Observer dependence of the quasi-local energy and momentum in Schwarzschild space-time

Abstract

The observer dependence of the quasi-local energy (QLE) and momentum in the Schwarzschild geometry is illustrated. Using the Brown-York prescription, the QLE for families of non-geodesic and geodesic observers penetrating the event horizon is obtained. An explicit shell-building process is presented and the binding energy is computed in terms of the QLE measured by a static observer field at a radius outside the horizon radius. The QLE for a radially geodesic observer field freely-falling from infinity is shown to vanish. Finally, a simple relation for the dynamics of the quasi-local momentum density for a geodesic observer field is noted.