DOC.

4

THEORY OF STATIC GRAVITATIONAL

FIELD

107

Doc.

4

On the

Theory

of the Static Gravitational

Field

and

Note

Added

in

Proof

by

A.

Einstein

[Annalen

der

Physik

38

(1912):

443-458]

In

a

recently published paper I

derived the

equations

of motion

of

a

material

point

[1]

moving

in such

a

field from

a

hypothesis

that I called

the

equivalence principle.

In

what follows

I

will

present an

exact

derivation

of

the kind of influence that the static

gravitational

field has

on

electromagnetic

and thermal

processes according

to

the

equivalence principle.

The first

of

these

two

questions

I

have

already

treated earlier

to

a

first

degree

of

approximation.

At

the end

I

will derive the differential

equation

[2]

for the static

gravitational

field itself.

§1.

Derivation

of the

Electromagnetic

Equations

Taking

into Consideration

the

(Static)

Gravitational Field

The

path

we are

taking

here is

exactly

the

same as

the

one

that

yielded

the

equations

of motion

of

the material

point

in

the earlier

paper.

Namely,

we

seek the

electromag-

netic

equations

that hold relative

to

a

uniformly

accelerated

system

(in

Born's

sense)

[4]

K(x, y,

z,

t)

and

assume,

in accordance with the

equivalence principle,

that these

equations

also hold in the static

gravitational

field. In order to find the

equations

that

hold with

respect

to

K, we

start

out

from the familiar

equations

that hold with

respect

to

an

unaccelerated

system

E(£, n,

C,

T).

If

we

choose the time unit in the latter

such that the

velocity

of

light

becomes

1,

then these

equations

have for the

vacuum

the familiar form

[3]

(1)

rot'

0

=

div'

=

-rot'

@

[5]

p'

=

div'

e

The

symbols

for the

scalars, vectors,

and

operators appearing

in these

equations

are