4.2 Kaluza’s five-dimensional unification

What is now called Kaluza–Klein theory in the physics community is a mixture of quite different contributions by both scientists82. Kaluza’s idea of looking at four spatial and one time dimension originated in or before 1919; by then he had communicated it to Einstein:

“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 [238], 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.”84View original Quote

Kaluza’s paper was communicated by Einstein to the Academy, but for reasons unknown was published only in 1921 [181Jump To The Next Citation Point]. Kaluza’s idea was to write down the Einstein field equations for empty space in a five-dimensional Riemannian manifold with metric gαβ, i.e., Rαβ = 0, α, β = 1,...,5, where R αβ is the Ricci tensor of M5, and to look at small deviations γ from Minkowski space: gαβ = − δαβ + γαβ.85. In order to obtain a theory in space-time, he assumed the so-called “cylinder condition”

gαβ,5= 0, (109 )
equivalent to the existence of a spacelike translational symmetry (Killing vector). Equation (109View Equation) is used for all “functions of state” (Zustandsgrössen), i.e., also for the matter variables. Kaluza did not normalize the Killing vector to a constant, i.e., he kept
g55 ⁄= const. (110 )
Equation (110View Equation) is called the “sharpened cylinder condition” by some authors including Einstein. Of the 15 components of gαβ, five had to get a new physical interpretation, i.e. gα5 and g55; the components gik, i,k = 1,...,4, were to describe the gravitational field as before; Kaluza took gi5 proportional to the electromagnetic vector potential Ai. The component g55 turned out to be a (scalar) gravitational potential which, in the static case, satisfies the equation
∇2g55 = − κμ0, (111 )
with the constant matter density μ0.

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 u α: u1, u2,u3,u5 ≪ 1, u4 ≃ 1. The Lorentz force appears augmented by an additional term containing g55 of the order u 2 (c) which thus may be neglected. From the fifth equation of motion Kaluza concluded that the fifth component of momentum p5 ∼ e, with e 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 M5 is destroyed. The remaining covariance group G5 is given by

x5′ = x5 + f(xk), xl′ = xl(xm ), k,l,m = 1,...,4. (112 )
The objects transforming properly under (112View Equation) are: the scalar g5′5′ = g55, the vector-potential l 5 g5′k′ = g5l ∂∂xxk′-+ g55 ∂∂xxk′, and the projected metric
( ) l m g′ ′ − gi′5′gk′5′ = g − gl5gm5- ∂x--∂x---. (113 ) ik g5′5′ lm g55 ∂xi′∂xk ′
Klein identified the group; however, he did not comment on the fact that now further invariants are available for a Lagrangian, but started right away from the Ricci scalar of M5 [185Jump To The Next Citation Point]. The group G5 is isomorphic to the group H5 of transformations for five homogeneous coordinates X μ′ = f μ(X ν) with f ν homogeneous functions of degree 1. Here, contact is made to the projective formulation of Kaluza’s theory (cf. “projective geometry” in Sections 2.1.3 and 6.3.2).

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.”86View original Quote (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 [105Jump To The Next Citation Point]. 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 (109View Equation) 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 WeylEddington approach, Einstein kept silent on the five-dimensional theory until 1926.


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