identityof the equations forthe infinitely an identityof the rigorousequations,then the introductionof the strongerequation (9) wouldbe justified. A complicatedsystematicinvestigationhas shownthat no such rigorous identityexists. One may ask, ifnot despitethe absence oftheseidentities,theintroductionofequation (9) may be considered. This, too, has to be answeredin the negative on the basis of a considerationwhichis applicable also in othercases. Let us assumethatwe have a systemofequationsG = 0 forwhichthereexistsa rigorous identity,whichis linear and homogeneousin the equations. Writtensymbolically L(G) _ 0 whereL is an operatorwhichis linear and homogeneousin the G. Now L and G can be developedaccordingto the powersof the fieldquantitiesand theirderivatives. (Lo + LI + -..)(G,

+ G2+

= ?

-)

wherebytheidentitydividesaccordingto powersofthe fieldquantities. The firsttwo are 0

Lo(G1) Lo(G2) + Ll(G)

--0.

We nowassumethat we have a parametersolutionforG ofthe fieldquantitiesg 0

G(eor+e2g2+ or (G, +

G2 +

*-(.Eg,

+

E2g2+

)

0

or

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

RELATIVISTIC

THEORY

This shall be identically satisfied in e. GI(egi) =

OF GRAVITATION

737

This yields the firsttwo equations.

0 or GI(gi) = 0 (linear in g)

GI(e2g2) + G2(egl) = 0 or GI(g2) + G2(gl) = 0. _0 we get, applying Lo to the second

We now apply our identity. Since we had Lo(G) equation Lo(G2(gi)) = 0.

(a)

This is an equation of the second degree in g. Since we saw above that (b)

Lo(G2) + L1(G1)

0

we know that this quadratic equation is a result of the linear equations GI(gl) = 0. If a linear identity exists for the first approximation to which there corresponds no rigorous identity, (as in the case of our equations) then we derive equations (a) as before but since the identity (b) will not hold in general this equation is no longer a result of the linearized field equations G, (g1) = 0. They are thereforeadditional equations for the first approximation. In the case of the field equations under our consideration they are so constructed that each of their terms is a product of a symmetric by an antisymmetric (or derivatives of these quantities). 9 (Yik) If we interpret the symmetric 'Yik as an expression of the gravitation field and the 'Yil V

as an expression of the electromagnetic field, then for the firstapproximation of the field we get a dependence of the electric of the gravitation field which cannot be brought in of equations accord with our physical knowledge, therefore the considered strengthemnng (8) is out of the question.

The linearized equations whichaccordingto (8) hold foran antisymmetric (electromagnetic)field are 7ik,k = V ('Yikl + V

ykli + V

0

7li~k) ,,8 = V

0

If, in the second equation, the expressioninsidethe parentheseswould itself vanish,then we would have Maxwell's equations foremptyspace, whose solutions thereforesatisfyour equations. The latter seem to be too weak. This, however,is not a (justified)objectionto the theorysince we do not know to which solutionsof the linearizedequations there correspondrigorous solutions whichare regularin the entirespace. It is clear fromthe startthat in a consistent fieldtheorywhichclaims to be complete (in contraste.g. to the pure theoryofgravitation)onlythosesolutionsare to be consideredwhichare regular intheentirespace. Whethersuch (non-trivial)solutionsexistis as yetunknown. ?3. Conditionsforthe gik whichfollowfromequat on (2) We now wish to investigatewhat conditionsthe gikhave to satisfyin order that equations (2) determinethe r uniquelyand withoutsingularities. At each point we can transIn the followingwe write:gik = Sik; ik = ak .

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

738

formthe coordinatesso that Sik= (2) becomes: (2.3)

AND E. G. STRAUS

A. EINSTEIN

ask

sia +

ai8rask +

+

sArka

Sibik

(i not an indexof summation). Equatioir =

SiIak

(i, k not indices of summation).

gik,a

If we let i = a = k we get (2.4) Hence we musthave

giii.

2siri-

si $ 0 or, in otherwords,at everypoint we have forthe determinantI sik (10)

(10.1)

0.

|sih I|

This resultis importantsinceit impliesthe existenceof a "light cone" whose signatureis the same everywhere. The division of the line elements into spacelikeand timelikeelementsis therebysecured. If the signatureis now chosen as is customaryin the theoryof relativity, we can specialize the coordinatesfurtherso that /

Sik =

nikS

> O

r

Ii ==

and

ik i = 1, 2, 3\ bik

for i = 4

so that at our point all We can also performa (local) Lorentz transformation aik

except

a12

=

and a34

-a2

=

-a43 vanish.

We write

a12

ale12

=

; an

=

a2634

.

we shall use capital Roman indicesfor1, 2, and Greek indices In the following, 4. for3, Considerfirsttheequations (2.5)

al(eSK

rIA

+

eIS rI K)

+

+

S(PA

FAK)

=

IKA

Whereall indicesare 1, 2 and not all A, I, K are equal. We thenget six equations in six unknowns(ignoringF+A whichwe got from(2.2)) with the determninant

(AI , K)=

r= |

r2

ri2

r22

ri

(1, 1, 2)

s

0

0

s

a,

0

(1, 2, 1)

0

s

0

s

0

- a1

a,

0

0

8

8

(1, 2, 2) - a,

r,2 r2l

(2,1,1)

s

s

0

0

-a,

a,

(2, 1, 2)

a,

0

s

0

s

0

(2, 2, 1)

0

-a,

s

0

0

s

=

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

-4S2(S2

+

al)2.

