204
DOC. 32 INTEGRATION
OF FIELD
EQUATIONS
[8]
[9]
{6}
proportionality
factor;
the latter is found
easily
from
calculating
the
right-hand
side.
The
energy-momentum
theorem
of
matter,
without
neglecting
any terms,
is
yv
+
dg
~ a*_
^
po
2 ^
dx*
^Tp°
^
p
a
u-*n
Under the desired order
of
approximation
this
can
be
replaced by
V
+
1dgpy
0
f
dxa
2^
a*M
p-"_
'
(7a)
This formulation is
by
one
order
more
precise
than
equation (7).
It
follows that the
right-hand
side
of
(6) yields
under the here-considered
modification,
"
4k£
If-V
V
The conservation
theorem, therefore,
appears as
E
d(T/XV
+
t
)/
/XV
dx
=
0,
(10)
where
*
-±
"v
4k
£
ocß
dy'afl
dy
_
is
^(dy'rf
dxß
dxv
2
{
dxT
(11)
are
the
energy components
of the
gravitational
field.
As
the
simplest example
of
application, we
calculate
the
gravitational
field
of
a
mass
point
of
mass
M,
resting
at
the
point
of
origin
of
the coordinates. The
energy
tensor
of
matter,
neglecting
surface
forces,
is
T
-
^ ~
P
ds
ds'
(12)
considering hereby
that
in
the first
approximation
the covariant
energy
tensor
can
be
replaced by
the contravariant
one.
The scalar
p
is
the
(naturally measured)
mass
density.
It follows from
(9)
and
(12)
that all
y'uv
except
y'44 vanish,
and the latter
component
is
Y
= Y
44 2tt
r
(13)
[p. 692]
With the
help
of
(8)
and
(1)
one
obtains for the
guv
the values