DOC. 43 COSMOLOGICAL CONSIDERATIONS
425
for
the
components
of
momentum,
and
for
the
energy
(in
the
static
case)
m/B.
From the
expressions
for
the
momentum,
it
follows
that
m-A/B
plays
the
part of
the rest
mass.
As
m
is
a
constant
peculiar
to the
point
of
mass,
independently
of
its
position,
this
expression,
if
we
retain
the condition
/g
-
=
1
at
spatial infinity,
can
vanish
only
when
A
diminishes
to
zero,
while B increases
to
infinity.
It
seems,
therefore,
that such
a degeneration
of
the
co-efficients
guv
is
required by
the
postu-
late
of
relativity
of
all
inertia. This
requirement
implies
that
the
potential energy
m/B
becomes
infinitely great
at
infinity.
Thus
a
point
of
mass can never
leave
the
system;
and
a more
detailed
investigation
shows
that
the
same
thing
applies
to
light-rays.
A
system
of
the universe with such
behaviour
of
the
gravitational potentials
at
infinity
would
not
therefore
run
the
risk
of
wasting away
which
was
mooted
just
now
in connexion with the Newtonian
theory.
I wish to
point
out
that the
simplifying assumptions
as
to
the
gravitational
potentials
on
which this
reasoning
is
based,
have been introduced
merely
for the sake
of
lucidity.
It
is
possible
to
find
general
formulations for the behaviour
of
the
guv
at
infinity
which
express
the essentials
of
the
question
without further restrictive
assumptions.
At this
stage,
with the kind assistance
of
the
mathe-
matician
J.
Grommer,
I
investigated centrally symmetrical,
static
gravitational fields, degenerating
at
infinity
in
the
way
mentioned. The
gravitational potentials
guv
were applied,
and
from them the
energy-tensor
Tuv
of
matter
was
calculated
on
the basis
of
the
field
equations
of
gravitation.
But here
it
proved
that for the
system
of
the
fixed
stars
no
boundary
con-
ditions
of
the kind
can come
into
question
at
all,
as
was
also
rightly
emphasized
by
the astronomer
de
Sitter
recently.
For the contravariant
energy-tensor Tuv
of
ponderable
matter
is
given by
[7]
[8]
rrwr
_
dXfi
dXy
where
p
is
the
density
of
matter in natural
measure.
With