DOC.

30 FOUNDATION OF GENERAL RELATIVITY 151

recognize

that

the

path of

a

ray

of light

with

respect

to K'

must in

general

be curvilinear,

if with

respect

to K

light

is

propagated

in

a

straight

line with

a

definite

constant

velocity.

§

3.

The

Space-Time Continuum. Requirement

of

General

Co-Variance for the

Equations Expressing General

Laws

of

Nature

In

classical

mechanics,

as

well

as

in

the

special

theory

of

relativity,

the

co-ordinates

of

space

and time have

a

direct

physical meaning.

To

say

that

a

point-event

has

the

X1

co-

ordinate

x1

means

that the

projection

of

the

point-event

on

the

axis

of Xl,

determined

by rigid

rods

and

in accordance with the.

rules of

Euclidean

geometry,

is

obtained

by

measuring off

a

given

rod

(the

unit

of length)

x1

times

from

the

origin

of

co-

ordinates

along

the

axis of

X1.

To

say

that

a

point-event

has

the

X4

co-ordinate

x4

=

t, means

that

a

standard

clock,

made to

measure

time in

a

definite

unit

period,

and which is

stationary

relatively

to

the

system

of co-ordinates and

practic-

ally

coincident

in

space

with the

point-event,*

will

have

measured

off

x4

=

t

periods

at

the

occurrence

of

the

event.

This

view

of

space

and

time has

always

been

in the

minds

of

physicists,

even

if,

as a

rule, they

have been

unconscious

of it.

This

is

clear

from

the

part

which these

concepts

play

in

physical

measurements;

it

must also have

underlain the

reader's

reflexions

on

the

preceding paragraph

(§

2)

for

him

to

connect

any

meaning

with what he there

read.

But

we

shall

now

show

that

we

must

put

it

aside

and

replace

it

by

a

more

general

view,

in

order to be able to

carry through

the

postulate

of

general

relativity, if

the

special

theory

of

relativity

applies

to

the

special case

of

the

absence

of

a

gravi-

tational

field.

In

a space

which

is free of

gravitational

fields

we

introduce

a

Galilean

system

of

reference

K

(x,

y, z, t),

and also

a

system

of

co-ordinates

K'

(x',

y',

z',

t')

in uniform rotation

relatively

to

K. Let

the

origins

of

both

systems,

as

well

as

their

axes

*

We

assume

the

possibility of verifying "simultaneity"

for

events im-

mediately proximate

in

space,

or-to

speak more

precisely-for

immediate

proximity or

coincidence in

space-time,

without

giving a

definition

of

this

fundamental

concept.