422
DOC.
43 COSMOLOGICAL CONSIDERATIONS
limit
at
infinity,
the
mean
density
p
must decrease toward
zero more rapidly
than
1/r2 as
the distance
r
from
the
centre
increases.*
In this
sense,
therefore,
the
universe
according
to Newton
is
finite,
although
it
may
possess
an
infinitely
great
total
mass.
From this
it
follows in
the
first
place
that the radiation
emitted
by
the
heavenly
bodies
will,
in
part,
leave
the
Newtonian
system
of the
universe, passing radially
outwards,
to become ineffective and lost
in
the infinite.
May
not
entire
heavenly
bodies
fare
likewise?
It
is
hardly
possible
to
give
a
negative
answer
to
this
question.
For it
follows
from
the
assumption
of
a
finite limit for
Q
at
spatial infinity
that
a
heavenly body
with
finite
kinetic
energy
is able
to
reach
spatial infinity
by overcoming
the Newtonian
forces of
attraction.
By
statistical mechanics this
case
must
occur
from
time to
time,
as
long
as
the total
energy
of the stellar
system-transferred
to
one
single
star-is
great enough
to
send
that star
on
its
journey
to
infinity,
whence
it
never can
return.
We
might try
to avoid
this
peculiar
difficulty
by
assuming
a
very
high
value for
the
limiting potential
at
infinity.
That
would be
a possible way,
if the
value of
the
gravitational
potential
were
not itself
necessarily
conditioned
by
the
heavenly
bodies.
The truth
is
that
we are compelled
to
regard
the
occurrence
of
any
great
differences
of
potential
of
the
gravitational field
as
contradicting
the
facts.
These
differences
must
really
be of
so
low
an
order of
magnitude
that the stellar
velocities
generated by
them
do
not
exceed
the
velocities
actually
observed.
If
we
apply
Boltzmann's law
of
distribution
for
gas
molecules to
the
stars, by
comparing
the stellar
system
with
a
gas
in thermal
equilibrium,
we
find
that the Newtonian
stellar
system
cannot
exist
at all.
For there is
a
finite ratio
of densities
corresponding
to
the
finite difference of
potential
between the
centre
and
spatial infinity.
A
vanishing
of
the
density
at
infinity
thus
implies
a
vanishing
of
the
density
at
the
centre.
*
p
is the
mean density
of
matter, calculated for
a region
which is
large as
compared
with the distance between
neighbouring
fixed
stars,
but small in
comparison
with
the
dimensions
of
the
whole stellar
system.