Positive energy from timelike singularities

Abstract

The energy of various asymptotically AdS and asymptotically flat solutions to the Einstein equations with time-translational invariance and a timelike singularity is computed, spaces with Kasner timelike singularities and black holes with negative masses among them. The definition used for gravitational energy is the on-shell Hamiltonian of the physical classical action proposed by Hawking and Horowitz thirty years ago and which agrees with the Abbott-Deser and ADM expressions. To deal with the singularity, a timelike regulating cut-off boundary surrounding the singularity compatible with a well-defined foliation in constant-time slices is introduced, so calculations are performed at a finite value of the regulating parameter, which is set to zero in the final expression for the energy. In all instances considered, the energy contribution from the singularity is finite and positive, its value depends on the geometry around the singularity and is such that the total gravitational energy is positive. Of the solutions reviewed, only the AdS soliton (which has no singularity) has negative gravitational energy relative to AdS space. The AdS and the Schwarzschild black holes with negative masses have respectively the same energies as AdS and Minkowski spaces.