Lorentz Connections and Gravitation

The different roles played by Lorentz connections in general relativity and in teleparallel gravity are reviewed. Some of the consequences of this difference are discussed.Comment: Lecture presented at the Sixth International School on Field Theory and Gravity, Petropolis, Brazil, 201

Spin Connection and Renormalization of Teleparallel Action

In general relativity, inertia and gravitation are both included in the Levi-Civita connection. As a consequence, the gravitational action, as well as the corresponding energy-momentum density, are in general contaminated by spurious contributions coming from inertial effects. In teleparallel gravity, on the other hand, because the spin connection represents inertial effects only, it is possible to separate inertia from gravitation. Relying on this property, it is shown that to each tetrad there is naturally associated a spin connection that locally removes the inertial effects from the action. The use of the appropriate spin connection can be viewed as a renormalization process in the sense that the computation of energy and momentum naturally yields the physically relevant values. A self-consistent method for solving field equations and determining the appropriate spin connection is presented.Comment: v2: 13 pages, minor corrections, new section about solving the field equations include

Gravitation and the Local Symmetry Group of Spacetime

According to general relativity, the interaction of a matter field with gravitation requires the simultaneous introduction of a tetrad field, which is a field related to translations, and a spin connection, which is a field assuming values in the Lie algebra of the Lorentz group. These two fields, however, are not independent. By analyzing the constraint between them, it is concluded that the relevant local symmetry group behind general relativity is provided by the Lorentz group. Furthermore, it is shown that the minimal coupling prescription obtained from the Lorentz covariant derivative coincides exactly with the usual coupling prescription of general relativity. Instead of the tetrad, therefore, the spin connection is to be considered as the fundamental field representing gravitation.Comment: 8 pages, no figures. Some signs and references corrected; version to appear in Int. J. Theor. Phy

Spacetime: Arena or Reality?

For small values of the mass (in relation to the angular momentum and electric charge), the Kerr-Newman (KN) solution of Einstein equation reduces to a naked singularity of circular shape. By considering the Hawking and Ellis extended interpretation of the KN spacetime, as well as Wheeler's idea of "charge without charge", the non-trivial topological structure of the extended KN spatial section is found to represent gravitational states with half-integral angular momentum. As a consequence, it can be consistently interpreted as a model for the electron-positron system, in which the concepts of mass, charge and spin emerge from the spacetime geometry. According to this model, therefore, instead of a simple arena, spacetime must have a concrete existence, being responsible -- through its highly non-trivial topological structures -- for the building blocks of (at least some of) the existing matter in the universe.Comment: Chapter in the book "Relativity and the Dimensionality of the World", Springer series "Fundamental Theories of Physics", Vol. 153 (2007). Volume Editor: Vesselin Petko