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DOC. 21 GENERAL RELATNITY

[p. 778] Plenary Session of November 4, 1915

Doc. 21

On the General Theory of Relativity

My efforts in recent years were directed toward basing a general theory of relativity, (1) also for nonuniform motion, upon the supposition of relativity. I believed indeed to
have found the only law of gravitation that complies with a reasonably formulated postulate of general relativity; and I tried to demonstrate the truth of precisely this solution in a paper1 that appeared last year in the Sitvmgsberichte.
Renewed criticism showed to me that this truth is absolutely impossible to show in the manner suggested. That this seemed to be the case was based upon a misjudgment. The postulate of relativity-as far as I demanded it there-is always satisfied if the Hamiltonian principle is chosen as a basis. But in reality, it provides no tool to establish the Hamiltonian function H of the gravitational field. Indeed, equation (77) I.e. which limits the choice of H says only that H has to be an invariant toward linear transformations, a demand that has nothing to do with the relativity of accelerations. Furthermore, the choice determined by equation (78) I.e. does not [3] determine equation (77) in any way.
For these reasons I lost trust in the field equations I had derived, and instead looked for a way to limit the possibilities in a natural manner. In this pursuit I arrived at the demand of general covariance, a demand from which I parted, though with a heavy heart, three years ago when I worked together with my friend GROSSMANN. As a matter of fact, we were then quite close to that solution of the problem, which will be given in the following.
Just as the special theory of relativity is based upon the postulate that all equations have to be covariant relative to linear orthogonal transformations, so the [p. 779) theory developed here rests upon the postulate of the covariance of all systems of equations relative to transformations with the substitution determinant 1.
Nobody who really grasped it can escape from its charm, because it signifies a real triumph of the general differential calculus as founded by GAUSS, RIEMANN, CHRISTOFFEL, RICCI, and LEVI-ClvITA.

1"Die formale Grundlage der Relativitatstheorie," Sitzungsberichte 41 (1914), pp. [2] 1066-1077. Equations of this paper are quoted in the following with the additional note
"I.e." in order to keep them distinct from those in the present paper.

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§1. Laws of Forming Covariants

I can be brief on the laws of forming covariants since I gave an elaborate description of the methods of absolute differential calculus in my paper of last year; thus we need only to investigate what will change in the theory of covariants if only substitutions of determinant 1 are permitted. The equation

d7' ,

=

a(x\ ... X14)d
- - - - - 7',

a(x1 .. . x4)x

which is valid for any substitutions, becomes, due to the premise in our theory, i.e.,

a(x'1 ... x'4) = 1

(1)

a(x1 .. ,x4)x

now

(2)
and the four-dimensional volume element dr is therefore an invariant. Since furthermore (equation (17) I.e.) Fgdr is an invariant toward arbitrary substitutions, it follows for the group that interests us now

J-g' = Fg.

(3)

The determinant of the 8µ.v is therefore an invariant. Because of the scalar character of Fg one can simplify the basic formulas of the formation of covariants, as
compared to those of general covariance; which in short means, the factors Fg and 1 / n
no longer occur in the basic formulas, and the distinction between tensors and Vtensors drops out. Specifically one gets:

J. The place of the tensors Giklm = Fg 8;1c1m and Giklm = _l_ Swm (as in (19) [p. 780)
Fi
and (21a) Le.) is now taken by the tensors

G;klm = G iklm = 8;1c1m

(4)

which are of a simpler structure. 2. The basic formulas (29) Le. and (30) Le. for the extension of tensors can,
under our premise, not be replaced by simpler ones, but the equations that define divergence (representing a combination of the equations (30) I.e. and (31) Le.) can be simplified. This can be written as

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DOC. 21 GENERAL RELATIVITY

But according to (24) I.e. and (24a) I.e.

E {STl E E .! .! =

g $QC ( agsa + ag,,a - agff ) =

g sa ag$0, = o(lgFi). (6)

,, s

2 as

ax,, axs axa 2

ax,,

ax,,

And this quantity has the characteristics of a vector, due to (3). Consequently, the last term on the right-hand side of (5) is itself a contravariant tensor of rank l. We are therefore entitled to replace (5) by the simple definition of divergence, viz.,

(Sa)

and we shall do so throughout. For example, the definition (37) I.e.

ct>=

_l_L
Fi ,.

~(HA")
ax,.

has to be replaced by the simpler definition

L ct> = aA,.'

