152

DOC. 30 FOUNDATION OF GENERAL RELATIVITY

of

Z,

permanently

coincide.

We shall

show

that

for

a space-

time

measurement

in the

system

K' the

above definition

of

the

physical meaning

of

lengths

and

times

cannot be

main-

tained.

For

reasons

of

symmetry

it

is clear

that

a

circle

around the

origin

in the

X,

Y

plane

of K

may

at

the

same

time

be

regarded

as a

circle

in the

X',

Y'

plane

of

K'.

We

suppose

that the circumference and diameter

of

this

circle

have

been

measured with

a

unit

measure

infinitely

small

compared

with the

radius,

and

that

we

have the

quotient

of

the

two results.

If this

experiment

were

performed

with

a

measuring-rod

at rest

relatively

to

the

Galilean

system

K,

the

quotient

would be

r.

With

a

measuring-rod

at rest

relatively

to

K',

the

quotient

would be

greater

than

r.

This

is

readily

understood

if

we

envisage

the

whole

process

of

measuring

from

the

"stationary" system

K,

and take into consideration

that the

measuring-rod

applied

to

the

periphery

undergoes

a

Lorentzian

contraction,

while

the

one applied

along

the

radius

does not.

Hence Euclidean

geometry

does not

apply

to K'.

The notion

of

co-ordinates

defined

above,

which

pre-

supposes

the

validity

of

Euclidean

geometry,

therefore breaks

[10]

down

in relation

to

the

system

K'.

So,

too,

we are

unable

to

introduce

a

time

corresponding

to

physical

requirements

in

K',

indicated

by

clocks at rest

relatively

to

K'. To

convince ourselves

of

this

impossibility,

let

us

imagine

two

clocks of

identical constitution

placed,

one

at

the

origin

of

co-ordinates,

and the other

at

the circumference

of

the

circle,

and both

envisaged

from

the

"stationary" system

K.

By

a

familiar result

of

the

special

theory

of

relativity,

the

clock at

the

circumference-judged

from K-goes

more

slowly

than

the

other,

because the former is

in motion

and

the

latter

at rest. An observer at

the

common

origin

of

co-ordinates,

capable

of

observing

the

clock at

the

circum-

ference

by

means

of

light,

would

therefore

see

it

lagging

be-

hind the

clock beside him. As

he

will not

make

up

his mind

to

let the

velocity

of light

along

the

path

in

question

depend

explicitly on

the

time,

he

will

interpret

his observations

as

showing

that the

clock at

the

circumference

"really"

goes

more slowly

than the

clock

at the

origin.

So

he

will be

obliged

to

define

time in such

a

way

that the rate

of

a

clock

depends

upon

where the

clock

may

be.