D O C . 1 G R A V I T A T I O N A L W A V E S 1 3
and (of the gravitational field) satisfy, according to the general theory, the re-
lations
,
.
From these follows
.
We find the if we bring the right-hand side in the form of the left-hand side,
thereby using the from the field equations. In case of the approximation con-
sidered here, the two factors on the right-hand side of this equation are small quan-
tities of the first order. In order to get the accurately as quantities of the second
order, one only needs to substitute the two factors on the right accurately in terms
of quantities of the first order. One, therefore, can replace
by
and by .
For we introduce the which, under the desired approximation, deviate in
value from the only in sign. With respect to the character of their indices, the
are quantities analogous to the . We have to determine the from the
equation
. (8)
We transform the right-hand side by observing that, due to (3),
tμ
σ
[p. 157]
∂Tσ
μ
∂xσ
-
σ
∑-----------
1
2∑∂gρσTρσ
-- -
∂xμ
-----------
ρσ
+ 0 =
[9]
∂( Tμ
σ
tμ
σ
+ )
∂xσ
σ
∑------------------------------
0 =
∂tμ
σ
∂xσ
-
σ
∑---------
1
2∑∂gρσTρσ
-- -
∂xμ
-----------
ρσ
=
tμ
σ
[10]
Tρσ
[11]{1}
tμ
σ
∂gρσ
∂xμ
-----------
∂γ
∂xμ
-----------ρσ
–
Tρσ Tρσ
tμ
σ
tρσ
tρ
σ
tρσ Tμσ tμσ
∂tμσ
∂xσ
σ
∑----------
1
2∑---------- -- -
∂γρσTρσ
∂xμ
-
ρσ
=