Geometrization of the physics with teleparallelism. I. The classical interactions
- 78 Downloads
A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.
The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational contribution.
Nonlinear field equations for the torsion result. In a “low-energy” approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys.21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.
A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.
Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector.
KeywordsManifold Field Equation Dirac Equation Lorentz Force Bianchi Identity
Unable to display preview. Download preview PDF.
- 1.Y. H. Clifton,J. Math. Mech. 16, 569 (1966).Google Scholar
- 2.E. Cartan,C. R. Acad. Sci. (Paris) 174, 1104 (1922).Google Scholar
- 3.J. G. Vargas, D. G. Torr, and A. Lecompte,Found. Phys. 22, 527 (1992).Google Scholar
- 4.J. G. Vargas and D. G. Torr,Found. Phys. 19, 269 (1989).Google Scholar
- 5.J. G. Vargas and D. G. Torr,Nucl. Phys. B (Proc. Suppl.) 6, 115 (1989).Google Scholar
- 6.J. G. Vargas,Found. Phys. 21, 379 (1991).Google Scholar
- 7.A. Einstein,Ann. Inst. Henri Poincaré 1, 1 (1930).Google Scholar
- 8.E. Cartan and A. Einstein,Letters on Absolute Parallelism, 1929–1932, R. Debever, ed. (Princeton University Press, Princeton, New Jersey, 1979).Google Scholar
- 9.A. Lichnerowicz,Elements of Tensor Calculus (Wiley, New York, 1962).Google Scholar
- 10.J. G. Vargas and D. G. Torr,Gen. Relativ. Gravit. 23, 713 (1991).Google Scholar
- 11.E. Kaehler,Abh. Dtsch. Akad. Wiss. Berlin, Kl. für Math., Phys. Tech., No. 4, 1960.Google Scholar
- 12.E. Cartan,J. Math. Pures Appl. 1, 141 (1922).Google Scholar
- 13.E. Cartan,Exposés de Géométrie (Hermann, Paris, 1971), reprinted fromActualités Scientifiques et Industrielles 72 and79 (1933).Google Scholar
- 14.S. Chern,Am. Math. Mon. 97, 679 (1990).Google Scholar
- 15.E. Cartan, “Leçons sur les Invariants Intégraux” (Hermann, Paris, 1971).Google Scholar
- 16.A. Bejancu,Finsler Geometry and Applications (Horwood, Chichester, England, 1990).Google Scholar
- 17.E. Cartan,Bull. Soc. Math. 48, 294 (1924).Google Scholar
- 18.D. Hestenes,Spacetime Algebra (Harper & Row, New York, 1967).Google Scholar
- 19.J. G. Vargas,Found. Phys. 12, 765 (1982).Google Scholar
- 20.J. G. Vargas and D. G. Torr,Found. Phys. 16, 1089 (1986).Google Scholar
- 21.J. G. Vargas,Found. Phys. 16, 1231 (1986).Google Scholar