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 3
function of the ’s and ’s, then one can express from (4) the ’s and the
’s by the ’s and the ’s and can insert the result into the left-hand sides
of (2) and (3).
One arrives more simply at the goal—as regards the left-hand sides of (2) and
(3)—in the following way. Since we have not yet introduced any assumptions with
regard to the choice of the scalar density H as a function of γ and ϕ, equations (4)
express nothing other than that
is a complete differential (with regard to the variables and ). This means
the same thing as that
is a complete differential, in that we now, conversely, view the ’s and the ’s
as functions of the ’s and the ’s. Our result then means that a function H*
of the ’s and the ’s exists, with the character of a scalar density, which sat-
isfies the conditions
(4a)
The choice of the function H* fully determines the left-hand side of (2) and (3). The
functions H and H* determine each other unequivocally. Namely,
(9)
or, if H* is a homogeneous quadratic function of the ’s and a homogeneous
function to zero order of the ’s,
(9a) .
Taking into account Maxwell’s theory, we set
(10) .
Here β is a constant,
g
the determinant , the normalized subdeterminants
for the
gσα’s.1)
From this follows by simple calculation
1)
One easily recognizes from (10) and (9) that equation (9a) is satisfied for this formula-
tion.
γμν ϕμν γμν
ϕμν gμν fμν
gμνdγμν fμνdϕμν +
γμν ϕμν
γμνdgμν ϕμνdfμν +
γμν ϕμν
gμν fμν [p. 370]
gμν fμν
γμν
∂gμν
∂
H* =
ϕμν
∂fμν
∂
H*. =
dH dH* + d(
γμνgμν ϕμνfμν)
+ =
fμν
gμν
H H* =
H*
β
2
--fαβfαβ - g ––
β
2
- gσαgτβfστfαβ g – –-- = =
gαβ gσα