D O C . 2 8 2 O N E D D I N G T O N ’ S T H E O R Y 2 7 1
Before we draw conclusions from this
axiom,[5]
let us introduce a logically ar-
bitrary constraint. The scalar density H is not supposed to depend on the Γ’s in the
most generally conceivable way, that is, not to be formed arbitrarily from the
’s (92,
41),[6]
but rather exclusively from their contraction (92, 42),
i.e. out of the symmetrical and asymmetrical components of this tensor:
(2) ,
(3) .
In accordance with this condition, one initially obtains instead of (1)
(1a) ,
where one sets
(4)
In (1a) the ’s and ’s are to be expressed, because of (2) and (3), as
and . Taking into account that the 40 variations of can be chosen inde-
pendently from one another, one obtains from (1a) 40 equations:
(1b)
There the tensor densities
(5)
(6)
are introduced. The 40 equations (1b) allow us to express the 40 quantities by
the ’s, ’s and their derivatives. In order to bring this about, one must switch
from the contravariant tensor densities to the contravariant tensors, and from these
to the covariant tensors. For this purpose we define the tensors and by the
equations
*Bμνσ ε *Gμν
[p. 368]
γμν
∂xα
∂
Γμν α – Γμβ α Γνα β
1
2©
--§
-
∂xν
∂
Γμα α
∂xμ
∂
Γνα¹ α +
·
Γμν α Γαβ β – + + =
ϕμν
1
2©
--§
-
∂xν
∂
Γμα α
∂xμ
∂
Γνα¹ α –
·
=
gμνδγμν fμνδϕμν)dτ
+
³(
0 =
∂γμν
∂H
gμν =
∂ϕμν
∂H
fμν =
δγμν δϕμν Γμνσ
δΓμν σ δΓμν α
gμν)α (
1
2
--(gμσ)σδα - ν –
1
2
--( - gνσ)σδα μ –
1
2
--iμδα - ν –
1
2
--iνδα - μ – 0. =
gμν)α (
∂xα
∂
gμν gσνΓσα μ gμσΓσα ν gμνΓασ,σ + + + =
fμ
∂xσ
∂
fμσ =
Γμν σ
gμν fμν
gν gμν