DOC. 43 COSMOLOGICAL
CONSIDERATIONS
429
our
assumption
as
to
the
uniformity
of
distribution
of
the
masses
generating
the
field,
it
follows
that the curvature
of
the
required space
must
be
constant. With this distribution
of
mass, therefore,
the
required
finite continuum
of
the
x1, x2,
x3,
with constant
x4,
will be
a
spherical space.
We
arrive
at
such
a
space,
for
example,
in the
following
way.
We start
from
a
Euclidean
space
of
four
dimensions,
s1, s2,
s3
s4,
with
a
linear element do;
let, therefore,
do2
=
ds21 +
ds22
+
ds23
+
ds24
...
(9)
In this
space
we
consider
the
hypersurface
R2
=
s21
+
s22
+
s23
+
s24
...
(10)
where R denotes
a
constant. The
points
of
this
hypersurface
form
a
threedimensional
continuum,
a
spherical
space
of
radius
of
curvature R.
The fourdimensional Euclidean
space
with which
we
started
serves
only
for
a
convenient
definition of
our
hyper
surface.
Only
those
points of
the
hypersurface
are
of
interest to
us
which
have
metrical
properties
in
agreement
with those
of
physical space
with
a
uniform distribution
of
matter.
For
the
description
of
this threedimensional
con
tinuum
we
may
employ
the
coordinates
s1,
s2,
s3
(the
pro
jection upon
the
hyperplane
s4
=
0) since,
by
reason
of
(10),
s4 can
be
expressed
in terms
of s1,
s2,
s3.
Eliminating
s4
from
(9), we
obtain
for
the
linear element
of
the
spherical
space
the
expression
do2
=
yuvdsudsv
yuv
=
duv
+ susv/R2p2 ...
(11)
where
duv =
1,
if
u
=
v;
=
0,
if
u
#
v,
and
p2
=
s21
+ s22
s23+
The
coordinates
chosen
are
convenient when it
is
a
question
of
examining
the environment
of
one
of
the
two
points
s1
=
s2
=
s3
=
0.
Now
the
linear
element
of
the
required
fourdimensional
spacetime
universe
is also
given
us.
For the
potential
guv,
both
indices of
which
differ
from
4,
we
have to
set
guv
=

(guv
+ xuxv/R2(x21
x22
+
x23))
.
(12)
[12]