DOC. 30 FOUNDATION

OF GENERAL RELATIVITY

153

We therefore

reach this

result:-In the

general theory

of

relativity,

space

and time cannot be defined

in

such

a

way

that

differences of

the

spatial

co-ordinates

can

be

directly

measured

by

the

unit

measuring-rod,

or

differences

in the

time

co-ordinate

by

a

standard

clock.

The method

hitherto

employed

for

laying

co-ordinates

into the

space-time

continuum

in

a

definite

manner

thus

breaks

down,

and

there

seems

to be

no

other

way

which

would allow

us

to

adapt systems

of

co-ordinates

to

the four-dimensional

universe

so

that

we

might expect

from

their

application

a

particularly simple

formulation

of

the

laws

of

nature.

So

there

is

nothing

for

it but

to

regard

all

imaginable systems

of co-ordinates,

on

principle,

as equally

suitable

for

the

description

of nature.

This

comes

to

requiring

that:-

The

general

laws

of

nature

are

to

be expressed

by

equations

which hold

good

for

all

systems

of co-ordinates,

that

is,

are

co-variant with

respect

to

any

substitutions

whatever

(generally

co-variant).

It

is clear

that

a

physical

theory

which

satisfies

this

postulate

will also be suitable

for the

general postulate

of

relativity.

For

the

sum

of

all

substitutions in

any

case

in-

cludes

those which

correspond

to all

relative motions

of

three-

dimensional

systems

of co-ordinates.

That

this

requirement

of

general

co-variance,

which

takes

away

from

space

and

time the last remnant

of

physical objectivity,

is

a

natural

one,

will be

seen

from

the

following

reflexion. All

our

space-time

verifications

invariably

amount to

a

determination

of

space-time

coincidences.

If,

for

example,

events consisted

[11]

merely

in the motion

of

material

points,

then

ultimately

nothing

would be observable

but

the

meetings

of two

or more

of

these

points.

Moreover,

the results

of

our

measurings

are

nothing

but

verifications of such

meetings

of the material

points

of

our

measuring

instruments with other material

points,

coincidences

between the hands

of

a

clock

and

points

on

the

clock

dial,

and

observed

point-events happening

at

the

same

place

at the

same

time.

The introduction

of

a

system

of

reference

serves

no

other

purpose

than

to facilitate the

description of

the

totality

of such

coincidences.

We

allot to

the universe

four

space-time

vari-

ables

x1, x2, x3,

x4

in such

a

way

that

for

every

point-event