196

DOC.

16

FOUNDATIONS OF GRAVITATION

(3).

We

are

therefore

looking

for

equations

that

express

the

equality

of

two tensors,

one

of which is the

given

tensor 3uv,

while the other

comes

from the fundamental

tensor

guv

through

differential

operations.

[9]

It has

now

turned

out

that the conservation laws

of

momentum and

energy

make

possible

the derivation of these

equations.

It has

already

been

emphasized

above that

the material

process

alone

cannot

satisfy

the conservation

laws;

but

we

must

demand

that the conservation laws be

satisfied for the material

process

and the

gravitational

field

together. According

to

the

arguments

presented above,

this

means

that there

must

exist four

equations

of the form

(4)

£

JL

($av

+ tov)

=

0. (0

=

1,2,3,4)

v

°XV

Here the tov characterize the

stress-energy

components

of the

gravitational

field in

a manner

analogous

to

the

way

in which the

quantities

$ov

characterize

those of the

material

process.

In

particular,

the

quantities

Sov

and

tov

must have the

same

invariant-theoretical character. It turned out

to

be

possible

to

show

by means

of

a

general

argument

that the

equations

that

completely

determine the

gravitational

field

[10] cannot

be covariant with

respect

to

arbitrary

substitutions. This fundamental

discovery

is

especially noteworthy

because all other

physical

equations,

such

as, e.g., equations

(2), possess general

covariance. In accordance with this

general result,

the

postulated

equations

(4) are

also covariant

only

with

respect

to

linear

substitutions,

but

are

not

[11]

so

with

respect

to

arbitrary

substitutions.

Hence,

we

will have

to

demand covariance

only

with

respect

to

linear transformations from the

gravitation equations

that

we

are

seeking.

It has turned

out that

one

is led

to

completely

determined

equations

if

one

adds

to

these considerations the demand that when

these

equations

are

applied

to

the

relevant

special

case

and

an approximate

solution is

sought, they

must

yield

Poisson's

[12] equation (3).

Using

the

way

indicated,

one

obtains the

following equations:

(5)

(6)

(V

9

7a

ß

ff

Oft Qßp ^

~

X

v

-f-

tn

v)

;

(O,

v =

1,2,3,4

Here

v=;

(.-fe),

k

is

a

universal constant that

corresponds

to the

gravitational constants;

6ov

is

1

or

0,

depending on

whether

o

and

v are

different

or equal.

One

can see

from the

system

of

equations (5),

which

corresponds

to

equation (3),

that

along

with the

stress-energy components

3ov

of the material

process,

those

of

the

gravitational

field

(namely, tov) appear

as an

equivalent field-inducing cause,

a

circumstance

that

obviously

must be

demanded;

for the

gravitational

effect

of

a

system

may

not

depend on

the

physical

nature

of

the

system's field-producing energy.