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 , 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 [81, 82]. He came too late: Klein had already shown the same before . 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:
He then referred to the papers of Klein [185, 186] and to “Fochs Arbeit” which is a paper by Fock 1926 , 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 . 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 , even in first approximation the new field will appear in the four-dimensional Einstein–Maxwell equations (, 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 . 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 into , 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 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.” (, p. 145)
“The importance of the additional coordinate parameter 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.”134 (, 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 . Neither Mandel nor Fock used the “sharpened cylinder condition” (110).
A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation. 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 , taken as periodic with period , is understood:
In his papers, Einstein took over Klein’s condition , 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 (109):
“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 (, dependence of the components of the fundamental [tensor] of is admitted) that the characteristic properties of the field equations are then conserved as well, i.e., they keep the form
Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail .
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 with the electromagnetic 4-potential, had failed. Beyond incredibly complicated field equations nothing much had been gained . Reichenbächer’s ansatz is well founded: As we have seen in Section 4.2, due to the violation of covariance in , 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 , but not without getting into a squabble with Klein, who felt misunderstood [188, 47]. 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 .
In 1929 Mandel tried to “axiomatise” the five-dimensional theory: His two axioms were the cylinder condition (109) and its sharpening, Equation (110). He then weakened the second assumption by assuming that “an objective meaning does not rest in the 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 .
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 (148) with where is the metric on space-time following from the 5-metric by . For he took the Reissner–Nordström solution and did not obtain a discrete spectrum . He continued his approach by trying to solve Schrödinger’s wave equation 
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 .
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 [216, 217, 218, 219].
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” :
In the eyes of Einstein, by avoiding the artificial cylinder condition (109), 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 is affixed to space-time , and “mixed” tensors are defined linking the tangent space of space-time with a such that138. Indices are raised and lowered with the metrics of or , respectively. There exists a “preferred direction of ” defined by , and which is the normal to a “preferred plane” 139. A consequence then is  and Bortolotti’s  papers. The autoparallels of lead to the exact equations of motion of a charged particle, not the geodesics of . Einstein and Mayer made three basic assumptions: symmetric tensor could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of into .
Einstein and Mayer introduced what they called “Fünferkrümmung” (5-curvature) via the three-index symbol given above by
It turns out that .
The field equations put forward in the paper by Einstein and Mayer now are.
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.”141 (, 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” (, 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.”142 
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 in fourdimensional space, which are bound to space by a linear mechanism. Let be the 4-vector belonging to ; then such a relation obtains. In the theory equations are meaningful which hold independently of the special relationship generated by . Infinitesimal transport of in fourdimensional space is defined, likewise the corresponding 5-curvature from which spring the field equations.”143 (, 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.” (, 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 [374, 376, 375]. 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 [381, 163]. 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 [...].”144 (, p. 307)
“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.” (, 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.”145 
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 .
In a second note, Einstein and Mayer extended the 5-vector-formalism to include Maxwell’s equations with a non-vanishing current density . Of the three basic assumptions of the previous paper, the second had to be given up. The expression in the middle of Equation (153) is replaced by
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.”146 
At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on projective geometry [361, 362, 360]. He used homogeneous coordinates , with , and the invariant , 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 [316, 317, 315, 318]147, which, according to Pauli, combine
Both the Einstein–Mayer 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.” (, p. 271)
In this paper (, 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 Mayer”150 . In a note added after proofreading, the authors showed that they had noted Schouten and Dantzig’s papers [316, 317]. 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.”151 (, p. 458)
The second order wave equation iterated from their form of Dirac’s equation, besides the spin term contained a curvature term , 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 Einstein–Mayer.[...] 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 [...].”152 (, 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 Einstein–Mayer 5-vector-formalism:
“The geometry considered by Einstein and Mayer in their ‘Unified field theory’ leads to the consideration of an -dimensional Riemannian space with a metric tensor , to each point of which is associated an -dimensional linear vector space , (), for which vector spaces a general linear connection is defined. For the general case () we find that the calculation of the ‘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 , , the Einstein–Mayer case.” 
Michal had come from Cartan and Schouten’s papers on group manifolds and the distant parallelisms defined on them . 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  (cf. Section 6.4.3).
Cartan wrote a paper on the Einstein–Mayer theory as well (, 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.
© Max Planck Society and the author(s)