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

TL;DR

This paper constructs spherically symmetric counterexamples to the Penrose inequality and positive mass theorem under the weak energy condition, challenging their validity in this context.

Contribution

It provides the first known counterexamples to these fundamental inequalities assuming only the weak energy condition in spherical symmetry.

Findings

Counterexamples violate the Penrose inequality under weak energy condition.

Counterexamples also violate the positive mass theorem with weak energy condition.

Construction is elementary but has significant theoretical implications.

Abstract

Of the various energy conditions which can be assumed when studying mathematical general relativity, intuitively the simplest is the weak energy condition $\mu\geq 0$ 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. The construction itself is quite elementary. However, the consequences of the counterexample are quite interesting. The Penrose inequality was conjectured by Penrose in \cite{Penrose} using certain heuristic arguments which, as we discuss, continue to hold in the case of the weak energy condition. Yet, a counter example exists. Moreover, the methods developed naturally led to the construction of a counter example to the positive mass theorem assuming the weak energy condition. The counter example can be constructed to be diffeomorphic to $\mathbb{R}^3$ and to contain no minimal surfaces and no future apparent horizons.