6.3 Kaluza’s idea taken up again

6.3.1 Kaluza: Act I

Einstein became interested in Kaluza’s theory again due to O. Klein’s paper concerning a relation between “quantum theory and relativity in five dimensions” (see Klein 1926 [185Jump To The Next Citation Point], received by the journal on 28 April 1926). Einstein wrote to his friend and colleague Paul Ehrenfest on 23 August 1926: “Subject Kaluza, Schroedinger, general relativity”, and, again on 3 September 1926: “Klein’s paper is beautiful and impressive, but I find Kaluza’s principle too unnatural.” However, less than half a year later he had completely reversed his opinion:

“It appears that the union of gravitation and Maxwell’s theory is achieved in a completely satisfactory way by the five-dimensional theory (Kaluza–Klein–Fock).” (Einstein to H. A. Lorentz, 16 February 1927)

On the next day (17 February 1927), and ten days later Einstein was to give papers of his own in front of the Prussian Academy in which he pointed out the gauge-group, wrote down the geodesic equation, and derived exactly the Einstein–Maxwell equations – not just in first order as Kaluza had done [8182]. He came too late: Klein had already shown the same before [185Jump To The Next Citation Point]. Einstein himself acknowledged indirectly that his two notes in the report of the Berlin Academy did not contain any new material. In his second communication, he added a postscript:

“Mr. Mandel brings to my attention that the results reported by me here are not new. The entire content can be found in the paper by O. Klein.”132View original Quote

He then referred to the papers of Klein [185Jump To The Next Citation Point186] and to “Fochs Arbeit” which is a paper by Fock 1926 [130Jump To The Next Citation Point], submitted three months later than Klein’s paper. That Klein had published another important clarifying note in Nature, in which he closed the fifth dimension, seems to have escaped Einstein133 [184Jump To The Next Citation Point]. Unlike in his paper with Grommer, but as in Klein’s, Einstein, in his notes, applied the “sharpened cylinder condition”, i.e., dropped the scalar field. Thus, the three of them had no chance to find out that Kaluza had made a mistake: For g55 ⁄= const., even in first approximation the new field will appear in the four-dimensional Einstein–Maxwell equations ([145], p. 5). Mandel of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. Klein’s results [216Jump To The Next Citation Point]. In a footnoote, Mandel stated that he had learned of Kaluza’s (whom he spelled “Kalusa”) paper only through Klein’s article. He started by embedding space-time as a hypersurface x5 = const. into M5, and derived the field equations in space-time by assuming that the five-dimensional curvature tensor vanishes; by this procedure he obtained also a matter-energy tensor “closely linked to the second fundamental form of this hypersurface”. From the geodesics in M5 he derived the equations of motion of a charged point particle. One of the two additional terms appearing besides the Lorentz force could be removed by a weakness assumption; as to the second, Mandel opinioned

“that the experimental discovery of the second term appears difficult, yet perhaps not entirely impossible.” ([216Jump To The Next Citation Point], p. 145)

As to Fock’s paper, it is remarkable because it contains, in nuce, the coupling of the Schrödinger wave function ψ and the electromagnetic potential by the gauge transformation ψ = ψ e2πip∕h 0, where h is Planck’s constant and p “a new parameter with the unit of the quantum of action” [130Jump To The Next Citation Point]. In Fock’s words:

“The importance of the additional coordinate parameter p seems to lie in the fact that it causes the invariance of the equations [i.e., the relativistic wave equations] with respect to addition of an arbitrary gradient to the 4-potential.”134View original Quote ([130Jump To The Next Citation Point], p. 228)

Fock derived the general relativistic wave equation and the equations of motion of a charged point particle; the latter is identified with the null geodesics of M5. Neither Mandel nor Fock used the “sharpened cylinder condition” (110View Equation).

A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation

( 2 ) aik --∂-U-- − {ik}∂U-- = 0, i,k,= 1,...,5 (148 ) ∂xi∂xk r ∂xr
and by neglecting the gravitational field, he arrived at the four-dimensional Schrödinger equation after insertion of the quantum mechanical differential operators ih--∂- − 2π ∂xi. It was Klein’s papers and the magical lure of a link between classical field theory and quantum theory that raised interest in Kaluza’s idea – seven years after Kaluza had sent his manuscript to Einstein. Klein acknowledged Mandel’s contribution in his second paper received on 22 October 1927, where he also gave further references on work done in the meantime, but remained silent about Einstein’s papers [189Jump To The Next Citation Point]. Likewise, Einstein did not comment on Klein’s new idea of “dimensional reduction” as it is now called and which justifies Klein’s name in the “Kaluza–Klein” theories of our time. By this, the reduction of five-dimensional equations (as e.g., the five-dimensional wave equation) to four-dimensional equations by Fourier decomposition with respect to the new 5th spacelike coordinate x5, taken as periodic with period L, is understood:
ψ (x,x5) = √1--Σ ψ (x),einx5∕R5 L n n
with an integer n. Klein had only the lowest term in the series. The 5th dimension is assumed to be a circle, topologically, and thus gets a finite linear scale: This is at the base of what now is called “compactification”. By adding to this picture the idea of de Broglie waves, Klein brought in Planck’s constant and determined the linear scale of x5 to be unmeasurably small (∼ 10− 30). From this, the possibility of “forgetting” the fifth dimension arose which up to now has not been observed.

