DOC.
32 INTEGRATION
OF
FIELD
EQUATIONS
205
1
k
M
4r
r
0
0
0
0
1
K
M
4v
r
0
0
0
1
0
K
M
4r
r
0
0
0
1
+
0
k
M
4r
r
(14)
Herr De
Sitter
sent
me
these values
by letter; they
differ from those
which
I
previously gave only
in the choice of the
system
of
reference.
They
led
me
to the
simple approximative
solution
given
above.
However,
one
has to
keep
in mind that
the choice
of
coordinates which has been made here has
no equivalent
in the
general
case,
as
the
yuv
the
y'uv
have
tensorial character
only
with
respect
to
linear,
orthogonal
substitutions,
but not under
general
substitutions.
§2.
Plane
Gravitational Waves
It follows from
equations (6)
and
(9)
that
gravitational
fields
always propagate
with
velocity
1,
i.e.,
with the
velocity
of
light.
Plane
gravitational
waves,
traveling along
the
positive
Xaxis,
can
therefore be found
by setting
Y'uv
=
«uvf(x1 +
ix4)
=
«uvf(x

t).
(15)
The
auv
are
constants
here; f is
a
function
of
the
argument x

t.
If
the
space
under consideration
is free
of
matter, i.e.,
if
the
Tuv
vanish,
equations
(6)
are
satisfied
by
(15).
Equations (4) yield
the
following
relations between the
auv:
a11
+
ia14 =
0
a12
+
ia24
=
0
a13
+
ia34
=
0
a14
+
ia44
=
0
(16)
Consequently, only
6
of
the
10
constants
auv can
be chosen
freely.
We
can
superpose
the most
general type
of
wave, among
those
considered,
from the
following
6
types
a)
a11
+
ia14
=
0
a14
+
ia44
=
0
b)
a12
+
ia24 =
0
c)
a13
+
ia34
=
0
d)
a22
#
0
e)
a23
# 0
f)
a33
# 0
(17)