Bounds on area and charge for marginally trapped surfaces with cosmological constant

TL;DR

This paper refines inequalities relating area, charge, and cosmological constant for marginally trapped surfaces, incorporating stability eigenvalues to establish bounds and explore implications for black hole horizons.

Contribution

It introduces bounds on area and charge for marginally trapped surfaces with a cosmological constant, including stability eigenvalues, extending previous inequalities.

Findings

Derived bounds on area and charge involving the eigenvalue of the stability operator.

Established conditions for equality in spherically symmetric, static cases.

Discussed implications for the merging and stability of charged MOTS.

Abstract

We sharpen the known inequalities $A \Lambda \le 4\pi (1-g)$ and $A\ge 4\pi Q^2$ between the area $A$ and the electric charge $Q$ of a stable marginally outer trapped surface (MOTS) of genus g in the presence of a cosmological constant $\Lambda$. In particular, instead of requiring stability we include the principal eigenvalue $\lambda$ of the stability operator. For $\Lambda^{*} = \Lambda + \lambda > 0$ we obtain a lower and an upper bound for $ \Lambda^{*} A$ in terms of $ \Lambda^{*} Q^2$ as well as the upper bound $ Q \le 1/(2\sqrt{\Lambda^{*}})$ for the charge, which reduces to $ Q \le 1/(2\sqrt{\Lambda})$ in the stable case $\lambda \ge 0$. For $\Lambda^{*} < 0$ there remains only a lower bound on $A$. In the spherically symmetric, static, stable case one of the area inequalities is saturated iff the surface gravity vanishes. We also discuss implications of our inequalities for "jumps" and mergers of charged MOTS.