In his papers, Einstein took over Klein’s condition g55 = 1, which removed the additional scalar field admitted by the theory. It was Reichenbächer who apparently first tried to perform the projection into space-time of the most general five-dimensional metric, and without using the cylinder condition (109View Equation):

“Now, a rather laborious calculation of the five-dimensional curvature quantities in terms of a four-dimensional submanifold contained in it has shown to me also in the general case (g55 ⁄= const., dependence of the components of the fundamental [tensor] of 5 x is admitted) that the characteristic properties of the field equations are then conserved as well, i.e., they keep the form

√ -- ik 1-ik ik ∂--gF-ik- i R − 2g R = T , ∂xk = s ,
only the ik T contain further terms besides the electromagnetic energy tensor ik S, and the quantities collected in si do not vanish. [...] The appearance of the new terms on the right hand sides could even be welcomed in the sense that now the field equations are obtained not only for a field point free of matter and charge135.”136View original Quote ([276Jump To The Next Citation Point], p. 426)

Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail [102].

It is likely that Reichenbächer had been led to this excursion into five-dimensional space, an idea which he had rejected before as unphysical, because his attempt to build a unified field theory in space-time through the ansatz for the metric γik = gik − 𝜖2ϕiϕk with ϕk the electromagnetic 4-potential, had failed. Beyond incredibly complicated field equations nothing much had been gained [275]. Reichenbächer’s ansatz is well founded: As we have seen in Section 4.2, due to the violation of covariance in M5, γik transforms as a tensor under the reduced covariance group.

Even L. de Broglie became interested in Kaluza’s “bold but very beautiful theory” and rederived Klein’s results his way [46], but not without getting into a squabble with Klein, who felt misunderstood [18847]. He also suggested that one should not accept the cylinder condition, a suggestion looked into by Darrieus who introduced an electrical 5-potential and 5-current, and deduced Maxwell’s equations from the five-dimensional homogeneous wave equation and the five-dimensional equation of continuity [43].

In 1929 Mandel tried to “axiomatise” the five-dimensional theory: His two axioms were the cylinder condition (109View Equation) and its sharpening, Equation (110View Equation). He then weakened the second assumption by assuming that “an objective meaning does not rest in the gik proper, but only in their quotients”, an idea he ascribed to O. Klein and Einstein. He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [220].

Klein’s lure lasted for some years. In 1930, N. R. Sen claimed to have investigated the “Kepler-problem for the five-dimensional wave equation of Klein”. What he did was to calculate the energy levels of the hydrogen atom (as a one particle-system) with the general relativistic wave equation in space-time (148View Equation) with aik = γik, where γik = gik + α2γ55 is the metric on space-time following from the 5-metric γαβ by 5 dx = 0. For gik he took the Reissner–Nordström solution and did not obtain a discrete spectrum [324]. He continued his approach by trying to solve Schrödinger’s wave equation [325]

( 2 ) 2 gik --∂-u--− {ik} ∂u-- = − 4π--m2 c2u. ∂xi∂xk r ∂xr h2 0

Presently, the different contributions of Kaluza and O. Klein are lumped together by most physicists into what is called “Kaluza–Klein theory”. An early criticism of this unhistorical attitude has been voiced in [210].

6.3.2 Kaluza: Act II

Four years later, Einstein returned to Kaluza’s idea. Perhaps, he had since absorbed Mandel’s ideas which included a projection formalism from the five-dimensional space to space-time [216217218Jump To The Next Citation Point219].

In a paper with his assistant Mayer, Einstein now presented Kaluza’s approach in the form of an implicit projective four-dimensional theory, although he did not mention the word “projective” [107Jump To The Next Citation Point]:

“Psychologically, the theory presented here connects to Kaluza’s well-known theory; however, it avoids extending the physical continuum to one of five dimensions.”137View original Quote

In the eyes of Einstein, by avoiding the artificial cylinder condition (109View Equation), the new method removed a serious objection to Kaluza’s theory.

