DOC.
43 COSMOLOGICAL CONSIDERATIONS
423
It
seems
hardly
possible
to
surmount
these
difficulties
on
the
basis of the Newtonian
theory.
We
may
ask ourselves
the
question
whether
they
can
be removed
by
a
modification
of
the Newtonian
theory.
First
of all
we
will
indicate
a
method which
does
not in itself claim
to be
taken
seriously;
it
merely serves as a
foil
for
what is
to
follow.
In
place
of
Poisson's
equation we
write
V2^
-
\£
= 4?TKp
. . .
(2)
where
A
denotes
a
universal
constant. If
p0
be
the uniform
density
of
a
distribution
of
mass,
then
*
-
-
T•
.
• •
(3)
is
a
solution
of
equation
(2).
This solution
would
correspond
to
the
case
in which the matter
of
the
fixed
stars
was
dis-
tributed
uniformly through space,
if
the
density
p0
is
equal
to the
actual
mean
density
of
the
matter
in
the
universe.
The solution then
corresponds
to
an
infinite extension
of
the
central
space,
filled
uniformly
with matter.
If,
without
making any
change
in the
mean
density,
we imagine
matter
to
be
non-uniformly
distributed
locally,
there
will
be,
over
and
above
the
Q
with the constant
value
of
equation
(3), an
additional
Q,
which in the
neighbourhood
of
denser
masses
will
so
much the
more
resemble
the
Newtonian field
as
AQ
is
smaller
in
comparison
with
4nkp.
[5]
A
universe
so
constituted
would have,
with
respect
to
its
gravitational
field, no
centre.
A decrease of
density
in
spatial
infinity
would
not have to
be
assumed,
but both the
mean
potential
and
mean density
would
remain constant
to
infinity.
The
conflict
with statistical mechanics which
we
found in
the
case
of
the
Newtonian
theory
is
not
repeated.
With
a
definite
but
extremely
small
density,
matter is in
equilibrium,
without
any
internal material
forces
(pressures)
being required
to maintain
equilibrium.
§
2.
The Boundary
Conditions
According
to the General
Theory
of
Relativity
In the
present paragraph
I
shall conduct the reader
over
the road that I
have
myself
travelled,
rather
a
rough
and
winding
road, because
otherwise I cannot
hope
that he
will