1 6 8 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 serving to overdetermine the field are derivable through the generaliza-

tion of (1). To do this, an obvious way is available. By the introduction of a local

“geodetic” system of coordinates, one easily proves that the covariant derivatives

of the Riemann tensor fulfill the identity (found by Bianchi)

(11)

From this follows that the general equations

(12)

are contained in (1).

Now, I consider it not unlikely that equations (12), together with equations (8)

of the existing general theory of relativity, which likewise follow out of (1), are the

sought-after system of equations for the overdetermination of the entire field.

The calculational proof that satisfies equation system (12) has not been

possible for me because its complexity is too great. However, this seems thorough-

ly plausible, as not only but also satisfy system (12). For the lat-

ter, it is the case because of the vanishing electric field intensities; for the former,

because is a solution to (1). By multiplying (12) by and summing

over the indices iklm, one obtains Maxwell’s equations.

Hence, there exists a certain probability that the connection between systems

(12) and (8) yields the sought-after overdetermination of the whole field. As a re-

sult, the following questions arise:

Does satisfy equation system (12)?

Does the double system (12) and (8) determine the mechanical behavior of sin-

gularities?

Do the processes according to (12) and (8) correspond to what we know from

quantum theory?

The last two questions pose great demands on the mathematician who wishes to

solve them; methods of approximation need to be invented to tackle the problems

of motion. But the circumstance that a possibility for a real scientific foundation of

quantum theory seems to open up here justifies the great efforts. In conclusion, one

should emphasize again that to me the main issue of this paper is the idea of over-

determination; I willingly concede that the derivation of equations (12) is not as

compelling as one would like.

Supplement added in proof. The first of the questions posed above has in the

meantime been answered. Mr. Dr. Grommer has established by direct calculation

that the solution satisfies the equation system

(12).[14]

Rik,

lm ;n

0 Rik

lm ;n ,

Rik

mn ;l ,

Rik

nl ;m ,

+ + ≡

Ψik

lmn ,

Ψik

lm ;n ,

Ψik,

mn ;l

Ψik,

nl ;m

+ + 0 = =

L m ε, ( )

L( 0 ε, ) L m, 0) (

L( 0 ε, ) gilgkm

[p. 364]

L m ε, ( )

L m ε, ( )