Another motivation is also put forward: The linearity of Maxwell’s equations “may not correspond to reality”; thus, for strong electromagnetic fields, Einstein expected deviations from Maxwell’s equations. After a listing of all the shortcomings of Kaluza’s theory, the new approach is introduced: At every event a five-dimensional vector space V5 is affixed to space-time V4, and “mixed” tensors k γι are defined linking the tangent space of space-time V4 with a V5 such that

gικγιlγ κm = glm, (149 )
where glm is the metric tensor of V4, and g ικ a non-singular, symmetric tensor on V5 with ι,κ = 1,...,5, and k,l = 1,...,4138. Indices are raised and lowered with the metrics of V5 or V4, respectively. There exists a “preferred direction of V5” defined by γιkA ι = 0, and which is the normal to a “preferred plane” γ kωk = 0 ι139. A consequence then is
τk k τ τ τ γσkγ = γσ γk = δσ − A σA . (150 )
A covariant derivative for five-vectors in V4 is defined with a “three-index-symbol” Γ ι πl with two indices in V 5, and one in V 4 standing in for the connection coefficients:
+ ∂X ι ∇l X ι =---l-+ Γ πιX π. (151 ) ∂x
The covariant derivative of 4-vectors is defined as usual,
i i ∂X-- i j ∇lX = ∂xl + {jl}X , (152 )
where i {jl} is calculated from the metric of V4 as given in Equation (149View Equation). Both covariant derivatives are abbreviated by the same symbol A;k. The covariant derivative of tensors with both indices referring to V5 and those referring to V4, is formed correspondingly. In this context, Einstein and Mayer mention an extension of absolute differential calculus by “WAERDEN and BARTOLOTTI” without giving any reference to their respective papers. They may have had in mind van der Waerden’s [368Jump To The Next Citation Point] and Bortolotti’s [24Jump To The Next Citation Point] papers. The autoparallels of V5 lead to the exact equations of motion of a charged particle, not the geodesics of V4. Einstein and Mayer made three basic assumptions:
gικ; l = 0, γιk ; l = A ιFkl, (153 ) F = − F , kl lk
where Aι is the preferred direction and Fkl an arbitrary 2-form, later to be interpreted as the electromagnetic field tensor. From them A σ;l = γσkFlk follows. They also noted that a symmetric tensor F kl could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V4 into V5.

Einstein and Mayer introduced what they called “Fünferkrümmung” (5-curvature) via the three-index symbol given above by

P σιkl = ∂kΓιlσ− ∂lΓ ιkσ + Γτk σΓιlτ− Γ τlσΓ ιkτ. (154 )
It is related to the Riemannian curvature r R mlk of V4 by
σ r r r P ιkl γ σm = A ι(Fmk;l − Fml;k) + γιr(R mlk + FmkF l − FmlF k ), (155 )
σ r r P ιklAσ = γιr(Fk ; l − F l ; k). (156 )
From (154View Equation), by transvection with τk γ, the 5-curvature itself appears:
P τιkl = γιrAτ(F k;rl − F l;rk) + γ τrA ι(Frk;l − Frl;k) + γιrγτs(Rrslk + FskF lr− FslF kr). (157 )
By contraction, r τ P ιk := γτ P ιrk and ιk P := γ Pιk. Two new quantities are introduced:
  1. Uιk := Pιk − 14(P + R ), where R is the Ricci scalar of the Riemannian curvature tensor of V4, and
  2. the tensor Nklm := F {kl;m }140.

It turns out that P = R − FkpF kp.

The field equations put forward in the paper by Einstein and Mayer now are

Uιk = 0, Nklm = 0, (158 )
and turn out to be exactly the Einstein–Maxwell vacuum field equations. Thus, by another formalism, Einstein and Mayer rederived what Klein had obtained in his first paper on Kaluza’s theory [185Jump To The Next Citation Point].

