This paper shows how a gravitational field generated by a given energy-momentum
distribution (for all realistic cases) can be represented by distinct
geometrical structures (Lorentzian, teleparallel, and nonnull nonmetricity
spacetimes) or that we even can dispense all those geometrical structures and
simply represent the gravitational field as a field in Faraday’s sense living in
Minkowski spacetime. The explicit Lagrangian density for this theory is given,
and the field equations (which are Maxwell’s like equations) are shown to be
equivalent to Einstein's equations. Some examples are worked in detail in order
to convince the reader that the geometrical structure of a manifold (modulus
some topological constraints) is conventional as already emphasized by Poincaré
long ago, and thus the realization that there are distinct geometrical
representations (and a physical model related to a deformation of the continuum
supporting Minkowski spacetime) for any realistic gravitational field strongly
suggests that we must investigate the origin of its physical nature. We hope
that this paper will convince readers that this is indeed the case.
P. G. Bergmann. The Riddle of Gravitation. Dover, New York, 1992.
E. Cartan. A generalization of the Riemann curvature and the spaces with torsion. Comptes Rendus Acad. Sci., 174 (1922), 593–596.
Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick. Analysis, Manifolds and Physics. 2nd ed. North-Holland Publishing, Amsterdam, 1982.
Mathematical Reviews (MathSciNet): MR685274
C. J. S. Clarke. On the global isometric embedding of pseudo-Riemannian manifolds. Proc. Roy. Soc. London Ser. A, 314 (1970), 417–428.
Mathematical Reviews (MathSciNet): MR259813
B. Coll. A Universal law of gravitational deformation for general relativity. Proc. of the Spanish Relativistic Meeting, EREs, Salamanca, Spain, 1998.
V. C. de Andrade, H. I. Arcos, and J. G. Pereira. Torsion as an alternative to curvature in the description of gravitation. Preprint arXiv:gr-qc/0412034.
R. Debewer. Élie Cartan- Albert Einstein: Letters on Absolute Parallelism. Princeton University Press, Princeton, 1979.
Mathematical Reviews (MathSciNet): MR543192
M. P. do Carmo. Riemannian Geometry. Birkhhäuser, Boston, 1992.
A. S. Eddington. The Mathematical Theory of Relativity. 3rd ed. Chelsea, New York, 1975.
A. Einstein. Unified field theory of gravitation and electricity. Session Report of Prussian Acad. Sci., (1925), 414–419.
A. Einstein. Unified field theory of gravitation and electricity. Session Report of the Prussian Acad. Sci., (1928), 217–221.
A. Einstein. New possibility for a unified field theory of gravitation and electricity. Session Report of the Prussian Acad. Sci., (1928), 224–227.
A. Einstein. Unified field theory based on Riemannian metrics and distant parallelism. Math. Ann., 102 (1930), 685–697.
V. V. Fernández and W. A. Rodrigues Jr. Gravitation as a Plastic Distortion of the Lorentz Vacuum. Fundamental Theories of Physics 168, Springer, Heidelberg, 2010.
R. P. Feynman, F. B. Morinigo, and W. G. Wagner. Feynman Lectures on Gravitation. B. Hatfield (Ed.), Addison-Wesley, Reading, MA, 1995.
T. Frankel. The Geometry of Physics. Cambridge University Press, Cambridge, 1997.
R. Geroch. Spinor structure of space-times in general relativity. I. J. Math. Phys. 9 (1968), 1739–1744.
Mathematical Reviews (MathSciNet): MR234703
H. F. M. Goenner. On the history of unified field theories. Living Rev. Relativity 7 (2004), http://relativity.livingreviews.org/Articles/lrr-2004-2.
L. P. Grishchuk. Some uncomfortable thoughts on the nature of gravity, cosmology, and the early universe. To appear in Space Sciences Reviews, arXiv:0903.4395.
M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer. Geometric Theory of Generalized Functions with Applications to General Relativity. Mathematics and Its Applications 537, Kluwer, Dordrecht, 2001.
R. Hermann. Ricci and Levi-Civita's Tensor Analysis Paper. Translation, Comments and Additional Material. Lie Groups: History, Frontiers and Applications vol. II, Math. Sci. Press, Brookline, MA, 1975.
Mathematical Reviews (MathSciNet): MR472453
D. Hestenes and G. Sobczyk, Clifford Algebra to Geometrical Calculus. D. Reidel Publishing Company, Dordrecht, 1984.
Mathematical Reviews (MathSciNet): MR759340
L. D. Landau and E. M. Lifshitz. The Classical Theory of Fields. 4th revised ed. Pergamon Presss, New York, 1975.
Mathematical Reviews (MathSciNet): MR475345
R. B. Laughlin. A Different Universe: Reinventing Physics from the Bottom Down. Basic Books, New York, 2005.
A. A. Logunov and M. A. Mestvirishvili. The Relativistic Theory of Gravitation. Mir Publishers, Moscow, 1989.
E. A. Notte-Cuello and W. A. Rodrigues Jr. A Maxwell like formulation of gravitational theory in Minkowski spacetime. Int. J. Mod. Phys. D, 16 (2007), 1027–1042.
