DOC. 10 RESEARCH

NOTES

247

servation for such

a

choice

of

gravitation tensor.

The left-hand side

is

proportional to

the

gravitational

force

density acting

on

the

source masses.

The

right-hand

side

can

be

expressed

as

the

divergence

of

a

second-rank tensor,

the

stress-energy tensor

of the

gravitational

field.

The role of identities such

as [eq.

145]

in

the formation of

field

equations

is

stressed

in Ein-

stein and Grossmann

1913 (Doc.

13), part

1,

§5.

[103]Einstein

confirms the

compatibility

of

two

conditions that

must apply to pressureless

dust: the

continuity equation

and

the

law

of conservation of

energy momentum.

The

continu-

ity

equation

for dust

with

rest density

p0

in

the weak

field

case

is

[eq.

147]

or

its

equivalent

[eq. 148]

(with

vi

=

dxi/dx

and

x

the

proper

time).

The

continuity equation

is

rewritten

as

[eq. 149].

The

vanishing

of the

first

line

is

equivalent to

the

energy-momentum

conservation

law in

the weak

field

case,

Tik, k

=

0,

with

Tik = p0vivk

the

stress-energy tensor.

The

two

conditions

are

compatible

if

Dvi/Dx

=

0.

Presumably

these conditions also

test

the coordi-

nate

condition

yklgiK l

=

0,

for this condition

together

with the

field

equation

ykigim,

kl = KTim

gives

the

energy-momentum

conservation

law

yklTik, l

=

0.

[p.

39]

dxK dxK

0

dl

dx

[eq.

151]

für

gleiche

i &

K.

I

/

d8iK

1

dg

kk

\

V

dxK

2

dxi

j

=

0

[eq. 152]

A^.

=

°im

[106]

XSkk

=

U

Gravitationsgleichungen

A^n-^)=7'n

A

gn

=

Tn

4

=

T

14

[eq.

153]

2^U

=

Xtkk

[eq.

154]

[104]

g~

= 0

[eq. 150]

[105]