208
DOC.
32
INTEGRATION OF FIELD
EQUATIONS
/
T22dV
=
a2
9*4
[p. 695] or,
after
introducing
real-valued
coordinates,
and
allowing oneself,
as an
approxima-
tion,
to
equate
the
energy density
(-T44)
with the
mass
density
p
of
arbitrarily
moving
masses:
!T*äV
-
i-£iiJ*y2jv)-
2 dt2
(22)
One
also
has
Y
22
K
(fPy2dv)
4ttR
dt2
(23)
In
an analogous manner one
calculates
K
Y33
j?_
(/
P*
2dv)
47TR
dt2
(23a)
[11]
{8}
K
Y
23
=
(|
p
yzdv).
AirR dt2
(23b)
The
integrals
in
(23), (23a),
and
(23b)
are
nothing more
than time-variable
momenta
of
inertia,
and shall be denoted
by
the abbreviations
J22,
J33,
J23.
The
intensity
Sx
of the
energy
radiation from
(18)
then
yields
{9}
\2
'
^
23
=
+
647r2/?2 dt3 dt3
.
(20)
From this
follows, furthermore,
that the
mean
value of
energy
radiation in all
directions is
given by
K
f£
ray2
64tt1R
^
aß
dt3
where the summation
of the indices
1
to
3
is
to be
extended
over
all
9
combinations,
the
reason being
that this
expression
is
on
the
one
hand invariant under
spatial
rotations
of the coordinate
system
(as
is
easily
seen
from the three-dimensional
tensorial character
of
and, on
the other
hand,
it coincides with
(20)
in
case
of
radial
symmetry
(J11
= J22 = J33; J23 =
J31
= J12 =
0).
Consequently, one
obtains the
radiation
A
of the
system per
unit time
by multiplying
with
4nR2: