Abstract
In this paper we formulate Einstein's gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime.
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References
R. K. Sachs and H. Wu,General Relativity for Mathematicians (Springer, New York, 1977).
W. A. Rodrigues, Jr., and M. A. Faria-Rosa, “The meaning of time in the theory of relativity and ‘Einstein's later view of the twin paradox’,”Found. Phys. 19, 705–724 (1989).
B. O'Neill,Elementary Differential Geometry (Academic Press, New York, 1966).
J. K. Beem and P. E. Ehrlich,Global Lorentzian Geometry (Marcel Dekker, New York, 1981).
H. Poincaré, “Sur la dynamique de l'electron”,Rend. C. Math. Palermo XXI, 129–175 (1906).
N. Rosen, “General relativity on flat space I”,Phys. Rev. 57, 147–150 (1940).
N. Rosen, “Flat-space metric in general relativity,”Ann. Phys. 22, 1–11 (1963).
A. Papapetrou, “Eine neue Theorie des Gravitationsfeldes”, Part I:Math. Nachr. 12, 129–141 (1954); Part II:Math. Nachr. 12, 143–154 (1954).
R. H. Kraichnan, “Special-relativistic derivation of generally covariant gravitational theory,”Phys. Rev. 98, 1118–1122 (1955).
S. N. Gupta, “Einstein's and other theories of gravitation,”Rev. Mod. Phys. 29, 334–336 (1957).
R. P. Feynman, “Quantum theory of gravitation,”Acta Phys. Pol. 24, 697–722 (1963).
F. J. Belinfante and J. C. Swihart, “Phenomenological linear theory of gravitation,” Part I:Ann. Phys. (Leipzig) 1, 168–195 (1957); Part II:Ann. Phys. (Leipzig) 1, 196–212 (1957); Part III:Ann. Phys. (Leipzig) 2, 81–99 (1957).
V. I. Ogievetsky and I. V. Polubarinon, “Interacting field of spin 2 and the Einstein equations,”Ann. Phys. (Leipzig) 35, 167–208 (1965).
S. Weinberg, “Photons and gravitons in perturbation theory: derivation of Maxwell's and Einstein's equations,”Phys. Rev. B 138, 988–1002 (1965).
Ya. B. Zel'dovich and I. D. Novikov,The Theory of Gravitation and the Evolution of Stars (Nauka, Moscow, 1971), p. 87.
L. P. Grishchuk, A. N. Petrov, and A. D. Popova, “Exact theory of (Einstein) gravitational field in an arbitrary background spacetime,”Commun. Math. Phys. 94, 379–396 (1984).
A. A. Logunov and M. A. Mestvirishvili, “Relativistic theory of gravitation,”Found. Phys. 16, 1–26 (1986).
A. A. Logunov, Yu. M. Loskutov, and M. A. Mestvirishvili, “Relativistic theory of gravity,”Int. J. Mod. Phys. 43, 2067–2099 (1988).
Ya. B. Zel'dovich and L. P. Grishchuk, “Gravitation, the general theory of relativity and alternative theories,”Sov. Phys. Usp. 29, 780–787 (1976).
D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966, 1987, 1992).
Q. A. G. de Souza and W. A. Rodrigues, Jr., “The Dirac operator and the structure of Riemann-Cartan-Weyl spaces,” submitted for publication, 1993.
A. Einstein, “Hamiltonsches Prinzip und allgemeine Relativitätstheorie,”Sitzungsber. Preuss. Akad. Wiss. (Berlin), 1111–1116 (1916).
M. Ferraris and J. Kijowski, “On the equivalence of relativistic theories of gravitation,”Gen. Relativ. Gravit. 14, 165–180 (1982).
W. Graf, “Differential forms as spinors,”Ann. Inst. H. Poincaré XXIX, 85–109 (1978).
R. Ablamowicz, P. Lounesto, and J. Maks, “Conference reports: Second workshop on Clifford algebra and their applications in mathematical physics,”Found. Phys. 21, 735–748 (1991).
P. Lounesto, “Clifford algebras and Hestenes spinors,”Found. Phys. 23, 1203–1237 (1993).
V. L. Figueiredo, E. C. Oliveira, and W. A. Rodrigues, Jr., “Covariant, algebraic and operator spinors,”Int. J Theor. Phys. 29, 371–395 (1990).
V. L. Figueiredo, E. C. Oliveira, and W. A. Rodrigues, Jr., “Clifford algebras and the hidden geometrical nature of spinors,”Algebras, Groups and Geometries 7, 153–198 (1990).
D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984, 1987).
W. A. Rodrigues, Jr., and V. L. Figueiredo, “Real spin Clifford bundle and the spinor structure of spacetime,”Int. J. Theor. Phys. 29, 413–424 (1990).
W. A. Rodrigues, Jr., and E. C. Oliveira, “Dirac and Maxwell equations in the Clifford and spin-Clifford bundles,”Int. J. Theor. Phys. 29, 397–412 (1990).
A. Maia, Jr., E. Recami, M. A. Faria-Rosa, and W. A. Rodrigues, Jr., “Magnetic monopoles without strings in the Kähler-Clifford algebra bundle: a geometrical interpretation,”J. Math. Phys. 31, 502–505 (1990).
A. Crumeyrolle,Orthogonal and Symplectic Clifford Algebras, Spinor Structures (Kluwer, Dordrecht, 1990).
L. Sédov,Mécanique des Milieux Continus (Mir, Moscow, 1975).
P. Baekler, F. W. Hehl, and E. W. Mielke, “Nonmetricity and torsion: Facts and fancies in gauge approaches to gravity,” inProceedings 4th Marcel Grossmann Meeting on General Relativity, R. Ruffini, (North-Holland, Amsterdam, 1986), pp. 277–316.
H. Kleinert,Gauge Fields in Condensed Matter, Vol. II (World Scientific, Singapore, 1989).
M. Göckeler and T. Schücker,Differential Geometry, Gauge Theories and Gravity (Cambridge University Press, Cambridge, 1988).
W. Thirring,Classical Field Theory (Springer, New York, 1980).
R. P. Wallner, “Notes on gauge theory and gravitation,”Acta Phys. Aust. 54, 165–189 (1982).
S. Weinberg,Gravitation and Cosmology (Wiley, New York, 1972).
L. P. Grishchuk and A. N. Petrov, “Closed worlds as gravitational fields,”Sov. Astron. Lett. 12, 179–181 (1986).
J. Schwinger,Particles, Sources and Fields (Addison-Wesley, Reading, Massachusetts, 1970).
W. A. Rodrigues, Jr., and Q. A. G. de Souza, “On the nature of the gravitational field,” submitted for publication, 1993.
J. F. Pommaret,Lie Pseudogroups and Mechanics (Gordon & Breach, New York, 1989).
D. Hestenes, “Spinor fields as distortions of space-time,”J. Math. Phys. 8, 1046–1050 (1967).
F. J. Chinea, “A Clifford algebra approach to general relativity,”Gen. Relativ. Gravit. 21, 21–44 (1989).
R. W. Tucker, “A Clifford calculus for physical field theories,” inClifford Algebras and Their Applications in Mathematical Physics, J. S. R. Chisholm and A. K. Common, eds. (Reidel, Dordrecht, 1986), pp. 177–179.
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Rodrigues, W.A., de Souza, Q.A.G. The Clifford bundle and the nature of the gravitational field. Found Phys 23, 1465–1490 (1993). https://doi.org/10.1007/BF01243942
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DOI: https://doi.org/10.1007/BF01243942