96 DOC.

3

STATICS OF GRAVITATIONAL FIELD

equivalent

to

a

system

at rest

in which there

is

a

mass-free static

gravitational

field3

of

a specific

kind. The

spatial

measurement

of K is carried

out

by

means

of

measuring

rods

that-when

compared

with

one

another

in

a

state

of

rest at

the

same

location in K-possess the

same

length;

the laws of

geometry

shall hold for

lengths

so

measured,

hence also for relations between the coordinates

x,

y,

z

and other

lengths.

This

stipulation

is

not

permitted

as a

matter

of

course;

in

fact, it

contains

physical assumptions

that

might possibly

prove to

be

wrong;

for

example,

they

most

probably

do

not

hold in

a

uniformly rotating system

in

which, owing

to

the Lorentz

contraction,

the ratio of the circumference

to

the diameter would have

to

be different

[5]

from

n

if

our

definition

were

applied.

The

measuring

rod

as

well

as

the coordinate

axes are

to

be conceived

as rigid

bodies. This is

permitted despite

the fact

that,

according

to

the

theory

of

relativity,

the

rigid body

cannot

possess

real existence. For

[6] one can imagine

the

rigid measuring body being replaced by

a

great

number of

nonrigid

bodies

arranged

in

a row

in such

a

manner

that

they

do

not exert any

pressure

on

each other in that each is

supported separately.

We

imagine

that the time

t

in

system

K is

measured

by

clocks

so

constituted and

so

rigidly arrayed

at

the

spatial points

of the

system

K that the time

span-measured

by

them-that

is needed

for

a

light ray

to

get

from

a

point A

to

a point

B

of the

system

K does

not

depend on

the

time

of

emission of the

light ray

at A.

Furthermore,

it will

turn out

that

simultaneity can

be

defined in

a

consistent

manner by postulating that,

with

respect

[7]

to

the

setting

of the

clocks,

all

light rays

that

pass

a

point A

of K have the

same

propagation velocity

at

A independently

of

their direction.

We

now

imagine

that the reference

system

K

(x, y, z, t)

is observed from

a

nonaccelerated

reference

system (of constant

gravitation potential)

2(£,

rj,

(, t). We

postulate

that the x-axis coincides

permanently

with the

£-axis

and that the

y-axis

is

permanently parallel

to the

n-axis,

while the z-axis is

permanently parallel

to

the

C-axis.

This

stipulation

is

possible

on

the

assumption

that the

state

of acceleration

has

no

influence

on

the

shape

of K with

respect

to

E.

We take this

physical

assumption

as our

basis. It follows from this that for

arbitrary

T

we

must

have

n

=

y,

(1)

C

=

z,

so

that it

only

remains for

us

to

find out the relation that obtains between

E

and

T

on

the

one

hand,

and

between

x

and t

on

the other. Let the

two

reference

systems

coincide

at

time

T

=

0;

then the

substitution

equations

that

we are

seeking

must be

of the form

3One has

to

imagine

that the

masses

that

generate

this field

are

situated

at

infinity.