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
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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
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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
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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.
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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.
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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
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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
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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 .
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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
=
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-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 +
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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
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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
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