206
DOC. 32 INTEGRATION OF FIELD
EQUATIONS
[p.
693]
These
statements
can
be understood such that for the
conditions
of
each
type
the
not
explicitly
named
auv
vanish;
e.g.,
for
type a)
only
a11, a14,
and
a44
differ from
zero,
etc.
Type a)
has the
symmetry properties
of
a
longitudinal wave; type b)
and
c)
are
transversal
waves,
while
types d), e),
and
f) correspond
to
a
new type
of
symmetry. Types b)
and
c)
differ
from one
another
only
in the orientations
of
their
y-
and
z-axes;
and the
same
is true for
types d),
e), f),
so
that there
are
essentially
only
three
wave types proper.
We
are
primarily
interested in the
energy transport
of
these
waves,
and this is
measured
by
the
energy
current
§x
=
1-it41.
One
gets
from
(11)
for the individual
types
[10] {7}
a)
"7*41
b)
c)
1
I
1
I
1
I
~tUI
"7*41
(a2n
+
«14
+
*44)
f-
(ai2 +
4,)
=
0
^-(«13
+
c4)
=
0
=
0
d)
1
I
"7*41
_/
2
e)
1
I
4x
/2
«22 =
1
7"
*41
f)
1
i
7"
*41
/"
2
f'2
2
4x
1
PW'
x2
22
dt
y
ay'
4x
1
\2
23
a?
J
ay'
4^
a33
~
4x
\2
33
dt
It
follows, therefore,
that
only waves
of
the last-named
type
do
transport energy;
and the
energy
transport
of
any plane
wave
is
given by
1
-
h
=
±
* i
41
Ax
dy'
22
v
dt
+
2
dy
23
dt
dy'
\2
33
3?
(18)
§3.
Energy
Loss
of Material
Systems
by
Emissions
of Gravitational Waves
Let the
system
whose
radiation
we
want to
investigate
be
permanently
in the
neighborhood
of
the
origin
of the
coordinates. We consider
the
gravitational
field,
generated by
the
system, only
in
a
field
point
whose distance R
from
the
origin
of
the
coordinates is
large compared
to the dimensions
of
the
system.
Let the field
point
be