1 6 6 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
equations set up by me really do have physical significance. My arguments will
have already reached their goal if they cause mathematicians to collaborate and if
they convince them that the path embarked upon is pursuable and absolutely must
be thought through to the finish. As always in the general theory of relativity, in this
case, too, it is difficult to acquire from these equations insights about their solutions
that are in agreement with the results of experience, in this case specifically quan-
tum theory.
§2. Derivation of a System of Overdetermined Equations.
We start from the overdetermined equation system
(1)
In this system,
stands for the Riemann curvature tensor (with reversed sign as is conventional),
a tensor that is homogeneous and of second order in the electric field com-
ponents and has the same symmetry properties as . This
we achieve by setting equal to a linear combination of the tensors:
(3)
(4)
where, to abbreviate in (3),
(5)
and in (4),
(6)
is set. Thus it should be
(7)
For reasons that will soon be apparent, we give the constants A′, A″, A′″ the values
[p. 362]
Rik,
lm
Ψik
lm ,
=
Rik,
lm
gijRki,
lm
gij¨
∂xm
∂Γkl
j
∂xl
∂Γkm
j
– ΓσlΓkm
j σ
– Γσm j Γkl¸σ +
© ¹
§ ·
= =
Ψik
lm ,
φμν¨ =
∂xν
∂φμ
∂xμ¹
∂φν·
–
©
¸
§
Rik,
lm
Ψik
lm ,
Φ′ik
lm ,
φikφlm
1
2
--( - φilφkm φimφkl) – +=
Φ″ik,
lm
gilΦ′km gkmΦ′il gimΦ′kl – gklΦ′im – +=
Φ′″ik
lm ,
gilgkm gimgkl)Φ′ – ( =
gkmΦ′ik,
lm
Φ′il =
gilΦ′il Φ′ =
Ψik
lm ,
A′Φ′ik
lm ,
A″Φ″ik,
lm
A′″Φ′″ik,
lm
+ + =