1 6 D O C . 1 G R A V I T A T I O N A L W A V E S
depends upon the location but not upon direction. It also follows from (11a) that
small, rigid bodies remain similar under changes of their position, whereby their
linear extension, measured in coordinates, changes with .
In our case, equation (9) gives for the
(for indices 1 to 3)
(13)
The values of the definitely depend upon the choice of coordinates, a fact
Herr G. Nordström already pointed out to me in a letter some time
ago.5
If the
choice of coordinates is made with the condition , for which I previously
gave the in case of a mass point with the expressions
(for indices 1 to 3)
,
then all energy components of the gravitational field vanish when one calculates
them accurately to second-order quantities by means of the formula
One might suspect that a suitable choice of the system of reference would per-
haps always get all the energy components of the gravitational field to vanish—
5
See also E. Schrödinger, Phys. Zeitschr. 1 (1918), p. 4.
[16]
1
κM⎞
8πr⎠
-------- - –
⎝
⎛
tμσ
tμσ
κM2
32π2⎝
-----------⎛
-
xμxσ
r6
---------- -
1
2
-- -
δμσr4
1
----
–
⎠
⎞
=
t14 t24 t34 0 = = =
[17]
t44
κM2
64π2
----------- -
1
r4
---- ⋅ – =
tμσ
[18]
[19]
g 1 =
[20]
gμν
{3}
gμσ
κM
4π
-------- –
xμxσ
r3
---------- -
–δμσ =
g14 g24 g34 0 = = =
g44 1–
κM
4π
--------
1
r
-- -
⋅ =
κtσ
α
1
2
--δσ -
α
g
μν⎧
μνλβ
∑
μλ
β
⎩ ⎭
⎨ ⎬
⎫ νβ
λ
⎩ ⎭
⎨ ⎬
⎧ ⎫
g
μν⎧
μνλ
∑
μλ
α
⎩ ⎭
⎨ ⎬
⎫ νσ
λ
⎩ ⎭
⎨
⎬.
⎧ ⎫
– =