DOC. 43 COSMOLOGICAL

CONSIDERATIONS

421

Doc.

43

Cosmological

Considerations

in the General

Theory

of

Relativity

This translation

by

W.

Perrett and G. B.

Jeffery

is

reprinted

from H. A. Lorentz

et

al.,

The

Principle

of

Relativity

(Dover, 1952), pp.

175-188.

[1]

IT

is well

known

that

Poisson's

equation

V2£

=

4ttKp

.

. .

.

(1)

in combination with the

equations

of

motion

of

a

material

point

is

not

as yet

a

perfect

substitute

for Newton's

theory

of action at

a

distance. There

is

still to

be

taken into account

the condition

that at

spatial infinity

the

potential

Q

tends

toward

a

fixed

limiting

value.

There

is

an analogous

state

of

things

in the

theory

of

gravitation

in

general

relativity.

Here,

too, we

must

supplement

the

differential

equations

by

limiting

conditions

at

spatial infinity,

if

we really

have to

regard

the universe

as

being

of

infinite

spatial

extent.

[2]

In

my

treatment of

the

planetary problem

I

chose

these

limiting

conditions

in the form

of

the

following assumption:

it

is

possible

to

select

a

system

of reference

so

that at

spatial

infinity

all

the

gravitational potentials

guv

become

constant.

[3]

But

it

is

by

no

means

evident

a

priori

that

we may lay

down

the

same

limiting

conditions

when

we

wish to take

larger

portions

of

the

physical

universe into consideration. In the

following pages

the

reflexions will be

given

which,

up

to the

present,

I

have made

on

this

fundamentally

important

question.

§

1.

The

Newtonian

Theory

It

is well

known that Newton's

limiting

condition

of

the

constant limit for

Q

at

spatial infinity

leads to

the

view

that

the

density

of

matter

becomes

zero

at

infinity.

For

we

imagine

that there

may

be

a place

in universal

space

round

about which the

gravitational

field of matter, viewed

on a

large scale,

possesses

spherical symmetry.

It then

follows

from

Poisson's

equation

that,

in order that

Q

may

tend to

a

177