Conserved charges in general relativity

TL;DR

This paper introduces a precise, covariant method to define conserved charges like energy and momentum in curved spacetimes, applicable to black holes and stars, and discusses related generators on manifolds.

Contribution

It provides a new covariant definition of conserved charges in curved spacetime, enabling volume integral calculations for black holes and stars, including gravitational binding energy.

Findings

Computed charges for Schwarzschild and BTZ black holes.

Calculated total energy of a static compact star, including gravitational binding energy.

Found gravitational binding energy can be a significant negative contribution, up to 68% of gravitational mass.

Abstract

We present a precise definition of a conserved quantity from an arbitrary covariantly conserved current available in a general curved spacetime with Killing vectors. This definition enables us to define energy and momentum for matter by the volume integral. As a result we can compute charges of Schwarzschild and BTZ black holes by the volume integration of a delta function singularity. Employing the definition we also compute the total energy of a static compact star. It contains both the gravitational mass known as the Misner-Sharp mass in the Oppenheimer-Volkoff equation and the gravitational binding energy. We show that the gravitational binding energy has the negative contribution at maximum by 68% of the gravitational mass in the case of a constant density. We finally comment on a definition of generators associated with a vector field on a general curved manifold.