Gravitational energy in spherical symmetry.

Various properties of the Misner-Sharp spherically symmetric gravitational energy E are derived, and known properties reviewed. In the Newtonian limit, E yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. In vacuo, E reduces to the Schwarzschild parameter. At null and spatial infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies respectively. In the small-sphere limit, the leading term in E is the product of volume and the energy density of the matter. A sphere is trapped if E > 1 2 r, marginal if E = 1 2 r and untrapped if E < 1 2 r, 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 ≥ 1 2 r0 on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius r0. All these inequalities extend to the asymptotic energies, recovering the BondiSachs energy loss and the positivity of the asymptotic energies, as well as proving the conjectured Penrose inequality in spherical symmetry. Implications for general definitions of gravitational energy are discussed. PACS: 04.70.Bw, 04.20.Dw, 04.20.Ha, 04.25.Nx