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