DOC. 43 COSMOLOGICAL
CONSIDERATIONS
421
Doc.
43
Cosmological
Considerations
in the General
Theory
of
Relativity
This translation
by
W.
Perrett and G. B.
Jeffery
is
reprinted
from H. A. Lorentz
et
al.,
The
Principle
of
Relativity
(Dover, 1952), pp.
175-188.
[1]
IT
is well
known
that
Poisson's
equation
V2£
=
4ttKp
.
. .
.
(1)
in combination with the
equations
of
motion
of
a
material
point
is
not
as yet
a
perfect
substitute
for Newton's
theory
of action at
a
distance. There
is
still to
be
taken into account
the condition
that at
spatial infinity
the
potential
Q
tends
toward
a
fixed
limiting
value.
There
is
an analogous
state
of
things
in the
theory
of
gravitation
in
general
relativity.
Here,
too, we
must
supplement
the
differential
equations
by
limiting
conditions
at
spatial infinity,
if
we really
have to
regard
the universe
as
being
of
infinite
spatial
extent.
[2]
In
my
treatment of
the
planetary problem
I
chose
these
limiting
conditions
in the form
of
the
following assumption:
it
is
possible
to
select
a
system
of reference
so
that at
spatial
infinity
all
the
gravitational potentials
guv
become
constant.
[3]
But
it
is
by
no
means
evident
a
priori
that
we may lay
down
the
same
limiting
conditions
when
we
wish to take
larger
portions
of
the
physical
universe into consideration. In the
following pages
the
reflexions will be
given
which,
up
to the
present,
I
have made
on
this
fundamentally
important
question.
§
1.
The
Newtonian
Theory
It
is well
known that Newton's
limiting
condition
of
the
constant limit for
Q
at
spatial infinity
leads to
the
view
that
the
density
of
matter
becomes
zero
at
infinity.
For
we
imagine
that there
may
be
a place
in universal
space
round
about which the
gravitational
field of matter, viewed
on a
large scale,
possesses
spherical symmetry.
It then
follows
from
Poisson's
equation
that,
in order that
Q
may
tend to
a
177