A MASS BOUND FOR SPHERICALLY SYMMETRICAL BLACK-HOLE SPACETIMES

Requiring that the matter fields are subject to the dominant energy co ndition, we establish the lower bounds root A/16 pi and (4 pi)(-1)kapp a 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 th e horizon, respectively.) Together with the fact that the Komar integr al provides a simple relation between M - (4 pi)(-1)kappa A and the st rong energy condition, this enables us to prove that the Schwarzschild metric represents the only static, spherically symmetric black hole s olution of a self-gravitating matter model satisfying the dominant, bu t 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 t his result to scalar fields, we recover the fact that the only black h ole configuration of the spherically symmetric Einstein-Higgs model wi th arbitrary non-negative potential is the Schwarzschild spacetime wit h constant Higgs field. In the presence of electromagnetic fields, we also derive a stronger bound for the total mass, involving the electro magnetic potentials and charges. Again, this estimate provides a simpl e tool to prove a 'no hair' theorem' for matter fields violating the s trong energy condition.