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

ds2

= gμνdxμdxν (1)

whose coefficients (gμν = gνμ)

g11 g12 g13 g14 g15

g21 g22 g23 g24 g25

— — — — —

— — — — —

g51 g52 g53 g54 g55

should not, according to what has been said, depend on x5. The components g11 ...

g44 should describe the gravitational field; g15, g25, g35, g45 would be the electric

potentials, g55 a field value that still awaits interpretation and may perhaps be

related to the Poincaré pressure that played a kind of awkward stand-in role in the

theory of the electron.

Kaluza’s essential hypothesis, now, consists of the assumption that the laws of

nature should be generally covariant in this five-dimensional world. Thus the ways

and means by which the electromagnetic potentials occur in the laws of nature are

necessarily connected with the ways and means by which the gravitational poten-

tials occur, which signifies a trenchant limitation of the possibilities. Thus the pos-

sibility arises for us to construct the physical worldview on a uniform Hamiltonian

function that does not contain heterogeneous terms superficially welded together

by a plus sign. Mr. Kaluza introduces another tensor for the material current besides

the magnitudes gμν, however. But it is clear that the introduction of such a tensor

only serves to give a preliminary, merely phenomenological description of matter,

whereas the ultimate goal we envision today is a pure field theory in which the field

variables represent the field of “empty space” as well as the electric elementary

particles that make up “matter.”

Nonetheless, the fundamental weak points of Kaluza’s idea must not be left

unmentioned. In the general theory of relativity, which operates with the four-

dimensional continuum,

ds2

= gμνdxμdxν

means a directly measurable magnitude for a local inertial system using measuring

rods and clocks, whereas the

ds2

of the five-dimensional manifold in Kaluza’s ex-

tension initially stands for a pure abstraction that seems not to deserve direct met-

rical significance. Therefore, from the physical point of view, the requirement of

general covariance of all equations in the five-dimensional continuum appears

completely

unfounded.[7]

Moreover, it is a questionable asymmetry that the re-

quirement of the cylinder property distinguish one dimension above the others and

yet with reference to the structure of the equations all five dimensions should be

equivalent.

[p. 3]