306 THE

RELATIVITY

PRINCIPLE

According

to

§17,

equation

(30)

is also

applicable to

a

coordinate

system

in

which

a

homogeneous

gravitational

field is

acting.

In that

case we

have

to

put

$

=

?£,

where $

is the gravitational

potential,

so

that

we

obtain

a

=

r

1

(30a)

We

have

defined

two

kinds

of

times for

S.

Which

of the

two

definitions

do

we

have to

use

in the various cases?

Let

us assume

that

at

two

locations

of different gravitational potentials

(y£)

there exists

one

physical

system

each, and

we

want

to

compare

their

physical

quantities.

To

do

this,

the

most

natural

procedure

might

be

as

follows: First

we

take

our

measuring

tools

to

the

first

physical

system

and

carry

out

our

measurements there;

then

we

take

our

measuring

tools

to

the

second

system

to

carry

out

the

same

measurement

here.

If

the

two sets

of measurements

give

the

same

results,

we

shall denote

the

two

physical

systems

as

"equal."

The

measuring

tools include

a

clock with

which

we

measure

local

times

a.

From

this it follows that

to

define the

physical

quantities at

some

position

of the

gravitational

field, it is natural

to

use

the time

a.

However,

if

we

deal with

a

phenomenon

in

which

objects situated

at

posi-

tions with different

gravitational

potentials

must

be

considered simultan-

eously,

we

have to

use

the time

r

in those

terms

in which

time

occurs

explicitly

(i.e.,

not

only

in the definition

of

physical

quantities), because

otherwise the

simultaneity of the events

would not

be

expressed

by

the

equal-

ity

of the

time values

of

the

two events.

Since in the definition

of

the

time

r a

clock situated

in

an

arbitrarily

chosen position

is

used,

but

not

an

arbitrarily

chosen

instant,

when using

time

r

the

laws

of

nature

can

vary

with

position

but

not

with time.

§19. The

effect

of

the gravitational

field

on

clocks

If

a

clock

showing

local time is located in

a

point

P

of gravitational

potential $,

then,

according to (30a),

its

reading

will

be

(1

+ o\2)

times

greater

than the time

r,

i.e.,

it

runs (1

+

o/c2) times faster than

an