154

DOC. 30 FOUNDATION OF GENERAL RELATIVITY

there

is

a

corresponding

system

of

values of

the

variables

x1

...

x4.

To two

coincident

point-events

there

corre-

sponds one

system

of

values of

the

variables

x1

...

x4,

i.e.

coincidence is

characterized

by

the

identity of

the co-ordinates.

If,

in

place

of

the

variables

x1

...

x4,

we

introduce

functions

of them,

x'1,

x'2, x'3,

x'4,

as a new

system

of

co-ordinates,

so

that the

systems

of

values

are

made to

correspond

to

one

another without

ambiguity,

the

equality

of all

four

co-ordin-

ates in the

new

system

will also

serve as an

expression

for

the

space-time

coincidence of

the

two

point-events.

As all

our physical experience

can

be

ultimately

reduced to such

coincidences,

there

is

no

immediate

reason

for

preferring

certain

systems

of

co-ordinates to

others,

that

is

to

say,

we

arrive

at

the

requirement

of

general

co-variance.

§

4.

The

Relation of

the

Four

Co-ordinates

to

Measure-

ment in

Space

and Time

It

is

not

my purpose

in this

discussion to

represent

the

general theory

of

relativity

as a

system

that

is

as simple

and

logical as possible,

and

with the minimum number

of axioms;

but

my

main

object

is to

develop

this

theory

in such

a way

that the reader

will feel

that the

path

we

have

entered

upon

is

psychologically

the natural

one,

and that

the

underlying

assumptions

will

seem

to have

the

highest

possible degree

of

security.

With this

aim in

view

let it

now

be

granted

that:-

For

infinitely

small four-dimensional

regions

the

theory

of

relativity

in the restricted

sense

is

appropriate,

if

the

co-

ordinates

are

suitably

chosen.

For this

purpose

we

must choose

the

acceleration

of

the

infinitely

small ("local")

system

of

co-ordinates

so

that

no

gravitational

field

occurs;

this

is

possible

for

an

infinitely

small

region.

Let

X1,

X2,

X3,

be

the

co-ordinates of

space,

and

X4

the

appertaining

co-ordinate

of

time

measured

in the

appropriate

unit.* If

a

rigid

rod is

imagined

to be

given

as

the unit

measure,

the

co-ordinates,

with

a

given

orientation

of

the

system of co-ordinates,

have

a

direct

physical meaning

*

The unit

of

time

is

to be chosen

so

that the

velocity

of light

in

vacuo as

measured in

the

"local"

system

of

co-ordinates is to

be

equal

to

unity.