DOC.

43

COSMOLOGICAL CONSIDERATIONS 427

In the

first

place

those

boundary

conditions

pre-suppose

a

definite choice

of

the

system

of

reference,

which

is

contrary

to

the

spirit of

the

relativity

principle. Secondly,

if

we

adopt

this

view,

we

fail to

comply

with the

requirement

of

the

relativity

of

inertia. For the inertia

of

a

material

point

of

mass m

(in

natural

measure)

depends

upon

the

guv;

but

these

differ but little

from

their

postulated values,

as

given

above,

for

spatial infinity.

Thus inertia

would indeed be

influenced,

but

would not be conditioned

by

matter

(present

in finite

space).

If

only

one single

point of mass

were

present,

according

to

this

view,

it would

possess

inertia,

and in fact

an

inertia almost

as

great

as

when

it

is

surrounded

by

the

other

masses

of

the actual

universe.

Finally,

those statistical

objections

must be

raised

against

this

view

which

were

mentioned in

respect

of

the Newtonian

theory.

From

what has

now

been said

it

will be

seen

that I have

not succeeded

in

formulating boundary

conditions

for

spatial

infinity.

Nevertheless,

there

is

still

a

possible

way

out,

without

resigning

as suggested

under

(b).

For

if

it

were

possible

to

regard

the universe

as a

continuum which

is

finite

(closed)

with

respect

to

its

spatial

dimensions,

we

should

have

no

need

at all

of any

such

boundary

conditions.

We

shall

proceed

to show

that both the

general

postulate

of

relativity

and the fact

of

the

small

stellar

velocities

are com-

patible

with

the

hypothesis

of

a

spatially

finite universe;

though certainly,

in

order

to

carry

through

this

idea, we

need

a generalizing

modification of

the

field equations of gravitation.

§

3.

The

Spatially

Finite Universe with

a

Uniform

Distribution of Matter

According

to

the

general theory

of

relativity

the metrical

character

(curvature)

of

the four-dimensional

space-time con-

tinuum

is defined at

every point

by

the matter at

that

point

and the

state of

that matter.

Therefore,

on

account

of the

lack of

uniformity

in

the distribution

of

matter,

the metrical

structure

of

this

continuum must

necessarily

be

extremely

complicated.

But

if

we are

concerned with the

structure

only

on a large

scale,

we may

represent

matter to ourselves

as

being uniformly

distributed

over

enormous spaces, so

that

its

density of

distribution

is

a

variable

function

which

varies