2 0 D O C . 1 G R A V I T A T I O N A L W A V E S
infinitesimally short. We ask for the gravitational waves of the system emitted in
the direction of the positive -axis of the system.
The last-named restriction implies that we can replace (7) for a sufficiently large
distance R of a point under consideration from the coordinate origin, with the equa-
tion
. (7a)
We can limit our consideration to energy-transporting waves; we then only have to
form, due to the results of §3, the components and . The space
integrals on the right-hand side of (7a) can be rewritten in a way which M. Laue
has devised. We only want to give a detailed calculation of the integral
Multiplying the two momentum equations
by and , resp., then integrating both over the entire material system and add-
ing them together, one gets, by partial integration after a simple rearrangement,
We transform the latter integral again with the help of the energy equation
.
x
[p. 162]
γμν
′
κ
2πR∫Tμν(
----------
x
°
° °
R)dV – ,t ,z ,y
°
– =
[26]
γ23
′
γ22
′
γ33
′
– ( )
2
-------------------------
[27]
∫T23dV°.
[28]
∂T21
∂x1
---------- -
∂T22
∂x2
---------- -
∂T23
∂x3
---------- -
∂T24
∂x4
---------- - + + + σ =
∂T31
∂x1
---------- -
∂T32
∂x2
---------- -
∂T33
∂x3
---------- -
∂T34
∂x4
---------- - + + + σ =
[29]
x3
2
---- -
x2
2
---- -
∫T23dV°
1-------
2dx4⎩
-- - +
d
-⎨ x3T24 x2T34)dV° +
∫(
⎭
⎬
⎧ ⎫
– 0. =
∂T41
∂x1
---------- -
∂T42
∂x2
---------- -
∂T43
∂x3
---------- -
∂T44
∂x4
---------- - + + + 0 =