D O C . 5 2 O N A F F I N E F I E L D T H E O R Y 5 1
These field equations can be put into the form of a Hamiltonian principle in
which one performs the variation for the metric tensor and the electric field inten-
sity. Its Hamiltonian function is given by
... (13)
R here means the Riemannian curvature scalar formed out of the ’s. As a proof
one expresses most conveniently with the tensor densities and and the
normalized subdeterminants belonging to the latter quantities and varied for
and .
For the physical interpretation of the field equations (11a) and (12a), it is surely
most practical to introduce the electromagnetic potential
(14)
which, according to (12a), is related to the current density by the equation
. (15)
(11a) then takes the form
. (16)
One obtains the former theory of gravitation and of the electromagnetic field in
the absence of electric densities by letting the factor β vanish. In any event, in order
not to come into contradiction with experience, one must set β very small, since
otherwise, according to (15), electricity-free electric fields could not exist. Then for
finite electrical and metric fields the ’s must always be vanishingly small. Except
for the sign of the constant β, equations (15) and (16) coincide with the field equa-
tions that H.
Weyl[9]
derived from a special action principle on the basis of his the-
ory. These equations do not yield a singularity-free
electron.[10]
53. Calculations R Tensor
ca. 31 May
1923[1]
[Not selected for translation.]
H
1
2
-- -
fστfστ
1
β
-- -
iσiσ¹

§ ·
+ =
gμν
H fμν gμν
fμν
gμν
–fμ
1
β
-- - =
βgμνfν –=
Rμν αgμν κ fμσfν
σ

1
4
--gμνfστfστ¹
- +