I N T R O D U C T I O N T O V O L U M E 1 6 l x v approach was the notion of a field of mutually orthogonal, normal vectors, defined on the space-time manifold. This was a so-called n-Bein-Feld, or, in more modern terminology, for n = 4, a field of tetrads. Such a theory admits the definition of a natural notion of distant parallelism by identifying vectors on this orthonormal frame field. Two vectors at distant points of the manifold are parallel, by definition, if they are represented by the same vector of the orthonormal frames at the respec- tive points. The manifold also carries a Riemannian metric, which can be expressed in terms of the tetrad field. Since the tetrad field determines the metric field, but not the other way around, the tetrads provide more degrees of freedom, which Einstein hoped could be put to use to provide a representation of the electromagnetic field. At Einstein’s request, on 7 June 1928, Max Planck presented a brief note on this “Riemannian Geometry Retaining the Concept of Distant Parallelism” to the Prus- sian Academy for publication in its Proceedings (Doc. 216). Since Einstein was not sure at the time whether the notion of Fernparallelismus, that is, distant parallelism or teleparallelism, and its associated geometric concepts, were known in the math- ematical literature, he asked Planck to inquire among his mathematician colleagues whether any of this was known before submitting the paper for publication. Planck did not find the occasion to do as requested but nevertheless submitted the paper (Doc. 218). Only a week later, Einstein realized how to put the geometry of distant parallel- ism to use for his project of a unified theory of both the gravitational and the elec- tromagnetic fields. The idea was to postulate a variational principle for an invariant action integral that depended on the tetrad field as the dynamical variable. From this perspective, the problem presented itself as a fairly well-defined mathematical problem, but posed difficulties of interpretation in terms of physical concepts. From the mathematical side, the required invariance of the variational integral created a clearly defined problem. One needed to identify all possible in- variants that can be constructed from the tetrads as well as a combination of these invariants that would be suitable as a Lagrangian for the variational integral. Second, variation with respect to the tetrad field would produce differential equa- tions that had to be associated with the known field equations of gravitation and electromagnetism in certain limiting cases. Third, solutions for the differential equations had to be found. Finally, Einstein later would become interested in find- ing identities that would be satisfied by the tetrads by virtue of general covariance, or that might be postulated to derive field equations. As far as the physical inter- pretation was concerned, the metric field would take on its old role, as in the general theory, namely corresponding to the gravitational field. But the electro- magnetic field also had to be identified with quantities occurring in the geometric framework. As a first step, Einstein identified the relevant possible invariants to be con- structed from the tetrads. He also realized that in addition to the possibility of

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l x i v I N T R O D U C T I O N T O V O L U M E 1 6 to generalize the schema in such a way that it would not rely on a counterpart of the equilibrium condition that he and Grommer had employed. And third, he hoped that he would derive results beyond just a derivation of the classical Lorentz force law, results that would say something about the quantum nature of matter. Yet again, we mostly learn of these hopes from deleted passages in the manu- script that were not included in the published paper and from Einstein’s correspondence after the paper was completed. The paper’s conclusion emphasizes only the importance of the classical result, namely, that the equations of motion of a charged particle subject to both gravita- tional and electromagnetic fields had now been derived from the Einstein-Maxwell equations rather than being assumed independently, and that higher orders in the approximation schema might bring about constraints on the motion of the charged particle. The manuscript, however, was much more explicit about the possible link of these considerations to quantum theory. Here Einstein wrote that the constraints that might occur at higher orders could correspond to “quantum conditions.” One such quantum condition can be found in another passage of the introduction to the manuscript, where Einstein made, and later deleted, the announcement that the present paper would deliver a generalization of the equations of motion as derived from the field equations, a generalization that would make possible the attribution of a characteristic frequency and a characteristic magnetic field to the electron.[42] In the published paper, however, Einstein ends up with classical equations of mo- tion, without any hint of implied quantum conditions. He voiced satisfaction with the positive result but also disappointment with its purely classical nature in a letter to Paul Ehrenfest two months later, on 21 January 1928. He noted that in order to proceed any further, he was turning his attention back to the Kaluza-Klein theory, proclaiming: “Long live the fifth dimension” (Doc. 137). Einstein had thus returned to modifying the field equations themselves, hoping, as before, to find field equations that would describe the gravitational and the elec- tromagnetic fields as a truly unified field. He also hoped that they would allow a solution that could be interpreted as an electron. However, he did not stay focused on the five-dimensional approach to unified field theory for long. Neither did he return to it later, when engaged in correspondence with Heinrich Mandel and Raschko Zaycoff, both of whom had written to Einstein about results in five- dimensional theory. Whereas his work in this domain had thus far relied on gener- alizations of pseudo-Riemannian geometry introduced by Weyl, Kaluza, and Eddington, Einstein in 1929 began exploring a new approach to unified field theory that would keep him engaged for the next few years. At some point in May 1929, while convalescing at home in Berlin,[43] Einstein had an idea for what he thought was “an entirely new way of realizing the general theory of relativity and that may be groundbreaking” (Doc. 217). Key to this new

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l x v i I N T R O D U C T I O N T O V O L U M E 1 6 constructing a metric-compatible Levi-Civita connection from the metric, as well as the associated notion of parallel transport, the tetrad field allowed the definition of another connection with its notion of parallelism. In contrast to the Levi-Civita connection, the teleparallel connection is asymmetric and describes a geometry that has vanishing Riemann curvature. Instead, it is characterized by the nonvanishing of a tensorial quantity constructed from the teleparallel connection that is now known as the torsion tensor. Taking the mathematical expression of torsion, a third-rank tensor, Einstein tentatively identified its contraction with the electromagnetic four-potential. And settling on what seemed to be the simplest in- variant to be taken as a basis for a tentative field theory, Einstein succeeded in de- riving, to first approximation in the field components, both the gravitational field equations of general relativity as well as an equivalent version of the Maxwell equations. Again, a brief note on this work was presented by Planck to the Academy, on 14 June 1928 (see Abs. 593), and was published in July under the title “New Pos- sibility for a Unified Field Theory of Gravitation and Electricity” (Doc. 219). These two notes mark the beginning of a search for a unified field theory in this teleparallel framework that would preoccupy Einstein for the next two or three years, beyond the end of the time period covered by this volume. The elaboration of the implications and consequences of the framework of dis- tant parallelism for a unified theory was explored by Einstein in intense coopera- tion with a number of correspondents and coworkers who either reacted to the new approach or carried out calculations at Einstein’s request. Early correspondents include a student named Curt Schwarz (Doc. 229), as well as Einstein’s longtime collaborator Jakob Grommer. The latter soon pointed out to Einstein some difficul- ties for the physical interpretation of the geometric framework caused by identify- ing the contracted torsion tensor with the potential vector (Doc. 232). Most critical of these difficulties was Grommer’s warning that no solution of the full equations may exist for every solution of the linearized equation (Doc. 91). The most intense correspondence on the teleparallel approach in this phase of the project took place with the mathematician Chaim Herman Müntz. Müntz was born in Lodz, Poland, in 1884.[44] His family was bourgeois and Jewish, though not religious. After studying mathematics, sciences, and philosophy in Berlin under Georg Frobenius and Hermann Amandus Schwarz, among others, Müntz obtained his Ph.D. with a dissertation on the boundary-value problem of partial differential equations of minimal surfaces. At the time of the collaboration with Einstein, Müntz was living again in Berlin and, in spite of having published a number of re- search papers in various mathematical fields, had not obtained a Habilitation and

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