DOC. 43
COSMOLOGICAL CONSIDERATIONS
431
universal
constant;
and Hamilton's
principle,
of
course,
guarantees
the
validity
of laws of
conservation. It
will be
shown
in
§
5
that
field
equation (13a)
is
compatible
with
our conjectures
on
field
and
matter.
§
5.
Calculation
and
Result
Since all
points
of
our
continuum
are
on an
equal footing,
it
is sufficient
to
carry through
the calculation
for
one
point,
e.g.
for
one
of
the
two
points
with the
co-ordinates
x1
=
x2
=
x3
=
x4
=
0.
Then
for
the
guv
(13a) we
have to
insert the
values
-1
0
0
0
0 -1
0 0
0
0 -1
0
0 0 0
1
wherever
they appear
differentiated
only once or
not at
all.
We thus obtain
in
the
first
place
Guv
= d/dx1[uv,1]
+
d/dx2[uv,2]
+
d/dx3[uv,3]
+
d2log
-g/dxudxv.
From this
we
readily
discover, taking
(7), (8),
and
(13)
into
account,
that
all
equations
(13a)
are
satisfied if
the
two
relations
-2/R+x=-kp/2, -x=
kp/2,
or
x
=
kp/2
=
1/R .
.
. .
(14)
are
fulfilled.
Thus the
newly
introduced universal
constant
X
defines
both the
mean
density
of
distribution
p
which
can
remain
in
equilibrium
and
also
the radius
R
and the
volume
2n2R3
of
spherical
space.
The total
mass
M of the
universe,
accord-
ing
to
our
view,
is
finite,
and
is
in fact
M
=
p
.
2n2R3
=
4n2R/k
= n232/k3p
.
.
(15)
Thus the theoretical
view of
the actual
universe,
if
it
is
in
correspondence
with
our
reasoning,
is
the
following.
The
[15]