3 0 D O C U M E N T 1 2 K A L U Z A ’ S T H E O R Y

12. “Proof of the Non-Existence of an Everywhere

Regular, Centrally Symmetric Field According to the

Field Theory of Kaluza”

[Einstein and Grommer 1923a, 1923b]

Received 10 January 1922

Published 1923

In: Scripta Universitatis atque Bibliothecae Hierosolymitanarum. Mathematica et

Physica 1 (1923), VII: 1–5; Kitvei ha-Universita ve-Beth-ha-Sfarim bi-Yerushalayim.

Mathematica u’Fisica. A (5684), VII: 1–4

(Hebrew).[1]

Surely the most important current issue of the general theory of relativity today is

the essential unity of the gravitational field and the electromagnetic field. Although

the essential unity of both kinds of fields cannot, by any means, be required a priori,

it would undoubtedly be a great advance in the theory if this dualism could be over-

come. Until a short while ago the sole attempt in this direction has been Weyl’s

theory.[2]

Considerable misgivings about it exist, however. It does not do justice to

the independence of the measuring rods and clocks, or atoms, from their

prehistories.[3]

Furthermore, it does not remove this dualism to the extent that its

Hamiltonian function is composed additively of two parts, an electromagnetic one

and a gravitational one, which are not independent of each other. Furthermore, this

theory leads to differential equations of fourth order while we have no indication

that equations of second order would work out.

A short while ago a draft of a theory was presented to the Academy of Science

in Berlin by Mr. Th[eodor] Kaluza that avoids all these troubles and is formally of

astonishing

simplicity.[4]

Let us first sketch Mr. Kaluza’s thoughts and then move

on to the question we wish to

examine.[5]

A five-dimensional manifold whose field variable does not depend on the fifth

variable is (with a suitable choice of coordinates) equivalent to a four-dimensional

continuum. It therefore does not signify any special physical hypothesis if we inter-

pret the four-dimensional space-time manifold of physical experience as such a

five-dimensional manifold, which we will call “cylindrical” with reference to

x5.[6]

This is what Kaluza does. He furthermore assumes that physical reality in this con-

tinuum is characterized by a quadratic line element

[p. 1]

[p. 2]