On the selection of triads in the teleparallel geometry and Bondi's radiating metric

TL;DR

This paper investigates how to select a unique set of triads in teleparallel gravity to accurately describe gravitational energy, especially in radiating spacetimes, and introduces a new expression for Bondi's news function facilitating numerical analysis.

Contribution

It proposes a criterion based on minimizing gravitational energy to uniquely determine triads in TEGR within radiating spacetimes, supported by a new expression for Bondi's news function.

Findings

A simple expression for Bondi's news function was derived.

Triads with specific asymptotic behavior minimize gravitational energy.

The energy minimization criterion uniquely determines the correct triads.

Abstract

A consistent Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) requires the theory to be invariant under the global SO(3) symmetry group, which acts on orthonormal triads in three-dimensional spacelike hypersurfaces. In the TEGR it is possible to make definite statements about the energy of the gravitational field. In this geometrical framework two sets of triads related by a local SO(3) transformation yield different descriptions of the gravitational energy. Here we consider the problem of assigning a unique set of triads to the metric tensor restricted to the three-dimensional hypersurface. The analysis is carried out in the context of Bondi's radiating metric. A simple and original expression for Bondi's news function is obtained, which allows us to carry out numerical calculations and verify that a triad with a specific asymptotic behaviour yields the minimum gravitational energy for a fixed space volume containing the radiating source. This result supports the conjecture that the requirement of a minimum gravitational energy for a given space volume singles out uniquely the correct set of orthonormal triads.