DOC. 30 FOUNDATION OF GENERAL RELATIVITY

155

in the

sense

of

the

special

theory

of

relativity. By

the

special theory

of

relativity

the

expression

ds2

=

- dX21 - dX22 -

dX23

+

dX24

. .

(1)

then has

a

value which

is

independent

of

the orientation

of

the

local

system

of

co-ordinates,

and

is

ascertainable

by

measurements

of

space

and time.

The

magnitude

of

the

linear element

pertaining

to

points

of

the

four-dimensional

continuum

in infinite

proximity,

we

call ds.

If the

ds

belong-

ing

to

the

element

dX1

...

dX4

is

positive,

we

follow

Minkowski in

calling

it

time-like;

if

it is

negative,

we

call

it

space-like.

To

the

"linear

element" in

question,

or

to the two

infin-

itely proximate point-events,

there will also

correspond

definite differentials

dx1

...

dx4

of

the

four-dimensional

co-ordinates

of

any

chosen

system

of reference.

If this

system, as

well

as

the

"local"

system,

is

given

for

the

region

under

consideration,

the

dXv will allow

themselves

to be

represented

here

by

definite

linear

homogeneous expressions

of

the dXo:-

dXv

=

XavadXa

....

(2)

Inserting

these

expressions

in

(1),

we

obtain

ds2

=

2g

ardxadxr,.

.

.

.

(3)

where the

gar

will be

functions

of

the

xo.

These

can no

longer

be

dependent

on

the orientation and

the

state of

motion

of

the

"local"

system

of co-ordinates,

for

ds2

is

a

quantity

ascertainable

by

rod-clock

measurement

of

point-

events

infinitely proximate

in

space-time,

and defined inde-

pendently

of

any particular

choice

of

co-ordinates.

The

gor

are

to

be chosen

here

so

that

gor

=

gra;

the summation

is

to

extend

over

all values of

o

and

t, so

that the

sum

consists

of

4

x

4 terms,

of which twelve

are

equal

in

pairs.

The

case

of

the

ordinary

theory

of

relativity

arises out of

the

case

here

considered,

if

it

is

possible,

by

reason

of

the

particular

relations

of

the

gor

in

a

finite

region,

to

choose

the

system

of reference

in the finite

region

in such

a

way

that

the

gor assume

the

constant values