Tetrad Fields, Reference Frames, and the Gravitational Energy-Momentum in the Teleparallel Equivalent of General Relativity

The concept and definitions of the energy-momentum and angular momentum of the gravitational field in the teleparallel equivalent of general relativity (TEGR) are reviewed. The importance of these definitions is justified by three major reasons. First, the TEGR is a well established and widely accepted formulation of the gravitational field, whose basic field strength is the torsion tensor of the Weitzenbock connection. Second, in the phase space of the TEGR there exists an algebra of the Poincare group. Not only the definitions of the gravitational energy-momentum and 4-angular momentum satisfy this algebra, but also the first class constraints related to these definitions satisfy the algebra. And third, numerous applications of these definitions lead to physically consistent results. These definitions follow from a well established Hamiltonian formulation, and rely on the idea of localization of the gravitational energy. In this review, the concept of localizability of the gravitational energy is revisited, in light of results obtained in recent years. The behavior of free particles is studied in the space-time of plane fronted gravitational waves (pp-waves). Free particles are here understood as particles that are not subject to external forces other than the gravitational acceleration due to pp-waves. Since these particles acquire or loose kinetic energy locally, the transfer of energy from or to the gravitational field must also be localized. This theoretical result is considered an important and definite argument in favor of the localization of the gravitational energy-momentum, and by extension, of the gravitational 4-angular momentum. A review is presented on the problem of defining gravitational energy. The analysis is carried out in the context of the teleparallel equivalent of general relativity. The localization of the gravitational energy is demonstrated by means of recent results regarding the velocity memory effect due to gravitational pp-waves. The gravitational energy definition is successfully applied to three gravitational energy configurations.image