(7)

,. ax,.

and equation (40) I.e. for the divergence of the contravariant six-vector by the simpler

(8)

In place of (41a) I.e. we have, due to our assumption,

(9)

[p. 781) A comparison with (41b) reveals that under our assumption the law of divergence is the same as that for the divergence of V-tensors in the general differential calculus. This remark applies to any divergence of tensors, as can be derived from (5) and (Sa). 3. Our limitation to transformations of determinant 1 brings the farthest-reaching simplification for those covariants which are formed only from the g,.v and their derivatives. It is shown in mathematics that these covariants can all be derived from the RIEMANN-CHRISTOFFEL tensor of rank four, which (in its covariant form) reads:

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It is in the nature of gravitation that we are most interested in tensors of rank two, which can be formed by inner multiplication of this tensor of rank four with the Kµ.v . Due to the symmetry properties of the RIEMANNIAN tensor, apparent from (10), viz.,

(ik, lm) = (lm, ik)

(ik, lm) = -(ki, Im),

(11)

this multiplication can be formed only in one way; whereby one obtains the tensor

L Gim = gld (ik, lm).

(12)

Id

It is more advantageous for our purposes to derive this tensor from a different fonn of (10) which CHRISTOFFEL has given,2 i.e.,

{m, bn) = ~ g''(ip, hn) = ~-~ • ~ [t}{P;}- {:}{P:}]- (13)

When this tensor is multiplied (inner multiplication) with the tensor
I: s~ = K1uicKa1
a
one obtains Gun, viz.,

(13) [p. 782]
{11

(13a) (4]

2A simple proof of the tensorial character of this expression can be found on page 1053 of my repeatedly quoted paper.

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DOC. 21 GENERAL RELATIVITY

.lL s~

a Xm

{ipm}{pl/.}.

(13b)

Under the constraint to transformations with determinants 1, not only (G;m) is a tensor, but (R;m) and (S;m) also have tensorial character. It follows indeed from
flil) the fact that Fg is a scalar, and because of (6), that i is a covariant four-

vector. (Sim), however, is, due to (29) I.e., nothing other than the extension of this four-vector, which means it is also a tensor. From the tensorial character of (Gim) and (S;m) follows the same for (R;m), from (13). The tensor (R;m) is of utmost importance for the theory of gravitation.

§2. Notes on the Differential Laws of "Material" Processes

J. The energy-momentum theorem for matter (including electromagnetic processes in a vacuum.
According to the general considerations of the previous paragraph, equation (42a) I.e. has to be replaced by

:E L ar: = .!. gTP. ag,,.v TV + K

(14)

V axy

2µ.-tY

axa T

at

where r; is an ordinary tensor, Ka an ordinary four-vector (not a V-tensor, V-vector, resp.). We have to attach a remark to this equation, because it is important for the following. The equations of conservation led me in the past to view the quantities

1.Lg TP. og,,.v 2 ,,. iha.
as the natural expressions of the components of the gravitational field, even though the formulas of the absolute differential calculus seem to suggest the CHRISTOFFEL

symbols {:") instead, as being the more natural quantities. The fonner view was a
fateful prejudice. The preference for the CHRISTOFFEL symbols justifies itself [p. 783) especially because of the symmetry in their covariant indices (here v and u) and,
furthermore, because they occur in the fundamentally important equations of the geodesic line (23b) I.e.; and these latter are-from a physical point of view-the equations of motion of a material point in a gravitational field. Equation (14) cannot

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serve as a counterargument because the first term on its right-hand side can be brought into the form

Therefore, from now on we shall call the quantities

1 l -E l r:v .fµ.v

=

-

=

u

g-[µv

.! Lg-( ogµa

=

-

+

a

a

2 a

oxv

OKva - dKµ.v) oxµ. axa

(15)

r; the components of the gravitational field. Kv vanishes when denotes the energy

tensor of all "material" processes, and the conservation theorem (14) takes the form

(14a)

We note that the equations of motion (23b) l.c. of a material point in a gravitational field take the form

r.: d2x,, = ~ dxµ. dxv.

ds 2

L µ.V ds ds
µ.v

(15) (2)

2. The considerations in paragraphs 10 and 11 of the quoted paper remain unchanged, except that the structures which were there called V-scalars and V-tensoIS are now ordiruuy scalars and tensors, respectively.

