Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition

Abstract

Of the various energy conditions which can be assumed when studying mathematical general relativity, intuitively the simplest is the weak energy condition which simply states that the observed mass-energy density must be non-negative. This energy condition has not received as much attention as the so-called dominant energy condition. When the natural question of the Penrose inequality in the context of the weak energy condition arose, we could not find any results in the literature, and it was not immediately clear whether the inequality would hold, even in spherical symmetry. This led us to constructing a spherically symmetric asymptotically flat initial data set satisfying the weak energy condition which violates the ‘usual’ formulations of the Penrose conjecture. This does not contradict (Malec and Murchadha 1994 Phys. Rev. D 49 6931–4), as there the data is assumed to be maximal. Our result is unexpected and interesting because the heuristic argument behind the Penrose inequality relies on the black hole area law which holds when the weak energy condition is satisfied. Our methods also lead to a counter example to the positive mass theorem assuming the weak energy condition.