DOC. 17 GRAVITY AND MATTER 87
A.
EINSTEIN
197
and
therefore
1'ík
-
b s/yyik, T44
= Jy
In consideration
of
what has
been
shown
hitherto,
we
obtain
in
place
of
(1a)
Pik
-
\yikP
=
-
ik
Ty
iP=
-
Jy
(13)
(14)
The
scalar
of
equation
(13) agrees
with
(14).
It
is
on
this
account
that
our
fundamental
equations permit
the
idea of
a
spherical
universe.
For
from
(13)
and
(14)
follows
_
4
k%\
n
P
ik
+
ä
--¡-y
ik
-
d
3
v
7
[22]
(15)
and
it
is
known* that this
system
is satisfied
by
a
(three-
dimensional)
spherical
universe.
But
we
may
also base
our
reflexions
on
the
equations
(9).
On
the
right-hand
side of
(9)
stand those terms
which,
from
the
phenomenological
point
of
view,
are
to be
replaced
by
the
energy-tensor of
matter; that
is, they
are
to be
replaced
by
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 p
where
p
denotes the
mean
density
of
matter
assumed
to be
at rest.
We
thus obtain the
equations
Pík
-
byikP
-
Í7*O0
=
0
. .
(16)
èP
+
¿Go
=
-
kP
. . . .
(17)
From the
scalar
of equation
(16)
and
from
(17)
we
obtain
G0
=
-
fP
=
2*p,
. . .
(18)
and
consequently
from
(16)
Pik
-
Kpyik
=0
. . .
(19)
*Cf. H.
Weyl,
“Raum,
Zeit, Materie,”
§
33.
[24]
[23]