Equivalence principle in the new general relativity

TL;DR

This paper investigates the equivalence of gravitational mass and energy in the tetrad theory of gravitation, deriving conditions under which they are equal, and analyzing solutions in spherical symmetry.

Contribution

It derives the superpotential and energy expressions in the tetrad theory, establishing conditions for mass-energy equivalence in spherically symmetric solutions.

Findings

Energy equals gravitational mass under certain parameter conditions.

In specific cases, the mass-energy equivalence does not hold for some solutions.

The superpotential is derived from a parity-invariant gravitational Lagrangian.

Abstract

We study the problem of whether the active gravitational mass of an isolated system is equal to the total energy in the tetrad theory of gravitation. The superpotential is derived using the gravitational Lagrangian which is invariant under parity operation, and applied to an exact spherically symmetric solution. Its associated energy is found equal to the gravitational mass. The field equation in vacuum is also solved at far distances under the assumption of spherical symmetry. Using the most general expression for parallel vector fields with spherical symmetry, we find that the equality between the gravitational mass and the energy is always true if the parameters of the theory $a_1$, $a_2$ and $a_3$ satisfy the condition, $(a_1+ a_2) (a_1-4a_3/9)\neq0$. In the two special cases where either $(a_1+a_2)$ or $(a_1-4a_3/9)$ is vanishing, however, this equality is not satisfied for the solutions when some components of the parallel vector fields tend to zero as $1/\sqrt{r}$ for large $r$.