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

TL;DR

This paper reviews the definitions and properties of gravitational energy-momentum and angular momentum in the teleparallel equivalent of general relativity (TEGR), emphasizing their localization, algebraic structure, and physical consistency.

Contribution

It provides a comprehensive review of energy-momentum and angular momentum definitions in TEGR, highlighting their algebraic properties and physical implications for gravitational energy localization.

Findings

Energy-momentum and angular momentum satisfy Poincaré algebra in TEGR.

Localized energy transfer is demonstrated through particle behavior in gravitational waves.

Definitions lead to physically consistent results across applications.

Abstract

We review the concept and definitions of the energy-momentum and angular momentum of the gravitational field in the teleparallel equivalent of general relativity (TEGR). 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 Weitzenb\"ock connection. Second, in the phase space of the TEGR there exists an algebra of the Poincar\'e 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 we revisit the concept of localizability of the gravitational energy, in light of results obtained in recent years. We have studied the behaviour of free particles 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. We consider this theoretical result an important and definite argument in favour of the localization of the gravitational energy-momentum, and by extension, of the gravitational 4-angular momentum.