DOC.

17

PROBLEM

OF

GRAVITATION

199

basis of

Coulomb's

law.

Even

if

confidence in the

comprehensive significance

of Newton's

ac-

tion-at-a-distance law

had thus been

shaken,

there

were

still

no

direct

reasons

necessitating

an

extension of Newton's

theory. However,

such

a

direct

reason

does

[4]

exist

today

for those who subscribe

to

the

correctness

of the

theory

of

relativity.

For,

according

to

the

the

theory

of

relativity, no means

exist in

nature

that would allow

us

to

send out

signals

with

a

velocity

greater

than that of

light.

On the other

hand,

however,

it

is obvious that

if

Newton's law

were

strictly

valid,

we

would be able

to

use

gravitation

for

sending

out

instantaneous

signals

from

a

place A

to

a

distant

place

B;

for the

motion

of

a gravitating mass

at

A

would have

to

result in simultaneous

changes

of

the

gravitational

field

at

B-in

contradiction

to

the

theory

of

relativity.

But

the

theory

of

relativity

not

only compels

us

to

modify

Newton's

theory;

fortunately,

it also

greatly

limits the

possibilities

for such

a

modification. If this

were

not

the

case,

the endeavor

to

generalize

Newton's

theory

would be

a

hopeless

undertaking.

In order

to

see

this

clearly,

one

need

only imagine

oneself in the

following analogous

situation: Assume that

of

all

electromagnetic phenomena, only

those of electrostatics

are

known

experimentally.

But

one

knows that electrical effects

cannot

propagate

with

superluminal velocity.

Who would have been able

to

develop

Maxwell's

theory

of

electromagnetic processes

from these data? And

yet, our

knowledge

in the

area

of

gravitation corresponds exactly

to

the

case we

have

just

invented;

all

we

know is the interaction between

masses

at rest,

and

even

this

we

probably

know

only

to

a

first

degree

of

approximation.

The

theory

of

relativity

limits

the

confusing

manifold of

possible

generalizations through

the fact

that,

according

to

it,

the time

coordinate

occurs

in all

systems

of

equations

in the

same

manner, up

to

certain differences in

sign, as

the three

spatial

coordinates. This

great

formal

discovery

of

Minkowski's,

which

is

here

only

roughly

indicated,

has

proved

to

be

a

tool

of

the

greatest importance

in the search for

equations

that accord with the

theory

of

relativity.

[5]

§2.

Plausible

Physical Hypotheses Concerning

the Gravitational Field

In

the

following

we

shall

put

forward

a

few

general postulates

that

can

be

imposed

on a theory

of

gravitation,

but

not

all of which

must

be

imposed:

1.

Satisfaction of the

laws

of

conservation of

momentum

and

energy. [6]

2.

Equality

of

the inertial and the

gravitational

mass

of isolated

systems. [7]

3.

Validity

of the

theory

of

relativity (in

the

narrower

sense); i.e.,

the

equation

systems

are

covariant with

respect

to

linear

orthogonal

substitutions