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).
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