A Mass Bound for Spherically Symmetric Black Hole Spacetimes

Details A Mass Bound for Spherically Symmetric Black Hole Spacetimes A Mass Bound for Spherically Symmetric Black Hole Spacetimes

1994-11-21

arxiv.org/abs/gr-qc/9411054v1

Class.Quant.Grav.12:779-790,1995

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

16 pages, LATEX, no figures

Scientific paper

10.1088/0264-9381/12/3/015

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.

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