Schwarzschild Space-Time in Gauge Theories of Gravity

TL;DR

This paper establishes conditions under which Schwarzschild space-time can be realized as a torsionless solution in gauge theories of gravity, and explores the implications for energy-momentum and angular momentum in these models.

Contribution

It provides necessary and sufficient conditions for Schwarzschild space-time to be a solution in Poincaré gauge theories, linking vierbein asymptotics to mass and angular momentum.

Findings

Schwarzschild space-time can be obtained as a torsionless solution under specific conditions.

Asymptotic vierbein forms restrict active gravitational mass and inertial mass equality.

Certain vierbein choices violate the mass equality despite reproducing Schwarzschild metric.

Abstract

In Poincar\'e gauge theory of gravity and in $\overline{\mbox{Poincar\'e}}$ gauge theory of gravity, we give the necessary and sufficient condition in order that the Schwarzschild space-time expressed in terms of the Schwarzschild coordinates is obtainable as a torsionless exact solution of gravitational field equations with a spinless point-like source having the energy-momentum density $\widetilde{\mbox{\boldmath $T$}}_\mu^{~\nu}(x) = - Mc^2 \delta_\mu^{~0} \delta_0^{~\nu} \delta^{(3)}(\mbox{\boldmath $x$})$. Further, for the case when this condition is satisfied, the energy-momentum and the angular momentum of the Schwarzschild space-time are examined in their relations to the asymptotic forms of vierbein fields. We show, among other things, that asymptotic forms of vierbeins are restricted by requiring the equality of the active gravitational mass and the inertial mass. Conversely speaking, this equality is violated for a class of vierbeins giving the Schwarzschild metric.