RELATIVISTIC

THEORY

739

OF GRAVITATION

In an analogous mannerthe determinantof the equations (2.6)

a2(e~KF6, +

e.,

1r) +

s(71,

r

+

rP?)

77.a

a

=

is

+

4S2(_2

a2)2.

Hence: (11)

(S2

+

a S)(-S2

+

2) iZ 0

or 9

(11.1)

0 .

=giAk

We now know that the gik exist (a fact whichwe had tacitlyassumed before). We have in fact: (12)

9g

=

2

+

a2 (sarK

+aieix); er)+

ga g'

=

(-8?7,,

+ a2e

,).

Then, if in the threeequations: +

g9krFl

g8Irks + +

g8Prlk

=

gik.

gkra =

gkli

g1Prks

9gi.k

gt8'rlk

=

we multiplythe firstby gmglak the second by -gkagml and the thirdby glmgka and add, we get: (2.7)

(

ml

=g

+

r

gkaggjm

_

+

ml9akgik,1

g9m9kagSik

_

9imka gkzt-

Let us firstconsiderthe case s=

o;l = L;k

=

K;i = L;m = M;a = A

using (12) the left-sideof (2.7) becomes (2.8)

(,

+a2)2

[(8aAXOML

+

a2eA

eML)711f

+

ala2(SAK eML +

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

5MLeAK)esIreP

740

A. EINSTEIN

AND E. G. STRAUS

The determinantis rl

r (i,

A,

M)=

(3,1,

1)

r,2 0

s2

r2l

r22

aja2

aja2

0

a2

0

0

-aia2

0

0

aia2

-aja2

0

0

aja2

0

-aja2

-aja2

0

0

0

-al

2

0

-a,

(3, 2, 1) 0

-ai 2

82

0

(3, 2, 2) a2

0

0

82

-aja2

- aa2

0

(4, 1, 2) ala2

0

0

-aja2

0

-s_'2

(4, 2, 1) aja2

0

0

-aja2

0

a

aja2

ala2

0

-a2

0

0

(4, 2, 2) 0 8s

a2)16

0

r12

s2 2

(4,1,1)

(2s)8

rl1

r22

0

(3, 1, 2)

(82 +

r12

2

al2

ai

al a2

S

a, a2

-aja2

a2a2

-aa2

-s

al a2

-aja2

al

2

2

-

0

0 82

2

-aa2

(2s)8

a2

2

al

+

(s2

a2)16

82

(2s)8 (2+ a)12 [(s2 -a2)2

+ 4a2 aI2J

which gives us the condition (13)

(s2 -

a2)2

+

4a2a2

5

0

or, in other words, we cannot at the same time have

a,

=

and

s8

a2

0-

=

We see immediatelythat the equations for F'K and r' have the determinants (2s)8 (s2

+

a2)4(-s2

)2

+

a2)8 [(R2 -al)2

22

+

4ala22

and thereforeyield no new inequalities. In an analogous mannerthe equations of rs, have the determinant (2s)8 2

2

i-)12 [(8

+

2)2

a)2

22a2

+ 4al222

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions

OF GRAVITATION

THEORY

RELATIVISTIC

741

whichgives us the condition (14)

(s2

+

a2)2

i 0 + 4a42a2

or in otherwordswe cannot at the same time have a2

and

hjis

=

a,

=

0.

The equationsforrFKand TL respectivelyhave thedeterminants: (S2

+

(2s)822 2)4(82+

222

+

a)8 [(2

2 + 4a44a]2

which again yields no new condition. If we introducethe covariant expressions(scalar densities) 1I =

ISik I

' slips

eij

i2

=

we

13

=

I aik I.

akkn al

Then we can sum up the conditions(10), (11), (13) and (14) as follows: conditionsfor the existenceof a unique nonThe necessaryand sufficient singularsolutionof equations (2) are (A) (B) (C)

g= UIl

h1 s 0 I1 + I2 + I3 I2) 2 +

IS

#0

0

O

(A) and (B) implyin the physicallymeaningfulcase the inequalities I gik I < 0

and

I Sik I < 0

where the latter guarantees the existence of a non-degenerate"light cone" in every point. Equation (C) states that in no point can the two equations II = I2 and I3 = 0 be satisfiedsimultaneously. In orderthat this be excluded fieldbe restricted it is sufficientthat e.g. everywherein space the antisymmetric by the inequality

I III >

I1 2

( i i stands here forthe absolute values). THE INSTITUTE

FOR ADVANCED STUDY

This content downloaded from 129.110.33.9 on Fri, 24 May 2013 08:55:16 AM All use subject to JSTOR Terms and Conditions