The teleparallel equivalent of general relativity

TLDR

General relativity can be reformulated using tetrad fields and torsion, providing alternative geometric insights while remaining equivalent to the standard metric–curvature formulation at the level of field equations. This review surveys the teleparallel equivalent of general relativity. It traces the historical development of teleparallel theories, discusses the interpretation of tetrad fields as reference frames for arbitrary observers, and derives the inertial acceleration tensor for such frames. The Lagrangian and Hamiltonian field equations yield surface‑integral expressions for the gravitational field’s energy, momentum, and angular momentum, which obey the Poincaré algebra in phase space.

Abstract

A review of the teleparallel equivalent of general relativity is presented. It is emphasized that general relativity may be formulated in terms of the tetrad fields and of the torsion tensor, and that this geometrical formulation leads to alternative insights into the theory. The equivalence with the standard formulation in terms of the metric and curvature tensors takes place at the level of field equations. The review starts with a brief account of the history of teleparallel theories of gravity. Then the ordinary interpretation of the tetrad fields as reference frames adapted to arbitrary observers in space–time is discussed, and the tensor of inertial accelerations on frames is obtained. It is shown that the Lagrangian and Hamiltonian field equations allow us to define the energy, momentum and angular momentum of the gravitational field, as surface integrals of the field quantities. In the phase space of the theory, these quantities satisfy the algebra of the Poincaré group.