D O C . 1 7 0 O N O V E R D E T E R M I N A T I O N 1 6 7
(7a)
We notice the following about the properties of system (1). If one multiplies by
and sums over the indices i and l, one obtains the equations
(8)
These are the well-known field equations of the general theory of relativity contain-
ing the Maxwell equations where besides the gravitational field only the electro-
magnetic field exists. System (8) is known to have the centrally symmetric static
solution1)
(9)
This solution, exhibiting a singular point (or a singular world line), which repre-
sents the negative, or positive electron, we want to designate symbolically, in ac-
cordance with the constants m (ponderable mass) and ε (electrical mass) occurring
in it, as
(10)
The system of overdetermined field equations we are looking for must likewise
have the solution .
Equations (1) themselves cannot yet be the system of equations sought by us.
For, according to them the metric field for a vanishing electric field is necessarily
a Euclidean one. Accordingly, Schwarzschild’s solution already does not
agree with equation system (1). On the other hand, through calculation I convinced
myself that the “mass-free” electron constitutes a solution of (1), i.e., that
satisfies system (1). For this reason, it seems to me that the sought-after
1)
Cf. H. Weyl, Space–Time–Matter,
§32.[13]
A′ 2 –=
A″
2
3
-=--+
A′″
1
6
-–=--
gil
Rkm
1
4
--gkmφαβφαβ - φkαφm¹
α·
–
©
§
–=
ds2 f2dt2 h2dr2 r2( dϑ2 cos2ϑdψ2)] + + [ –=
f2
1
h2
---- - 1
2m
r
------- –
ε2
2r2
------- -+ = =
φ4α
∂xn©
∂ ± -----¹·-εr
§
=
φ23 φ31 φ12 0. = = =
[p. 363]
L m ε, ( )
L m ε, ( )
L m, 0) (
L( 0 ε, )