Inequalities Relating Area, Energy, Surface Gravity, and Charge of Black Holes

The Penrose-Gibbons inequality for charged black holes is proved in spherical symmetry, assuming that outside the black hole there are no current sources, meaning that the charge $e$ is constant, with the remaining fields satisfying the dominant energy condition. Specifically, for any achronal hypersurface which is asymptotically flat at spatial or null infinity and has an outermost marginal surface of areal radius $r$, the asymptotic mass $m$ satisfies $2m\ge {r+e}^2/r$. Replacing $m$ by a local energy $\mu$, the inequality holds locally outside the black hole. A recent definition of dynamic surface gravity $\kappa$ also satisfies inequalities $2\kappa \le 1/{r-e}^2/r^3$ and $m\ge \mu \ge r^2\kappa {+e}^2/r$. All these inequalities are sharp in the sense that equality is attained for the Reissner-Nordström black hole.