Quasilocal gravitational energy

Abstract

A dynamically preferred quasilocal definition of gravitational energy is given in terms of the Hamiltonian of a 2+2 formulation of general relativity. The energy is well defined for any compact orientable spatial two-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, reduces to the Hawking mass in the absence of shear and twist, and reduces to the standard gravitational energy in spherical symmetry. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the ADM mass at spatial infinity, taking the limit along a foliation parametrized by the area radius. The energy is calculated for the Schwarzschild, Reissner-Nordström, and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasilocal energy are discussed.