§3. The Field Equations of Gravitation

From what has been said, it seems appropriate to write the field equations of gravitation in the form

Rµ.v = -KTµ.v

(16)

since we already know that these equations are covariant under any transformation of a determinant equal to 1. Indeed, these equations satisfy all conditions we can demand. Written out in more detail, and according to (13a) and (15), they are

(16a)
We wish to show now that these field equations can be brought into the HAMILTO- [p. 784) NIAN form

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DOC. 21 GENERAL RELATIVITY

g'' B{f(t\- - K~ T,,)<»-)j, (17) u - E r¼3~« urafJ
where the gµ.v have to be varied while the Tµ.v are to be treated as constants. The reason is that (17) is equivalent to the equations

(18)

a µv
where U has to be thought of as a function of the gµ.v and the _ag (= g:v). On the xu

other hand,. a lengthy but uncomplicated calculation yields the relations

au r« rP = _ ~

agµv

L..., µp va
afJ

{19)

au = ra

agaµv

µ.v•

(19a)

These together with (18) provide the field equations (16a).

It can now also be easily shown that the principle of the conservation of energy

g:V and momentum is satisfied. Multiplying (18) by

with summation over the

indices µ and v, one obtains after customary rearrangement

L KL ~(g:v aUV) _ a9 = _ T,,.vg:v.

ag:: aµv axa

dXa

µv

According to (14), on the other hand, for the total energy tensor of matter one has

E E ar} = _.!_ ag,,.v r

i dXi

2 µ.v dxq µ.v·

From the last two equations follows

~ ~ (TA + tA) = 0

£..,a

"

}. Xi

(1

,

(20)

where

(20a)

[p. 785) denotes the "energy tensor" of the gravitational field which, by the way, has tensorial

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character only under linear transformations. After a simple rearrangement, one gets from (20a) and (19a)
(20b)

Finally, it is of interest to derive two scalar equations that result from the field equations. After multiplying (16a) by gµ.v with summation overµ and v, we get after simple rearranging

L r;. L "o2ga/J - g'"f~r~a + L _l_(gaJJ olgFg) = -K:E

(21)

ap OXaOXp uraJJ

ap axa

axp

a

On the other hand, multiplying (16a) by g vJ. and summing over v, we get

L aa (gv"r:v> - L gvP~,,.r~ = -x.r;,

av 'Xa

apv

or, also considering (20b),

ax ~
Lav;

_l_(gv' a

-rµa.v)

-

_2!c5µ'-.

~
µL.va;fJ

gJJ.vfa
µ/J

P?v. a

= - K(Tµ'.-

+

tµ>. .) •

-a1~ a a Taking (20) into account, and after simple rearranging, this yields

aX,,_ LafJ

a2gcxP 'Xa 'Xp

-

u~LraJJ g crrraafJrf.-la]

--

0.

(22)

However, we demand somewhat beyond that:

(22a)

whereupon (21) becomes

:E _l_(gcxP a1gFg) = -KE r;.

(21a)

afJ dxa

dXp

a

Equation (21a) shows the impossibility to choose the coordinate system such

that F8 equals 1, because the scalar of the energy tensor cannot be set to zero.

[5)

Equation (22a) is a relation of the 8,,_v alone; it would not be valid in a new

coordinate system which would result from the original one by a forbidden

transformation. The equation therefore shows how the coordinate system has to be

adapted to the manifold.

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DOC. 21 GENERAL RELATIVITY

[p. 786)

§4. Some Remarks on the Physical Qualities of the Theory

A first approximation of the equations (22a) is

:E "iflgoJJ = o.
op OXaOXp

This does not yet fix the coordinate system, because this would require 4 equations. We are therefore entitled to put for a first approximation arbitrarily

{31

L oga/1 = o.

(22)

p OXp

For further simplification we want to introduce the imaginary time as a fourth variable. The field equations (16a) then take, as a first approximation, the form

(16b)

from which one sees immediately that it contains NEwroN's law as an approximation.. That the new theory complies with the relativity of motion follows from the fact that among the permissible transformations are those that correspond to a rotation of [6) the new relative to the old system (with arbitrarily variable angular velocity), and also those where the origin of the new system perfonns an arbitrarily prescribed motion relative to that of the old one.
Indeed, the substitutions
x ' = x cos -r + y sin -r
y ' = - x sin -r + y cos -r
z' = z t' = t
and
X1 = X - T1
y' =y - Tz
z' = Z - -r3
t' = t,
where -r and -r1, -r2, -r3 respectively are arbitrary functions oft and substitutions with the determinant 1.

