430 DOC.
43 COSMOLOGICAL CONSIDERATIONS
which
equation,
in
combination with
(7)
and
(8), perfectly
defines
the behaviour
of
measuringrods, clocks,
and
light
rays.
§
4.
On
an
Additional
Term for
the Field Equations of
Gravitation
My proposed
field
equations
of
gravitation
for
any
chosen
system
of
coordinates
run as
follows:
Guv =
K(Tuv

1/2guvT),
Guv =

d/dxa{uv,a}
+
{ua,
a}
+
{ua,
B}
{vB, a}
+
d2log g/dxudxv

{uv,a}d
log b/dxa
(13)
The
system
of
equations (13)
is
by no
means
satisfied
when
we
insert for the
guv
the
values
given
in
(7), (8),
and
(12),
and
for
the
(contravariant) energytensor
of
matter the
values indicated in
(6).
It
will be
shown in the next
para
graph
how this calculation
may conveniently
be made. So
that,
if it
were
certain that the
field
equations
(13)
which
I
have hitherto
employed were
the
only ones
compatible
with
the
postulate
of
general relativity,
we
should
probably
have
to conclude
that the
theory
of
relativity
does
not admit
the
hypothesis
of
a
spatially
finite universe.
However,
the
system
of
equations
(14)
allows
a
readily
suggested
extension which is
compatible
with
the
relativity
postulate,
and is
perfectly analogous
to
the extension
of
Poisson's
equation
given
by equation
(2).
For
on
the left
hand
side of field
equation
(13)
we
may
add
the fundamental
tensor guv,
multiplied
by a
universal
constant,

X,
at
present
unknown,
without
destroying
the
general
covariance.
In
place
of field
equation
(13)
we
write
Guv

Xguv
=

K(Tuv

1/2guvT)
. .
(13a)
This
field
equation,
with
X
sufficiently
small,
is
in
any
case
also
compatible
with the
facts
of
experience
derived
from
the
solar
system.
It
also satisfies laws of
conservation
of
momentum and
energy,
because
we
arrive
at
(13a)
in
place
of
(13)
by introducing
into Hamilton's
principle,
instead
of
the
scalar of
Riemann's
tensor,
this scalar
increased
by
a
[13]
[14]