INEQUALITIES RELATING AREA, ENERGY, SURFACE GRAVITY, AND CHARGE OF BLACK-HOLES

The Penrose-Gibbons inequality for charged black holes is proved in sp herical symmetry, assuming that outside the black hole there are no cu rrent sources, meaning that the charge e is constant, with the remaini ng fields satisfying the dominant energy condition. Specifically, for any achronal hypersurface which is asymptotically hat at spatial or nu ll infinity and has an outermost marginal surface of areal radius r, t he asymptotic mass m satisfies 2m greater than or equal to r + e(2)/r. Replacing m by a local energy mu, the inequality holds locally outsid e the black hole. A recent definition of dynamic surface gravity kappa also satisfies inequalities 2 kappa less than or equal to 1/r - e(2)/ r(3) and m greater than or equal to mu greater than or equal to r(2)ka ppa + e(2)/r. All these inequalities are sharp in the sense that equal ity is attained for the Reissner-Nordstrom black hole. [S0031-9007(98) 07700-X].