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Additional notes by translator

{1} The number (13) has already been assigned to an equation above. {2} The number (15) has already been assigned to an equation above. {3} The number (22) has already been assigned to an equation above.

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DOC. 22 ADDENDUM TO DOC. 21

[p. 799]

Doc. 22 On the General Theory of Relativity (Addendum)
by A. Einstein

In a recent investigation1 I have shown how RIEMANN'S theory of covariants in multidimensional manifolds can be utilized as a basis for a theory of the gravitational field. I now want to show here that an even more concise and logical structure of the theory can be achieved by introducing an admittedly bold additional hypothesis on the structure of matter.
The hypothesis whose justification we want to consider relates to the following
r; topic. The energy tensor of "matter" has a scalar ET/, whose vanishing for µ. the electromagnetic field is well known. In contrast, it seems to differ from zero for matter proper. Because, if we consider the most simple special case, that of an "incoherent" continuous fluid (with pres.5ure neglected), then we are used to writing

and we have

Lr: = L8µ.vTµ.v = Poh•

µ,

µ.v

The scalar of the energy tensor does not vanish in this approach.

One now has to remember that by our knowledge "matter" is not to be perceived

as something primitively given or physically plain. There even are those, and not just

a few, who hope to reduce matter to purely electrodynamic processes, which of

course would have to be done in a theory more completed than MA.XwELL's

electrodynamics. Now let us just assume that in such completed electrodynamics the

scalar of the energy tensor also would vanish! Would the result, shown above, prove

that matter cannot be constructed in this theory? I think I can answer this question

[p. 800] in the negative, because it might very well be that in "matter," to which the previous

E expres&on relates, gravitational fields do foan an important constituent In that case, r:

E can appear positive for the entire structure while in reality only (T: + t:) is

Er: positive and

vanishes everywhere. In the following we assume the

L conditions r: = 0 really to be generally true.

1These Sitzungsberichte, p. 778.

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Whoever does not categorically reject the possibility that gravitational fields could constitute an essential part of matter will find powerful support for this conception in the following.2

Derivation of the Field Equations

Our hypothesis allows us to take the last step that the idea of general relativity may consider as desirable. It allows us, namely, also to write the field equations of gravitation in a general covariant fonn. I have shown in the previous paper (equation (13)) that
(13)

is a covariant tensor. And we had

~ l R., . -L I

ax,

+

L
pl

lfPu)flpml

(13a)

(13b)

This tensor G1m is the only tensor available for the establishment of generally covariant equations of gravitation.
We have won generally covariant field equations if we agree that the field equations of gravitation should be

Gµ.v = -KTµ.v'

{16b)

These, together with the generally covariant laws, provided for by the absolute differential calculus, express the causal nexus for "material" processes in nature; and they express it in a form that emphasizes the fact that any special choice of coordinate system-which logically has nothing to do with nature's law anyway-is [p. 801] not used in the formulation of these laws.

21n writing this paper I was not yet aware that the hypothesis }: T: = 0 is, in
principle, admissible.

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DOC. 22 ADDENDUM TO DOC. 21

Based upon this system one can-by retroactive choice of coordinates-return to those laws which I established in my recent paper, and without any actual change in these laws, because it is clear that we can introduce a new coordinate system such that relative to it

Fg = 1
holds everywhere. S im then vanishes and one returns to the system of field equations

Rµ.v = - KTµv

(16)

of the recent paper. The formulas of absolute differential calculus degenerate exactly

in the manner shown in said paper. And our choice of coordinates still allows only

transformations of determinant 1.

The only difference in content between the field equations derived from general

Fi covariance and those of the recent paper is that the value of

could not be

prescribed in the latter. This value was rather determined by the equation

{l)

L .1._[gaJJ a1gFg) = - KE r;.

(21a)

aJJ axa

axp

q

This equation shows that here Fg can only be constant if the scalar of the energy
tensor vanishes.
Under our present derivation Fg = 1due to our arbitrary choice of coordinates.
The vanishing of the scalar of the energy tensor of "matter" follows now from our field equations instead of from equation (2 la). The generally covariant field equations (16b), which form our starting point, do not lead to a contradiction only when the hypothesis, which we explained in the introduction, applies. Then, however, we are also entitled to add to our previous field equation the limiting condition:

Fg = 1.

(21b)

Additional note by translator {l ) The " axa" inside of the parentheses has been corrected to " axp."

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