424
DOC.
43 COSMOLOGICAL CONSIDERATIONS
take much interest in the result
at
the
end
of
the
journey.
The
conclusion
I
shall
arrive at
is
that the
field
equations
of
gravitation
which I have
championed
hitherto still
need
a
slight modification, so
that
on
the
basis of
the
general theory
of relativity
those
fundamental
difficulties
may
be avoided
which
have been set
forth in
§
1
as
confronting
the Newtonian
theory.
This
modification
corresponds
perfectly
to
the transi-
tion
from
Poisson's
equation (1)
to
equation
(2)
of
§
1.
We
finally
infer that
boundary
conditions
in
spatial infinity fall
away altogether,
because
the universal continuum
in
respect
of
its
spatial
dimensions
is to be viewed
as a
self-contained
continuum
of
finite
spatial (three-dimensional)
volume.
[6]
The
opinion
which
I entertained
until
recently,
as
to
the
limiting
conditions to be laid down in
spatial
infinity,
took
its stand
on
the
following
considerations. In
a
consistent
theory of relativity
there
can
be
no
inertia
relatively to "space,"
but
only
an
inertia
of
masses relatively
to
one
another.
If,
therefore,
I
have
a mass
at
a
sufficient
distance
from all
other
masses
in the
universe,
its inertia must
fall
to
zero.
We
will
try
to
formulate this
condition
mathematically.
According
to
the
general theory
of
relativity
the
negative
momentum
is
given by
the
first
three
components,
the
energy
by
the last
component
of
the covariant
tensor
multiplied
by
/
-
g
/
dXa
-
9
' * •
(4)\j,
where,
as
always,
we
set
ds2
=
guvdxudxv
. . . .
(5)
In the
particularly
perspicuous
case
of
the
possibility
of
choosing
the
system
of co-ordinates
so
that the
gravitational
field
at
every
point
is
spatially
isotropic,
we
have
more
simply
ds2
= -
A(dx12
+
dx22
+
dx32)
+
Bdx24.
If,
moreover,
at
the
same
time
/
-g
=
1
=
/A3B
we
obtain
from
(4),
to
a
first
approximation
for small
velocities,
A
dxy
A
dxo
A
dx*