Inequalities relating area, energy, surface gravity and charge of black holes

TL;DR

This paper proves inequalities relating the area, energy, surface gravity, and charge of charged black holes under spherical symmetry, extending classical results and establishing sharp bounds that are attained by Reissner-Nordstrom black holes.

Contribution

It establishes new inequalities connecting key physical quantities of charged black holes, including a local energy version and bounds involving dynamic surface gravity.

Findings

Proved the Penrose-Gibbons inequality for charged black holes.

Derived bounds on surface gravity and mass in terms of charge and radius.

Inequalities are sharp, with equality for Reissner-Nordstrom black holes.

Abstract

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 >= r + e^2/r. Replacing m by a local energy, the inequality holds locally outside the black hole. A recent definition of dynamic surface gravity k also satisfies inequalities 2k <= 1/r - e^2/r^3 and m >= r^2 k + e^2/r. All these inequalities are sharp in the sense that equality is attained for the Reissner-Nordstrom black hole.