86 DOC. 17 GRAVITY AND MATTER
196
ATOMIC
STRUCTURE
occurs
Tuv
+
1/4guv(G
-
G0)
which is
independent
of
the
scalar
of
curvature. But the
new
formulation
has
this
great
advantage,
that the
quantity
A
appears
in
the fundamental
equations
as a
constant
of
integration,
and
no
longer
as a
[20]
universal
constant
peculiar
to
the fundamental
law.
d
3. On
the
Cosmological Question
[21]
The last result
already permits
the
surmise
that with
our
new
formulation the
universe
may
be
regarded
as
spatially
finite,
without
any necessity
for
an
additional
hypothesis.
As in
the
preceding
paper
I shall
again
show
that with
a
uniform
distribution
of
matter,
a spherical
world is
compatible
with the
equations.
In the
first
place
we
set
ds-
=
-
7ikdxidxk + dx\ (i,
A
=
1, 2,
3)
(11)
Then
if
Pik
and
P
are, respectively,
the
curvature
tensor of
the
second
rank
and
the
curvature scalar
in three-dimensional
space,
we
have
11 ik
=
Pik
(i,
A
=
1, 2,
3)
G*4 =
G4¿
=
G44
=
0
G
=
-
P
~
9
=
7-
It therefore
follows for
our case
that
G
ik
-
\(jik G
=
P
ik
-
i
y ik
I}
(i,
A
=
1, 2,
3)
G44
-
iy44G
=
¿P.
We
pursue
our
reflexions,
from
this
point
on,
in two
ways.
Firstly,
with the
support of equation
(1a).
Here
Tuv
denotes
the
energy-tensor of
the
electro-magnetic
field,
arising
from
the
electrical
particles constituting
matter.
For this
field
we
have
everywhere
X]
+
XI
+
XI
+
XI
=
0.
The
individual
Iuv are
quantities
which
vary rapidly
with
posi-
tion; but
for
our
purpose
we no
doubt
may
replace
them
by
their
mean
values.
We therefore
have to choose
X\
=
XI
=
XI
=
-
\X\
=
const.
X¿
=
0
(for
/i
=[=
v),
(12)