# On the charged Riemannian Penrose inequality with charged matter

## TL;DR

This paper investigates the validity of the charged Riemannian Penrose inequality in the presence of charged matter, identifying conditions under which it holds or fails, and providing counterexamples to clarify its limitations.

## Contribution

It extends the understanding of the charged Riemannian Penrose inequality to scenarios with charged matter, showing when the inequality is valid and constructing counterexamples when conditions are violated.

## Key findings

- The inequality holds if charge density does not change sign.
- Counterexamples show the inequality can fail if sign conditions are violated.
- Comparison with existing literature highlights the importance of charge distribution assumptions.

## 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.