On the charged Riemannian Penrose inequality with charged matter

Abstract

Throughout the literature on the charged Riemannian Penrose inequality, it is generally assumed that there is no charged matter present; that is, the electric field is divergence-free. The aim of this article is to clarify when the charged Riemannian Penrose inequality holds in the presence of charged matter, and when it does not.First we revisit Jang's proof of the charged Riemannian Penrose inequality to show that under suitable conditions on the charged matter, this argument still carries though. In particular, a charged Riemannian Penrose inequality is obtained from this argument when charged matter is present provided that the charge density does not change sign. Moreover, we show that such hypotheses on the sign of the charge are in fact required by constructing counterexamples to the charged Riemannian Penrose inequality when these conditions are violated. We conclude by comparing this counterexample to another version of the Riemannian Penrose inequality with charged matter existing in the literature.

Geometric Inequalities for Quasi-Local Masses

“…Furthermore, results are typically stated within the framework of initial data sets for the Einstein equations that satisfy the relevant energy condition, and the inclusion of angular momentum/charge requires the absence of angular momentum/charge density outside the horizon in addition to axisymmetry (which is only needed for angular momentum). The Penrose inequality has been established in the case of maximal data by Bray [8] and Huisken-Ilmanen [34], and charge was added in [40,44]. The inclusion of angular momentum is much more difficult and has not yet been established, although see [3,38] for partial results.…”