A Mass Bound for Spherically Symmetric Black Hole Spacetimes

Abstract

Requiring that the matter fields are subject to the dominant energy condition, we establish the lower bound $(4\pi {)}^{-1}\kappa \mathcal{A}$ for the total mass $M$ of a static, spherically symmetric black hole spacetime. ($\mathcal{A}$ 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, $R(K,K)\le 0$. 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.

16 pages, LATEX, no figures

References in corpus (4)

• General Laws of Black-Hole Dynamics
• Dirty blackholes: Thermodynamics and horizon structure
• On the topology of stationary black holes
• Maximal Hypersurfaces in Asymptotically Stationary Space-Times

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