“The idea of achieving [a unified field theory] by means of a five-dimensional cylinder world never dawned on me. [...] At first glance I like your idea enormously.” (letter of Einstein to Kaluza of 21 April 1919)
This remark is surprising because Nordström had suggested a five-dimensional unification of his scalar gravitational theory with electromagnetism five years earlier [237], by embedding space-time into a five-dimensional world in quite the same way as Kaluza did. In principle, Einstein could have known Nordström’s work. In the same year 1914, he and Fokker had given a covariant formulation of Nordström’s pure (scalar) theory of gravitation [104]. In a subsequent letter to Kaluza of 5 May 1919 Einstein still was impressed: “The formal unity of your theory is startling.” However, on 29 May 1919, Einstein became somewhat reserved83 :
“I respect greatly the beauty and boldness of
your idea. But you understand that, in view of the existing factual
concerns, I cannot take sides as planned
originally.” 84
Kaluza’s paper was communicated by
Einstein to the Academy, but for
reasons unknown was published only in 1921 [180]. Kaluza’s idea was to write down
the Einstein field equations for empty
space in a five-dimensional Riemannian manifold with metric
, i.e.,
,
, where
is the Ricci tensor of
, and to look at
small deviations
from Minkowski space:
.85
. In order to obtain a
theory in space-time, he assumed the so-called “cylinder
condition”
Kaluza also showed that the
geodesics of the five-dimensional space reduce to the equations of
motion for a charged point particle in space-time, if a weakness
assumption is made for the components of the 5-velocity :
,
. The Lorentz force appears
augmented by an additional term containing
of the order
which thus may be neglected.
From the fifth equation of motion Kaluza concluded that the fifth
component of momentum
, with
being the particles’ electric charge (up to a constant of
proportionality). From the equations of motion, charge conservation
also followed in Kaluza’s linear approximation. Kaluza was well aware that his
theory broke down if applied to elementary particles like electrons
or protons, and speculated about an escape in which gravitation had
to be considered as some “difference effect”, and the gravitational
constant given “a statistical meaning”. For him, any theory
claiming universal validity was endangered by quantum theory,
anyway.
From the cylinder condition, a grave objection
toward Kaluza’s approach results:
Covariance with regard to the diffeomorphism group of is destroyed. The remaining covariance group
is given by
While towards the end of May 1919 Einstein had not yet fully supported the publication of Kaluza’s manuscript, on 14 October 1921 he thought differently:
“I am having second thoughts about having kept
you from the publication of your idea on the unification of
gravitation and electricity two years ago. I value your approach
more than the one followed by H. Weyl. If you wish, I will present
your paper to the Academy after all.” 86
(letter from Einstein to Kaluza reprinted in [49],
p. 454)
It seems that at some point Einstein had set his calculational
aide Grommer
to work on regular
spherically symmetric solutions of Kaluza’s theory. This led to a
joint publication which was submitted just one month after Einstein had finally presented a
rewritten manuscript of Kaluza’s to the Berlin
Academy [105]. The negative
result of his own paper, i.e., that no
non-singular, statical, spherically symmetric exact solution
exists, did not please Einstein. He also thought that Kaluza’s assumption of general
covariance in the five-dimensional manifold had no support from
physics; he disliked the preference of the fifth coordinate due to
Equation (109
) which seemed to
contradict the equivalence of all five coordinates used by Kaluza in the construction of the
field equations [105].
In any case, apart from an encouraging letter to Kaluza in 1925 in which he called
Kaluza’s idea the only serious
attempt at unified field theory besides the Weyl-Eddington approach, Einstein kept silent on the
five-dimensional theory until 1926.