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Selected Papers on Teleparallelism Ii SELECTED PAPERS ON TELEPARALLELISM Edited and translated by D. H. Delphenich Table of contents Page Introduction ……………………………………………………………………… 1 1. The unification of gravitation and electromagnetism 1 2. The geometry of parallelizable manifold 7 3. The field equations 20 4. The topology of parallelizability 24 5. Teleparallelism and the Dirac equation 28 6. Singular teleparallelism 29 References ……………………………………………………………………….. 33 Translations and time line 1928: A. Einstein, “Riemannian geometry, while maintaining the notion of teleparallelism ,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 217- 221………………………………………………………………………………. 35 (Received on June 7) A. Einstein, “A new possibility for a unified field theory of gravitation and electromagnetism” Sitzber. Preuss. Akad. Wiss. 17 (1928), 224-227………… 42 (Received on June 14) R. Weitzenböck, “Differential invariants in EINSTEIN’s theory of teleparallelism,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 466-474……………… 46 (Received on Oct 18) 1929: E. Bortolotti , “ Stars of congruences and absolute parallelism: Geometric basis for a recent theory of Einstein ,” Rend. Reale Acc. dei Lincei 9 (1929), 530- 538...…………………………………………………………………………….. 56 R. Zaycoff, “On the foundations of a new field theory of A. Einstein,” Zeit. Phys. 53 (1929), 719-728…………………………………………………............ 64 (Received on January 13) Hans Reichenbach, “On the classification of the new Einstein Ansatz on gravitation and electricity,” Zeit. Phys. 53 (1929), 683-689…………………….. 76 (Received on January 22) Selected papers on teleparallelism ii A. Einstein, “On unified field theory,” Sitzber. Preuss. Akad. Wiss. 18 (1929), 2-7……………………………………………………………………………….. 82 (Received on Jan 30) R. Zaycoff, “On the foundations of a new field theory of A. Einstein; (Second part),” Zeit. Phys. 54 (1929), 590-593…………………………………………… 89 (Received on March 4) R. Zaycoff, “On the foundations of a new field theory of A. Einstein (Third part),” Zeit. Phys. 54 (1929), 738-740…………………………………………… 94 (Received on March 11) R. Zaycoff, “On the most recent formulation of Einstein’s unified field theory,” Zeit. Phys. 56 (1929), 717-726…………………………………………………... 97 (Received on June 6) R. Zaycoff, “Derivation of the Dirac-Whittaker wave equations from Einstein’s unified field theory,” Zeit. Phys. 56 (1929), 862-864……………………………. 108 (Received on June 21) A. Einstein, “Unified field theory and HAMILTON’s principle,” Sitzber. Preuss. Akad. Wiss. 18 (1929), 156-159……………………………………… 112 A. Einstein, “On the unified field theory that is based upon the Riemann metric and absolute parallelism,” Math. Ann. 102 (1929), 685-697…………………… 117 (Received on Aug 19) E. Cartan, “Historical survey of the notion of absolute parallelism,” Math. Ann. 102 (1929), 698-706…………………………………………………………… 130 R. Zaycoff, “Teleparallelism and wave mechanics. I,” Zeit. Phys. 58 (1929), 833-840…………………………………………………………………………... 138 (Received on October 21) R. Zaycoff, “Teleparallelism and wave mechanics. II,” Zeit. Phys. 59 (1929), 110-113…………………………………………………………………………... 145 (Received on November 4) A. Einstein, “Unitary theory of the physical field,” Ann. de l’Inst. Henri. Poincaré 1 (1930), 1-24…………………………………………………………. 150 (talk given November) A. Einstein, “The compatibility of the field equations in unified field theory,” Sitzber. Preuss. Akad. Wiss. 19 (1930), 18-23…………………….……………. 170 (Received on December 12) iii Table of Contents 1930 : A. Einstein and W. Mayer, “Two rigorous static solutions to the field equations of unified field theory,” Sitzber. Preuss. Akad. Wiss. 19 (1930), 110-120…….. 177 (Received on March 11) G. C. McVittie, “Solutions with axial symmetry of Einstein’s equations of teleparallelism,” Edinburgh Math. Soc. Proc. (2) 2 (1930), 140-150……………. 189 (Received on June 9) A. Einstein, “On the theory of spaces with a RIEMANN metric and teleparallelism,” Sitzber. Preuss. Akad. Wiss. 19 (1930), 401-402…………….... 200 Post-Einstein era: E. Cartan, “Absolute parallelism and unitary field theory,” Rev. Metaphys. Morale, 38 (1931), 13–28. Oeuvres Complètes, tome III/2, pp. 1167–1185……. 202 E. Stiefel, “Direction fields and teleparallelism in n-dimensional manifolds,” Comm. Math. Helv. 25 (1935) 305 – 353………………………………………... 