428

DOC. 43 COSMOLOGICAL

CONSIDERATIONS

extremely

slowly.

Thus

our

procedure

will

somewhat

re-

semble

that

of

the

geodesists

who, by means

of

an

ellipsoid,

approximate

to

the

shape

of

the earth's

surface,

which

on a

small

scale is

extremely

complicated.

The

most

important

fact that

we

draw

from

experience

as

to

the distribution

of matter is that

the

relative

velocities

of

the

stars

are very

small

as

compared

with the

velocity

of

light.

So

I think that

for

the

present

we

may

base

our

reasoning upon

the

following

approximative assumption.

There

is

a

system

of

reference

relatively

to

which matter

may

be looked

upon

as

being

permanently

at rest.

With

respect

to

this

system,

therefore,

the contravariant

energy-

tensor

Tuv

of

matter

is, by

reason

of

(5),

of

the

simple

form

[11]

0 0 0 0

0 0 0 0

0 0 0 0

0

0 0

p

.

(6)

The

scalar

p

of the

(mean) density

of

distribution

may

be

a

priori

a

function

of

the

space

co-ordinates.

But

if

we

assume

the universe

to be

spatially finite,

we

are

prompted

to

the

hypothesis

that

p

is

to

be

independent

of

locality.

On

this

hypothesis

we

base

the

following

considerations.

As

concerns

the

gravitational field,

it

follows

from the

equation

of

motion

of

the material

point

,

t

o

d/Xß

d?

*

{aß-

"'-ff

TT

~

0/%

that

a

material

point

in

a

static

gravitational

field

can

remain

at

rest

only

when

g44

is

independent of

locality.

Since,

further,

we

presuppose

independence of

the time co-ordinate x4

for

all

magnitudes,

we

may

demand for the

required

solution

that, for all

xv,

g44 =

1

. . . .

(7)

Further,

as

always

with static

problems,

we

shall

have

to

set

g14 = g24

=

g34

=

0

• • •

(8)

It

remains

now

to

determine those

components

of

the

gravitational potential

which

define the

purely

spatial-geo-

metrical relations

of

our

continuum

(g11,g12,

•••

g33).

From