GRAVITATIONAL ENERGY IN SPHERICAL-SYMMETRY

Various properties of the Misner-Sharp spherically symmetric gravitati onal energy E are established or reviewed. In the Newtonian limit of a perfect fluid, E yields the Newtonian mass to leading order and the N ewtonian kinetic and potential energy to the next order. For test part icles, the corresponding Hajicek energy is conserved and has the behav ior appropriate to energy in the Newtonian and special-relativistic li mits. 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 > 1/2 r, marginal if E = 1/2 r, and untrapped if E < 1/2 r, where r is the a real radius. A central singularity is spatial and trapped if E > 0, an d temporal and untrapped if E < 0. On an untrapped sphere, E is nondec reasing in any outgoing spatial or null direction, assuming the domina nt energy condition. It follows that E greater than or equal to 0 on a n untrapped Spatial hypersurface with a regular center, and E greater than or equal to 1/2 r(0) on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius r(0). All these ineq ualities 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 defi nitions of gravitational energy are discussed.