Conserved charges in general relativity

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.

16 pages (single column), v3 (major revision): more discussion on a compact star included, a comparison with previous results given in the appendix, more references added

References in corpus (2)

• Conformally-flat, non-singular static metric in infinite derivative gravity
• What does a quantum black hole look like?

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