Gravitational Energy in Spherical Symmetry

TL;DR

This paper reviews properties of the Misner-Sharp gravitational energy in spherical symmetry, demonstrating its consistency with Newtonian, relativistic, and asymptotic energies, and exploring its implications for black holes and cosmic censorship.

Contribution

It provides a comprehensive review of the Misner-Sharp energy, establishing new inequalities and connections with various energy concepts in spherical symmetry.

Findings

E reduces to Newtonian mass and energies in appropriate limits

E is conserved for test particles and matches relativistic energies

Inequalities relate E to trapped surfaces and support Penrose conjecture

Abstract

Various properties of the Misner-Sharp spherically symmetric gravitational energy E are established or reviewed. In the Newtonian limit of a perfect fluid, E yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. For test particles, the corresponding Hajicek energy is conserved and has the behaviour appropriate to energy in the Newtonian and special-relativistic limits. In the small-sphere limit, the leading term in E is the product of volume and the energy density of the matter. In vacuo, E reduces to the Schwarzschild energy. At null and spatial infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies respectively. The conserved Kodama current has charge E. A sphere is trapped if E>r/2, marginal if E=r/2 and untrapped if E<r/2, where r is the areal radius. A central singularity is spatial and trapped if E>0, and temporal and untrapped if E<0. On an untrapped sphere, E is non-decreasing in any outgoing spatial or null direction, assuming the dominant energy condition. It follows that E>=0 on an untrapped spatial hypersurface with regular centre, and E>=r_0/2 on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius r_0. All these inequalities extend to the asymptotic energies, recovering the Bondi-Sachs energy loss and the positivity of the asymptotic energies, as well as proving the conjectured Penrose inequality for black or white holes. Implications for the cosmic censorship hypothesis and for general definitions of gravitational energy are discussed.