212 Introduction 1. The unification of gravitation and electromagnetism. One of the articles of faith for many scientists (or at least the ones who regard science as a natural philosophy) is what one might call the principle of “maximal elegance.” That is, if natural law is truly worthy of reverence then that would be because in its purest form it must admit a truly elegant statement. Whenever scientists are confronted with a disorganized set of natural phenomena, their natural instinct it to organize them, and ideally one hopes to find some classification scheme that begins with the smallest number of first principles. For instance, consider the ongoing quest to understand the nature of matter. One of the earliest postulates about matter was the atomic hypothesis of Democritus, who felt that the reduction of matter to smaller pieces had to terminate after a finite number of steps. To him, the ultimate irreducible constituents would be “atoms,” which would then essentially “generate” the more complex states of matter by some process of combination. Of course, this concept eventually led to the periodic table of the elements, which once more initially appeared to be a complex and disorganized set of rules for atoms until quantum mechanics managed to find a more fundamental first principle that made things simple again. Similarly, there was once a time when electricity and magnetism seemed to be as distinct as lightning and lodestones. However, between the efforts of mostly Michael Faraday and James Clerk Maxwell, not only was it shown that electricity and magnetism were only two facets of a more general concept – viz., the electromagnetic field – but that one could also account for optical phenomena in the process by attributing electromagnetic waves to light. Since there are many other examples of how progress in science has often been associated with reducing the complexity of first principles by unifying disparate natural phenomena, once Einstein had formulated his theory of gravitation as a manifestation of spacetime geometry, he became convinced that the story did not stop there. His final – albeit, unsuccessful − quest in theoretical physics was to unify his theory of gravitation with Maxwell’s theory of electromagnetism by finding some more general – most likely, geometric – field on the spacetime manifold and a set of field equations for it that would subsume both the Einstein equations for gravitation and the Maxwell equations for electromagnetism in some limiting approximation. This problem is what we are calling the Einstein-Maxwell unification problem. Here, one must clearly distinguish between the unification of two field theories and their mere concatenation . In the latter case, all that one has really accomplished is to construct essentially a Cartesian product of the two theories, and in particular, the resulting field equations for the fields that are to be unified say nothing new about either. What one hopes for in a unification of field theories is a general set of field equations for the unified field that contains some sort of coupling between the two fields that was not present in the individual field theories. The best example of this situation is the fact that in Maxwell’s theory of electromagnetism, in addition to containing the equations of electrostatics and magnetostatics as special cases, the full set of Maxwell equations contains the far-reaching coupling of the electric and magnetic fields by electromagnetic induction. Consequently, one hopes that if gravitation and electromagnetism are unified 2 Selected papers on teleparallelism in a similar manner then perhaps there might be some sort of induction process whereby electromagnetism (under some unspecified circumstances) might induce a gravitational field and vice versa. The fact that none of the attempts at solving the Einstein-Maxwell unification problem contained such a mechanism was always regarded as a symptom that the unification was still just a glorified concatenation. Of course, one must objectively accept that there is nothing to say that a unification of two field theories exists, or at least in the form that one is expecting. Thus, one must treat the existence of unification as basically a conjecture that one is testing, and as such, a conjecture that might prove incorrect. Einstein made many attempts to solve the Einstein-Maxwell unification problem, but one of the common features that they had was the belief that first one had to increase the degrees of freedom in the unified field to something that equaled at least the sum of the other two degrees of freedom. Now, the spacetime metric tensor field g has components gµν that are symmetric 4 ×4 real matrices, and therefore represents ten degrees of freedom. By comparison, the electromagnetic field strength 2-form F has components Fµν that are anti-symmetric 4 ×4 matrices, and therefore represent six degrees of freedom. Thus, the unified field should probably include at least sixteen degrees of freedom. Since the vector space M(4; R) of 4 ×4 real matrices is itself sixteen-dimensional, one obvious first place to look for unification would be in the most general elements of M(4; R), or since the components of g are invertible, perhaps just the invertible elements, which then define GL (n). In fact, Einstein considered both possibilities,
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