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]
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