2 7 2 D O C . 2 8 2 O N E D D I N G T O N ’ S T H E O R Y
(7)
furthermore, iμ and iν by the equations
(8)
and obtain after an elementary
computation1)
(1c)
This result shows clearly that one has to conceive of the ’s as metric tensors.
The expression for the ’s resulting from the variation principle greatly resem-
bles the one resulting from Weyl’s theory; here, too, a four vector occurs besides
the metric tensor.
The field equations for the ’s and ’s are already incorporated in the re-
sults obtained so far, if one has adopted the Hamilton function H, because the field
equations result from equations (2) and (3) when one expresses their right-hand and
left-hand sides by the ’s and ’s. For the right-hand sides, this is done by
equation (1c); for the left-hand sides, by equations (4). Namely, if H is given as a
1)
By contraction of (1b) for the indices ν, α one initially gets
,
whereby one has instead of (1b):
In here one substitutes according to (5). Now, in accordance with the first of equa-
tions (7), one can switch to the contravariant form
After multiplying by , one recognizes that the parentheses of the last term equals
. After changing over to covariant indices, one then gets equation (1c) by solving for
the Γ’s in the familiar way.
gμν –g gμν, =
gμσgνσ δμ ν , =
g gμν , =
iμ –g iμ, =
iμ gμνiν =
[p. 369]
gμν)σ (
5
3
--iμ - –=
gμν)α (
1
3
--iμδα - ν
1
3
--iνδα - μ + + 0. =
gμν)α
(
∂xα
∂
gμν gμσΓσα ν gνσΓσα μ
1
3
--iμδα
-
ν
1
3
--iνδα
-
μ
gμν©§
∂xα
∂
lg –g Γασ¹
σ
–
·
+ + + + + 0. =
gμν
1
3
- –--iα
Γμν α
1
2
--gαβ¨ -
∂xν
∂gμβ
∂xμ
∂gνβ
∂xβ
∂gμν·
–+
© ¹
¸
§
1
2
--gμνiα - –
1
6
--δμ - αiν
1
6
--δν - αiμ. + + =
gμν
Γμν
α
gμν fμν
gμν fμν