Quasilocal energy in static gravitational fields

Abstract

The timelike Killing vector xi of any static spacetime defines a set of equipotentials on which mod xi mod =constant. The principle that flat spacetimes have zero energy allows a unique expression for the energy, E, within any equipotential. The energy, E, is the energy of the shell of stressed matter on the equipotential that gives the same external metric and a flat interior. The standard Arnowitt-Deser-Misner total energy is retrieved from this expression applied to a large equipotential. The authors compare E with the energy, E_N, obtained from Noether's theorem when the actual spacetime is mapped onto a flat background. The mapping is one in which the timelike Killing fields are identified and the normal congruence of spacelike lines orthogonal to mod xi mod =constant is mapped differentiably on a normal congruence of spacelike lines in the flat background. For a shell it is shown that E=E_N provided the mapping gives zero energy in the flat region. E is given explicitly for axially symmetric fields and the following examples are analysed: spherical symmetry, cylindrical symmetry, the metric of a rod and Lynden-Bell and Pineault's infinite disc of particles in oppositely rotating streams of constant speed. In each example the energy is positive whenever the field is generated by a physical source obeying the weak energy condition. It also increases as the equipotential is made larger.