ENERGY EXTREMALITY IN THE PRESENCE OF A BLACK-HOLE

We derive the so-called first law of black hole mechanics for variatio ns about stationary black hole solutions to the Einstein-Maxwell equat ions in the absence of sources. That is, we prove that delta M = kappa delta A + omega delta J + V dQ where the black hole parameters M, kap pa, A, omega, J, V and Q denote mass, surface gravity, horizon area, a ngular velocity of the horizon, angular momentum, electric potential o f the horizon and charge respectively. The unvaried fields are those o f a stationary, charged, rotating black hole and the variation is to a n arbitrary 'nearby' black hole which is not necessarily stationary. O ur approach is four-dimensional in spirit and uses techniques involvin g action variations and Noether operators. We show that the above form ula holds on any asymptotically Rat spatial 3-slice which extends from an arbitrary cross section of the (future) horizon to spatial infinit y. (Thus, the existence of a bifurcation surface is irrelevant to our demonstration. On the other hand, the derivation assumes without proof that the horizon possesses at least one of the following two (related ) properties: (i) it cannot be destroyed by arbitrarily small perturba tions of the metric and other fields which may be present, (ii) the ex pansion of the null geodesic generators of the perturbed horizon goes to zero in the distant future.)