156

DOC. 30 FOUNDATION OF GENERAL RELATIVITY

-1

0

0

0

0

-1

0 0

0

0 -1

0

. . .

(4)

0 0

0

+1

We

shall find

hereafter that the

choice of such

co-ordinates

is,

in

general,

not

possible

for

a

finite

region.

From the considerations

of

§

2

and

§

3

it

follows

that

the

quantities

gro

are

to

be

regarded

from

the

physical

stand-

point

as

the

quantities

which

describe

the

gravitational

field

in relation

to

the

chosen

system of

reference. For,

if

we now assume

the

special

theory

of

relativity

to

apply

to

a

certain four-dimensional

region

with the

co-ordinates

properly

chosen,

then the

gor

have the

values

given

in

(4).

A free

material

point

then

moves, relatively

to

this

system,

with

uniform

motion in

a

straight

line.

Then if

we

introduce

new

space-time

co-ordinates

x1, x2,

x3, x4,

by

means

of

any

substi-

tution

we

choose,

the

gor

in

this

new

system

will

no

longer

be

constants,

but functions

of

space

and time. At the

same

time

the

motion

of the

free

material

point

will

present

itself

in the

new

co-ordinates

as a

curvilinear non-uniform

motion,

and

the

law of

this motion

will be

independent

of

the

nature

of

the

moving particle.

We

shall

therefore

interpret

this

motion

as a

motion under the

influence

of

a

gravitational

field.

We thus

find

the

occurrence

of

a

gravitational

field

connected

with

a

space-time

variability

of

the

go

.

So,

too,

in the

general

case,

when

we are no longer

able

by a

suitable

choice

of

co-ordinates to

apply

the

special

theory

of

relativity

to

a

finite

region,

we

shall

hold

fast

to

the

view

that

the

gor

describe the

gravitational

field.

Thus,

according

to

the

general

theory

of

relativity,

gravi-

tation

occupies an exceptional position

with

regard

to

other

forces, particularly

the

electromagnetic forces,

since

the

ten

functions

representing

the

gravitational field

at the

same

time

define

the

metrical

properties

of

the

space

measured.

B. Mathematical Aids

to the

Formulation

of

Generally

Covariant

Equations

Having

seen

in

the

foregoing

that the

general

postulate

of relativity

leads to

the

requirement

that the

equations

of