A Mass Bound for Spherically Symmetric Black Hole Spacetimes

TL;DR

This paper establishes a lower mass bound for static, spherically symmetric black holes under energy conditions, leading to uniqueness and no-hair theorems for certain matter models including scalar and electromagnetic fields.

Contribution

It derives a new mass bound for black holes satisfying the dominant energy condition and proves uniqueness results for Schwarzschild solutions with specific matter fields.

Findings

Lower bound for black hole mass involving horizon area and surface gravity.

Uniqueness of Schwarzschild solution under dominant but violated strong energy condition.

Stronger mass bounds in electromagnetic cases supporting no-hair theorems.

Abstract

Requiring that the matter fields are subject to the dominant energy condition, we establish the lower bound $(4\pi)^{-1} \kappa {\cal A}$ for the total mass $M$ of a static, spherically symmetric black hole spacetime. (${\cal A}$ and $\kappa$ denote the area and the surface gravity of the horizon, respectively.) Together with the fact that the Komar integral provides a simple relation between $M - (4\pi)^{-1} \kappa A$ and the strong energy condition, this enables us to prove that the Schwarzschild metric represents the only static, spherically symmetric black hole solution of a selfgravitating matter model satisfying the dominant, but violating the strong energy condition for the timelike Killing field $K$ at every point, that is, $R(K,K) \leq 0$. Applying this result to scalar fields, we recover the fact that the only black hole configuration of the spherically symmetric Einstein-Higgs model with arbitrary non-negative potential is the Schwarzschild spacetime with constant Higgs field. In the presence of electromagnetic fields, we also derive a stronger bound for the total mass, involving the electromagnetic potentials and charges. Again, this estimate provides a simple tool to prove a ``no-hair'' theorem for matter fields violating the strong energy condition.