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
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