Self-renormalization of the classical quasilocal energy

Pointlike objects cause many of the divergences that afflict physical theories. For instance, the gravitational binding energy of a point particle in Newtonian mechanics is infinite. In general relativity, the analog of a point particle is a black hole and the notion of binding energy must be replaced by quasilocal energy (QLE). The QLE derived by York, and elaborated by Brown and York, is finite outside the horizon but it was not considered how to evaluate it inside the horizon. We present a prescription for finding the QLE inside a horizon, and show that it is finite at the singularity for a variety of types of black holes. The energy is typically concentrated just inside the horizon, not at the central singularity.

Figure 1

QLE computed inside and outside the event horizon for a Schwarzschild black hole. Both axes are in units of the mass $M$, and the horizon is at $2M$.

Figure 2

QLE of de Sitter space (positive cosmological constant). Both axes are in the same units, with a mass scale proportional to $1/\sqrt{\Lambda }$.

Figure 3

QLE of the de Sitter-Schwarzschild metric, a black hole in a spacetime with a positive cosmological constant. The units are the same as the plot of the de Sitter QLE, and the black hole has an unrealistically large mass so that details can be seen in the plot.

Figure 4

QLE of a Reissner-Nordstrom charged black hole. Both axes are in units of mass of the black hole, and the charge $e^2=0.8M^2$.