5 6 D O C . 9 E N E R G Y C O N S E RVAT I O N
where the corresponds to the -term; the are functions of the .
The formula
yields, in our case, for the the components
. (17)
For the , one obtains without difficulty from the field equations of gravita-
tion, considering the -term,
.
(18)
The calculation of the is far more cumbersome; it is best to base it upon
equation (20 l.c). But it turns out to be more practical to introduce instead of
and the quantities and as H. A.
Lorentz has done occasionally. The following relations hold
(19)
,
(19a)
t
σ
ν
( )1 λ t
σ
ν
( )2 gσ
μν
[p. 456]
Tσ
ν
g
–
gσα----------------ρ°νds-αds
dx dx
=
Tσ
ν
0 0 0 0
0 0 0 0
Tσ
ν
=) (
0 0 0 0
0 0 0 ρ
°
–g
tν
σ
( )1
{2}
λ
λ –g 0 0 0
0 λ –g 0 0
κ(
tν
σ
)1=
0 0 λ –g 0
0 0 0 λ –g
tν
σ
( )2
gμν
gμν
σ
g
μν
–g g
μν
=
∂
∂xσ
--------
g
μν
–g) (
gμν
σ
=
[12]
tσ
α
1⎛
2⎝
-- -
G*
δσ
α
∂G*-gσ
∂gα
μν
-------------
μν
–
⎠
⎜ ⎟
⎞
=
∂G*-
∂gα
μν
-------------
1
2κ⎝
------
μβ
β
⎩ ⎭
⎨
⎬δν
⎧ ⎫
α
νβ
β
⎩ ⎭
⎨
⎬δμ
⎧ ⎫
α
+
⎠
⎜ ⎟
⎛ ⎞
1⎧
κ⎩
-- -
–
μν
α
⎭
⎨ ⎬
⎫
=