1 0 D O C . 1 G R A V I T A T I O N A L W A V E S
§1. Solutions of the Approximation Equations for the Gravitational Field by Means
of Retarded Potentials
We start with the field equations
(2)
which are valid for arbitrary coordinate
systems.2
is the energy tensor of mat-
ter and is the associated scalar . One gets the approximation equa-
tions
(2a)
if one considers all terms which are of nth order in as small quantities of order
and, simultaneously, limits the evaluation of both sides of equation (2) to terms
of the lowest order. Multiplying this equation by with summation over
and , one next gets (by change of the indices) the scalar equation
2
We refrain here from an introduction of the “λ -term” (see these Sitzungsber. [1917],
p. 142).
x
--------
μν⎫
α
⎩ ⎭
⎨ ⎬
⎧
∂
∂xν
-
μα⎫
α
⎩ ⎭
⎨ ⎬
⎧ μα⎫⎧
β
⎩ ⎭⎩
⎬⎨
⎧ νβ⎫
α
⎭
⎬
μν⎫⎧αβ⎫
α
⎩ ⎭⎩
⎬⎨
⎧
β
⎭
⎬
αβ
∑⎨
–
αβ
∑⎨
+
α
∑-------
+
α
–∑∂∂α
Tμν
1
2
--gμνT⎠ - –
⎝
⎞
,
–κ⎛
=
[p. 155]
Tμν
T
αβ
∑gαβTαβ
[4]
[5]
∂2γμν
∂xα 2
-------------
∂2γαα
∂xμ∂xν
-----------------
∂2γμα
∂xν∂xα
-----------------
∂2γνα-⎞
∂xμ∂xα⎠
-----------------⎟ – – +
⎝
⎛
α
∑⎜
2κ⎛
Tμν
1
2
--δμν∑Tαα⎠
-
α
–
⎝
⎞
=
γμν
n
1
2
-- -
δμν – μ
ν
∂
2
γαα
∂xβ
2
------------- -
∂
2
γαβ
∂xα∂xβ⎠
-----------------⎟ + –
⎝
⎜
⎛ ⎞
αβ
∑
κ
α
∑Tαα
=