DOC.

16

FOUNDATIONS OF GRAVITATION

195

(1a)

i

(/V)

=

0,

where

dr

denotes the three-dimensional volume element.

In the

general theory,

the

following equations correspond

to

equations (1):

(2)

(0

=

1,2,3,4)

/XVT w"vV

Here

$ov

=

fg-V

g7/X6/XV

9

where

g

is the determinant

\gik\,

and

Yut

is the subdeterminant

adjoint

to gut

divided

by

this

determinant; 0uv is

the

symmetrical

second-rank contravariant tensor

that characterizes the behavior of

energy

in the domain of

phenomena

under

consideration. The

quantities

Sov have the

same

physical meaning

here

as

the

quantity

Tov

in the

original theory

of

relativity;

the

stress-energy components

of

the

gravitational

field

are

not

contained in them.

The

right-hand

side of

equations

(2)

vanishes if the

quantities guv

are

constant,

i.e.,

if

no

gravitational

field is

present.

In that

case,

equation

(2)

reduces

to

equation

(1)

and

can

therefore be

brought

into the

form

(1a);

in

other words:

the material

process

satisfies the conservation laws all

by

itself.

If, on

the

contrary,

the

guv

are

variable, i.e.,

if

a gravitational

field

is

present,

then the

right-hand

side

of

equations

(2)

expresses

the

energetic

influence of

the

gravitational

field

on

the material

process.

It

is clear that

no

conservation laws

can

be deduced from

equation

(2)

in that

case,

because the

stress-energy

components

of the material

process

cannot

satisfy any

conservation laws

all

by

themselves,

without the

components

of

the

gravitational

field.

The method sketched

up

to

this

point

shows how the

equation systems

of

physics

can

be obtained when the influence of

a given gravitational

field

on

the

processes

is

taken into

account.

But this does

not

solve

the

main

problem

of

the

theory

of

gravitation,

since the latter consists in

determining

the

quantities

gik

when

the field–

generating

material

processes (including

the electrical

ones)

are

to

be considered

as

given.

In other

words,

the

generalization

of

Poisson's

equation

(3)

A(p

=

4irkp

is

sought.

On

the

one

hand,

the

proportionality

of

energy

and inertial

mass

that is obtained

from the

ordinary theory

of

relativity,

and,

on

the

other

hand,

the

empirical

proportionality

of inertial and

gravitational

mass

lead

necessarily to

the view that the

same quantities

that determine the

energetic

behavior of

a

system

must

also determine

the

gravitational

effects of the

system.

From this

we

conclude that

tensor guv must

appear

in

equations

of

gravitation

we are

seeking,

in lieu of

the

density

p

of

equation

[8]