E. A. Notte-Cuello and W. A. Rodrigues Jr. Freud's identity of differential geometry, the Einstein-Hilbert equations and the vexatious problem of the energy-momentum conservation in GR. Adv. Appl. Clifford Algebr., 19 (2009), 113–145.
H. C. Ohanian and R. Ruffini. Gravitation and Spacetime. 2nd ed. W. W. Norton & Company, New York, 1994.
B. O'Neill. Elementary Differential Geometry. Academic Press, New York, 1966.
Mathematical Reviews (MathSciNet): MR203595
H. Poincaré. La Science et L' Hypothèse. Flamarion, Paris, 1902.
G. Ricci-Curbastro. Sulla Teoria degli Iperspazi. Rend. Acc. Lincei Serie IV, 232–237, 1895. Reprinted in: G. Ricci-Curbastro. Opere. vol. I, 431–437, Edizioni Cremonense, Roma, 1956.
G. Ricci-Curbastro. Dei sistemi di congruenze ortogonali in una varietà Qualunque. Mem. Acc. Lincei Serie 5 (vol. II) 276–322, 1896. Reprinted in: G. Ricci-Curbastro. Opere. Vol II, 1–61, Editore Cremonense, Roma, 1957.
G. Ricci and T. Levi-Civita. Méthodes de calcul différentiel absolu et leurs applications. Mathematische Annalen, 54 (1901), 125–201.
R. da Rocha and W. A. Rodrigues Jr. Gauge fixing in the Maxwell like gravitational theory in Minkowski spacetime and in the equivalent Lorentzian spacetime. Preprint arXiv:0806.4129.
W. A. Rodrigues Jr. and E. Capelas de Oliveira. A comment on the Twin paradox and the Hafele-keating experiment. Phys. Lett. A, 140 (1989), 479–484.
W. A. Rodrigues Jr. and E. Capelas de Oliveira. The Many Faces of Maxwell, Dirac and Einstein Equations. A Clifford Bundle Approach. Lecture Notes in Physics 722, Springer, Berlin, 2007.
W. A. Rodrigues Jr. and M. Sharif. Rotating frames in SRT: the Sagnac effect and related issues. Found. Phys., 31 (2001), 1767–1784.
W. A. Rodrigues Jr. and Q. A. G. Souza. An Ambiguous statement called `Tetrad Postulate' and the correct field equations satisfied by the Tetrad fields. Int. J. Mod. Phys D, 14 (2005), 2095–2150.
R. K. Sachs and H. Wu. General Relativity for Mathematicians. Graduate Texts in Mathematics 48, Springer-Verlag, New York, 1977.
Mathematical Reviews (MathSciNet): MR503498
E. L. Schücking. Einstein's apple and relativity's gravitational field. Preprint arXiv:0903.3768.
J. Schwinger. Particles, Sources and Fields: Vol. 1. Addison-Wesley, Reading, MA, 1970.
L. B. Szabados. Quasi-local energy-momentum and angular momentum in GR: a review article. Living Rev. Relativity, 7 (2004), 1–135.
W. E. Thirring. An alternative approach to the theory of gravitation. Ann. Phys., 16 (1961), 96–117.
Mathematical Reviews (MathSciNet): MR135564
A. Unzicker and T. Case. Translation of Einstein's attempt of a unified field theory with teleparallelism. Preprint arXiv:physics/0503046.
A. Unzicker. What can physics learn from continuum mechanics? Preprint arXiv:gr-qc/0011064v1.
A. Unzicker. Teleparallel space-time with defects yields geometrization of electrodynamics with quantized charges. Preprint arXiv:gr-qc/9612061v2.
J. G. Vargas. Geometrization of the Physics with teleparallelism. I. The classical interactions. Found. Phys. 22 (1992), 507–526.
J. G. Vargas, D. G. Torr, and A. Lecompte. Geometrization of the physics with teleparallelism. II. Towards a fully geometric Dirac equation. Found. Phys., 22 (1992), 527–547.
J. G. Vargas and D. G. Torr. Finslerian structures: the Cartan-Clifton method of the moving frame. J. Math. Phys., 34 (1993), 4898–4913.
J. G. Vargas and D. G. Torr. The cornerstone role of the torsion in Finslerian physical worlds. Gen. Relativity Gravitation, 27 (1995), 629–644.
J. G. Vargas and D. G. Torr. Is electromagnetic gravity control possible? In M. S. El-Genk (Ed.), AIP Conference Proceedings, Proceedings of the 2004 Space Technology and Applications International Forum (STAIF 2004), 1206–1213, 2004.
G. E. Volovik. The Universe in a Helium Droplet. Clarendon Press, Oxford, 2003,
J. R. Weeks. The Shape of Space. 2nd ed. Monographs and Textbooks in Pure and Applied Mathematics 249, Marcel Decker Inc., New York, 2002.
S. Weinberg. Gravitation and Cosmology. John Wiley & Sons, New York, 1972.
R. Weitzenböck. Differentialinvarianten in der Einsteinschen Theorie des Fernparallelismus. Sitzungsber. Preuss. Akad. Wiss., 26 (1928), 466–474.
M. Zorawski. Theorie Mathematiques des Dislocations. Dunod, Paris, 1967. }