Quasilocal Energy for a Kerr black hole

TL;DR

This paper calculates the quasilocal energy for Kerr black holes using a Hamiltonian approach, analyzing various boundary surfaces and their properties, including behavior at the horizon, ergosurface, and at infinity.

Contribution

It provides explicit formulas for quasilocal energy in Kerr spacetime considering different boundary surfaces, extending previous methods to rotating black holes.

Findings

Energy is positive and decreases with boundary radius.

Energy approaches ADM mass at infinity and irreducible mass at the horizon.

Energy at the ergosurface is compared with other surfaces.

Abstract

The quasilocal energy associated with a constant stationary time slice of the Kerr spacetime is presented. The calculations are based on a recent proposal \cite{by} in which quasilocal energy is derived from the Hamiltonian of spatially bounded gravitational systems. Three different classes of boundary surfaces for the Kerr slice are considered (constant radius surfaces, round spheres, and the ergosurface). Their embeddings in both the Kerr slice and flat three-dimensional space (required as a normalization of the energy) are analyzed. The energy contained within each surface is explicitly calculated in the slow rotation regime and its properties discussed in detail. The energy is a positive, monotonically decreasing function of the boundary surface radius. It approaches the Arnowitt-Deser-Misner (ADM) mass at spatial infinity and reduces to (twice) the irreducible mass at the horizon of the Kerr black hole. The expressions possess the correct static limit and include negative contributions due to gravitational binding. The energy at the ergosurface is compared with the energies at other surfaces. Finally, the difficulties involved in an estimation of the energy in the fast rotation regime are discussed.