Gravitational energy in spherical symmetry
Abstract
Various properties of the MisnerSharp 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 Hájíček energy is conserved and has the behavior appropriate to energy in the Newtonian and specialrelativistic limits. In the smallsphere 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 BondiSachs and ArnowittDeserMisner energies, respectively. The conserved Kodama current has charge E. A sphere is trapped if E>1/2r, marginal if E=1/2r, and untrapped if E<1/2r, 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 nondecreasing in any outgoing spatial or null direction, assuming the dominant energy condition. It follows that E>=0 on an untrapped spatial hypersurface with a regular center, and E>=1/2r_{0} 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 BondiSachs 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.
 Publication:

Physical Review D
 Pub Date:
 February 1996
 DOI:
 10.1103/PhysRevD.53.1938
 arXiv:
 arXiv:grqc/9408002
 Bibcode:
 1996PhRvD..53.1938H
 Keywords:

 04.70.Bw;
 04.20.Dw;
 04.20.Ha;
 04.25.Nx;
 Classical black holes;
 Singularities and cosmic censorship;
 Asymptotic structure;
 PostNewtonian approximation;
 perturbation theory;
 related approximations;
 General Relativity and Quantum Cosmology
 EPrint:
 23 pages. Belatedly replaced with substantially extended published version