The authors’ conclusion is:

“From the theory presented here, the equations for the gravitational and the electromagnetic fields follow effortlessly by a unifying method; however, up to now, [the theory] does not bring any understanding for the way corpuscles are built, nor for the facts comprised by quantum theory.”141View original Quote ([107Jump To The Next Citation Point], p. 19)

After this paper Einstein wrote to Ehrenfest in a letter of 17 September 1931 that this theory “in my opinion definitively solves the problem in the macroscopic domain” ([241Jump To The Next Citation Point], p. 333). Also, in a lecture given on 14 October 1931 in the Physics Institute of the University of Wien, he still was proud of the 5-vector approach. In talking about the failed endeavours to reconcile classical field theory and quantum theory (“a cemetery of buried hopes”) he is reported to have said:

“Since 1928 I also tried to find a bridge, yet left that road again. However, following an idea half of which came from myself and half from my collaborator, Prof. Dr. Mayer, a startlingly simple construction became successful. [...] According to my and Mayer’s opinion, the fifth dimension will not show up. [...] according to which relationships between a hypothetical five-dimensional space and the four-dimensional can be obtained. In this way, we succeeded to recognise the gravitational and electromagnetic fields as a logical unity.”142View original Quote [96]

In his letter to Besso of 30 October 1931, Einstein seemed intrigued by the mathematics used in his paper with Mayer, but not enthusiastic about the physical content of this projective formulation of Kaluza’s unitary field theory:

“The only result of our investigation is the unification of gravitation and electricity, whereby the equations for the latter are just Maxwell’s equations for empty space. Hence, no physical progress is made, [if at all] at most only in the sense that one can see that Maxwell’s equations are not just first approximations but appear on as good a rational foundation as the gravitational equations of empty space. Electrical and mass-density are non-existent; here, splendour ends; perhaps this already belongs to the quantum problem, which up to now is unattainable from the point of view of field [theory] (in the same way as relativity is from the point of view of quantum mechanics). The witty point is the introduction of 5-vectors σ a in fourdimensional space, which are bound to space by a linear mechanism. Let as be the 4-vector belonging to aσ; then such a relation as = γsσa σ obtains. In the theory equations are meaningful which hold independently of the special relationship generated by γs σ. Infinitesimal transport of σ a in fourdimensional space is defined, likewise the corresponding 5-curvature from which spring the field equations.”143View original Quote ([99Jump To The Next Citation Point], pp. 274–275)

In his report for the Macy-Foundation, which appeared in Science on the very same day in October 1931, Einstein had to be more optimistic:

“This theory does not yet contain the conclusions of the quantum theory. It furnishes, however, clues to a natural development, from which we may anticipate further developments in this direction. In any event, the results thus far obtained represent a definite advance in knowledge of the structure of physical space.” ([94Jump To The Next Citation Point], p. 439)

Unfortunately, as in the case of his previous papers on Kaluza’s theory, Einstein came in only second: Veblen had already worked on projective geometry and projective connections for a couple of years [374376375]. One year prior to Einstein’s and Mayer’s publication, with his student Hoffmann, he had suggested an application to physics equivalent to the Kaluza–Klein theory [381Jump To The Next Citation Point163Jump To The Next Citation Point]. However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:

“But these authors choose a formulation that, due to an unnecessary specialisation of the coordinate system, prefers the fifth coordinate relative to the remaining [coordinates] in much the same way as this had happened in Kaluza–Klein theory by means of the cylinder condition [...].”144View original Quote ([249Jump To The Next Citation Point], p. 307)

By using the idea that an affine (n + 1)-space can be represented by a projective n-space [413], Veblen and Hoffmann avoided the five dimensions of Kaluza: There is a one-to-one correspondence between the points of space-time and a certain congruence of curves in a five-dimensional space for which the fifth coordinate is the curves’ parameter, while the coordinates of space-time are fixed. The five-dimensional space is just a mathematical device to represent the events (points) of space-time by these curves. Geometrically, the theory of Veblen and Hoffmann is more transparent and also more general than Einstein and Mayer’s: It can house the additional scalar field inherent in Kaluza’s original approach. Thus, Veblen and Hoffmann also gained the Klein–Gordon equation in curved space, i.e., an equation with the Ricci scalar R appearing besides its mass term. Interestingly, the curvature term reads as 527R ([381], p. 821). In his note, Hoffmann generalised the formalism such as to include Dirac’s equations (without gravitation), although some technical difficulties remained. Nevertheless, Hoffman remained optimistic:

“There is thus a possibility that the complete system will constitute an improved unification within the relativity theory of the gravitational, electromagnetic and quantum aspects of the field.” ([163], p. 89)

In his book, Veblen emphasised

“[...] that our theory starts from a physical and geometrical point of view totally different from KALUZA’s. In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly four-dimensional.”145View original Quote [379]

Shortly after Einstein’s and Mayer’s paper had appeared, Schouten and van Dantzig also proved that the 5-vector formalism of this paper can be brought into a projective form [314].

