A MASS BOUND FOR SPHERICALLY SYMMETRICAL BLACK-HOLE SPACETIMES

Requiring that the matter fields are subject to the dominant energy condition, we establish the lower bounds root A/16 pi and (4 pi)(-1)kappa A for the total mass M of a static, spherically symmetric black hole spacetime. (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 self-gravitating 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) less than or equal to O. 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.