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 μ≥0 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 [18], 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.

Nonexistence of solutions to the coupled generalized Jang equation/zero divergence system

“…One of the key ingredients for the Penrose inequality is the black hole area law, which holds under the weaker assumption of the weak energy condition. However, in this setting there exists a counter-example satisfying all the assumptions of the above conjecture, with the dominant energy condition replaced by the weak energy condition, see [15].…”

“…We want our spherically symmetric initial data set to be asymptotically flat, satisfy the dominant energy condition, and have the boundary consist of an outermost apparent horizon. To do this, we use methods similar to those of [15].…”