In a second note, Einstein and Mayer extended the 5-vector-formalism to include Maxwell’s equations with a non-vanishing current density [109]. Of the three basic assumptions of the previous paper, the second had to be given up. The expression in the middle of Equation (153View Equation) is replaced by

ι ι ιr γ k;l = A Fkl + γ Vrlk, (159 )
where, again, Fkl = − Flk, and the new Vrlk = Vrkl are arbitrary tensors. The field equations were set up according to the method of the first paper; now the 5-curvature scalar was P = R − F F kp − V V rpq. kp rqp It also turned out that rpq lrpq V = 𝜖 ϕl with ∂ϕ- ϕl = ∂xl, i.e., that the introduction of Vrpq brought only one additional variable. The electric current density became ∼ VprqF rq.

In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:

“We note that Mr. Cartan, in a general and very illuminating investigation, has analysed more deeply the property of systems of differential equations that has been termed by us ‘compatibility’ in this paper and in previous papers.”146View original Quote [37Jump To The Next Citation Point]

At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on projective geometry [361362360]. He used homogeneous coordinates X α, with α = 1,...,5, and the invariant g X αX β αβ, and introduced projectors and covariant differentiation (cf. Section 2.1.3). Together with him, Schouten wrote a series of papers on projective geometry as the basis of a unified field theory [316Jump To The Next Citation Point317Jump To The Next Citation Point315Jump To The Next Citation Point318Jump To The Next Citation Point]147, which, according to Pauli, combine

“all advantages of the formulations of Kaluza–Klein and EinsteinMayer while avoiding all their disadvantages.” ([249Jump To The Next Citation Point], p. 307)

Both the EinsteinMayer theory and Veblen and Hoffmann’s approach turned out to be subcases of the more general scheme of Schouten and van Dantzig intending

“to give a unification of general relativity not only with Maxwell’s electromagnetic theory but also with Schrödinger’s and Dirac’s theory of material waves.” ([318Jump To The Next Citation Point], p. 271)

In this paper ([318], p. 311, Figure 2), we find an early graphical representation of the parametrised set of all possible theories of a kind148. The formalism of Schouten and van Dantzig allows for taking the additional dimension to be timelike; in their physical applications the metric of space- time is taken as a Lorentz metric; torsion is also included in their geometry.

Pauli, with his student J. Solomon149, generalised Klein, and Einstein and Mayer by allowing for an arbitrary signature in an investigation concerning “the form that take Dirac’s equations in the unitary theory of Einstein and Mayer150View original Quote [253Jump To The Next Citation Point]. In a note added after proofreading, the authors showed that they had noted Schouten and Dantzig’s papers [316317Jump To The Next Citation Point]. The authors pointed out that

“[...] even in the absence of gravitation we must pay attention to a difference between Dirac’s equation in the theory of Einstein and Mayer, and Dirac’s equation as it is written out, usually.”151View original Quote ([253Jump To The Next Citation Point], p. 458)

The second order wave equation iterated from their form of Dirac’s equation, besides the spin term contained a curvature term − 1R 4, with the numerical factor different from Veblen’s and Hoffmann’s. In a sequel to this publication, Pauli and Solomon corrected an error:

“We examine from a general point of view the theory of spinors in a five-dimensional space. Then we discuss the form of the energy-momentum tensor and of the current vector in the theory of EinsteinMayer.[...] Unfortunately, it turned out that the considerations of §in the first part are marred by a calculational error…This has made it necessary to introduce a new expression for the energy-momentum tensor and [...] likewise for the current vector [...].”152View original Quote ([254], p. 582)

In the California Institute of Technology, Einstein’s and Mayer’s new mathematical technique found an attentive reader as well; A. D. Michal and his co-author generalised the EinsteinMayer 5-vector-formalism:

“The geometry considered by Einstein and Mayer in their ‘Unified field theory’ leads to the consideration of an n-dimensional Riemannian space Vn with a metric tensor gij, to each point of which is associated an m-dimensional linear vector space Vm, (m > n), for which vector spaces a general linear connection is defined. For the general case (m − n ⁄= 1) we find that the calculation of the m − n ‘exceptional directions’ is not unique, and that an additional postulate on the linear connection is necessary. Several of the new theorems give new results even for n = 4, m = 5, the EinsteinMayer case.” [228]

Michal had come from Cartan and Schouten’s papers on group manifolds and the distant parallelisms defined on them [227]. H. P. Robertson found a new way of applying distant parallelism: He studied groups of motion admitted by such spaces, e.g., by Einstein’s and Mayer’s spherically symmetric exact solution [282] (cf. Section 6.4.3).

Cartan wrote a paper on the EinsteinMayer theory as well ([39], an article published only posthumously) in which he showed that this could be interpreted as a five-dimensional flat geometry with torsion, in which space-time is embedded as a totally geodesic subspace.

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