DOC.

23

PROPAGATION

OF

LIGHT

385

is

nothing

that

compels us

to

assume

that

the

clocks

U,

which

are

situated in

different

gravitational

potentials,

must

be

conceived

as

going

at

the

same

rate.

On the

contrary,

we

must

surely

define

the time in

K

in such

a manner

that the number of

wave

crests

and

troughs

between

S2

and

S1

be

independent

of the absolute

value

of the

time,

because

the

process

under consideration

is stationary

by

its nature.

If

we

did not

satisfy

this

condition,

we

would

arrive

at

a

definition of

time

upon

whose

application

time would

enter

explicitly

the

laws

of

nature,

which would

surely

be unnatural and

inexpedient.

Thus,

the

clocks in

S1

and

S2

do not

both

give

the "time"

correctly.

If

we measure

the

time at

S1

with

the

clock

U,

then

we

must

measure

the time

at

S2

with

a

clock

that

runs

1

+

9/c2 times slower than the clock U when

compared

with

the latter

at

one

and

the

same

location.

Because,

when

measured

with such

a clock,

the

frequency

of the

light ray

considered

above

is,

at its emission at

S2,

L

1

+

I

and

is

thus, according

to

(2a),

equal

to

the

frequency

v1

of the

same

ray

of

light on

its

arrival

at

S1.

From

this follows

a

consequence

of fundamental

significance

for this

theory. Namely,

if the

velocity

of

light

is

measured

at

different

places

in

the

accelerated, gravitation-free

system

K'

by means

of

identically

constituted

clocks

U,

the

values

obtained

are

the

same

everywhere.

According

to

our

basic

assumption,

the

same

holds also

for

K.

But,

according

to what has

just

been

said, we

must

use

clocks

of unlike constitution

to

measure

time

at

points

of different

gravitational potential.

To

measure

time

at

a

point

whose

gravitational

potential is

$

relative to

the coordinate

origin,

we

must

employ

a

clock

which,

when

moved to

the coordinate

origin, runs

(1 + O/c2)

times slower

than the

clock with which

time

is

measured

at

the coordinate

origin.

If

c0

denotes the

velocity

of

light

at

the

coordinate

origin,

then the

velocity

of

light c

at

a point

with

a

gravitational

potential

$

will

be

given by

the relation

[10]

(3)

c

=

c1[1

+

c2].

The

principle

of

the

constancy

of

the

velocity

of

light

does

not

hold

in this

theory

in

the

formulation

in which it

is

normally

used

as

the

basis

of the

ordinary theory

of

relativity.

§

4. Bending

of

Light Rays

in the Gravitational Field

From the

proposition just proved,

that the

velocity

of

light

in

the

gravitational

field

is

a

function of

place, one can easily

deduce,

via

Huygens'

principle,

that

light rays

propagated

across a gravitational

field must

undergo

deflection.

For let

c

be

a

plane