The generalized charged Penrose inequality for non-rotating black holes

Let $(M^3,g,K,E,B)$ be asymptotically flat initial data satisfying the dominant energy condition, with electric field $E$, magnetic field $B$, and a stable marginally outer trapped surface (MOTS) $\Sigma$ of area $A$. Define the total enclosed charge by

$$Q=\frac{1}{4\pi}\int_{\Sigma}E\cdot\nu\,d\sigma.$$

Generalized charged Penrose conjecture. The ADM mass should satisfy

$$M_{\mathrm{ADM}} \geq \sqrt{\frac{A}{16\pi}+\frac{Q^2}{4}}.$$

This is the charged, non-rotating extension of the Penrose inequality. The source explicitly states that a complete proof remains an important open problem in quasi-local mass theory.