D O C . 1 G R A V I T A T I O N A L W A V E S 1 7
which would be quite remarkable. But it can easily be shown that this is not gener-
ally true.
§3. The Plane Gravitational Wave
In order to find plane gravitational waves, we start from the ansatz
(14)
which satisfies the field equations (6). The are here real-valued constants and
is a real-valued function of . The equations (5) produce the relations
. (15)
When the conditions (15) are met, (14) represents a possible gravitational wave. We
calculate the density of its energy current in order to get a better understanding
of its physical nature. Putting into equation (9) the , which are given in (15),
one gets
. (16)
This result seems strange insofar as of six arbitrary constants, which occur in
(14) if (15) is used, only two remain in (16). A wave for which and
vanish does not transport energy. This phenomenon can be deduced from the fact
that such wave, in a certain sense, does not have any real existence at all, as can be
derived in the simplest way from the following consideration.
[21]
γ′μν αμν f x1 ix4), + ( =
[p. 160]
αμν
f x1 ix4) + (
α11 iα14 + 0 =
α21 iα24 + 0 =
α31 iα34 + 0 =
α41 iα44 + 0 =
t41-
i
-----
γ′μν
{4}
[22]
[23]
t41
i
----- -
1
4κ
------f
′
2
α33 –
2
----------------------⎞
⎠
2
α23
2
+
⎝
⎛α22
=
α22 α33 – α23