A Mass Bound for Spherically Symmetric Black Hole Spacetimes

Requiring that the matter fields are subject to the dominant energy condition, we establish the lower bound (4π)−1κA for the total mass M of a static, spherically symmetric black hole spacetime. (A and κ 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 − (4π)κA 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) ≤ 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.