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

H –g R 2α – κ©

1

2

-- -

fστfστ

1

β

-- -

iσiσ¹

–

§ ·

+ =

gμν

H fμν gμν

fμν

gμν

–fμ

1

β

-- - iμ =

iμ βgμνfν –=

Rμν αgμν – κ fμσfν

σ

–

1

4

--gμνfστfστ¹

- +

©

§ ·

βfμfν + –=

[p. 140]

iμ