D O C . 1 G R A V I T A T I O N A L W A V E S 1 9
and from these for the
.
(20)
If we furthermore fix the connection between the function in (19) and the func-
tion in (14) by the relation
, (21)
then it is seen that, aside from the naming of the constants, the of (20) and the
of (14) and (17) completely agree with each other.
Those gravitational waves which transport no energy can, therefore, be generat-
ed from a field-free system by a mere coordinate transformation; their existence is
(in this sense) only an apparent one. Real in the sense proper are, therefore, only
those waves, traveling along the -axis, whose propagation corresponds to the
quantities and (or to the quantities and resp.).
The two types are of the same nature and differ only in their mutual orientation. The
wave field acts in deforming angles in a plane which is perpendicular to the
direction of propagation. Density of the energy current, momentum, and energy are
given by (16).
§4. The Emission of Gravitational Waves by Mechanical Systems
We consider an isolated mechanical system whose center of gravity shall perma-
nently coincide with the coordinate origin. Changes within this system shall occur
so slowly and its spatial extension shall be so small that the light-time correspond-
ing to the distance between any two material points in it can be considered as
γμν′
[24]
λ1 iλ4 – λ2 λ3 iλ1 λ4 +
1
φ'
---γμν -
′
=⎞
⎝ ⎠
⎛
λ2 –λ1 iλ4 – 0 iλ2
λ3 0 –λ1 iλ4 – iλ3
iλ1 λ4 +
{5}
iλ2 iλ3 –λ1 iλ4 +
φ
f
φ' f =
γμν
′
γμν
′
[25]
x
γ22
′
γ33
′
– ( )
2
-------------------------
γ23
′
γ22 γ33) – (
2
------------------------
γ23