Energy and Momentum in the Tetrad Theory of Gravitation

TL;DR

This paper investigates the energy and momentum in the tetrad theory of gravitation, deriving solutions for isolated systems, and finds that total energy equals gravitational mass with momentum related to a constant vector, especially in weak fields.

Contribution

It provides an exact solution for the linearized vacuum field equations in tetrad gravity and clarifies the relation between energy, momentum, and the source's properties.

Findings

Total energy equals the gravitational mass of the source.

Spatial momentum coincides with a constant vector in the solution.

The constant vector vanishes for weakly gravitating sources.

Abstract

We study the energy and momentum of an isolated system in the tetrad theory of gravitation, starting from the most general Lagrangian quadratic in torsion, which involves four unknown parameters. When applied to the static spherically symmetric case, the parallel vector fields take a diagonal form, and the field equation has an exact solution. We analyze the linearized field equation in vacuum at distances far from the isolated system without assuming any symmetry property of the system. The linearized equation is a set of coupled equations for a symmetric and skew-symmetric tensor fields, but it is possible to solve it up to $O(1/r)$ for the stationary case. It is found that the general solution contains two constants, one being the gravitational mass of the source and the other a constant vector ${\grave B_\alpha}$. The total energy is calculated from this solution and is found to be equal to the gravitational mass of the source. We also calculate the spatial momentum and find that its value coincides with the constant vector ${\grave B_\alpha}$. The linearized field equation in vacuum, which is valid at distances far from the source, does not give any information about whether the constant vector ${\grave B_\alpha}$ is vanishing or not. For a weakly gravitating source for which the field is weak everywhere, we find that the constant vector ${\grave B_\alpha}$ vanishes.