Energy extremality in the presence of a black hole

We derive the so-called first law of black hole mechanics for variations about stationary black hole solutions to the Einstein–Maxwell equations in the absence of sources. That is, we prove that δM = κδA+ωδJ+V dQ where the black hole parameters M,κ,A, ω, J, V and Q denote mass, surface gravity, horizon area, angular velocity of the horizon, angular momentum, electric potential of the horizon and charge respectively. The unvaried fields are those of a stationary, charged, rotating black hole and the variation is to an arbitrary ‘nearby’ black hole which is not necessarily stationary. Our approach is 4-dimensional in spirit and uses techniques involving Action variations and Noether operators. We show that the above formula holds on any asymptotically flat spatial 3-slice which extends from an arbitrary cross-section of the (future) horizon to spatial infinity. (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 perturbations of the metric and other fields which may be present, (ii) the expansion of the null geodesic generators of the perturbed horizon goes to zero in the distant future.)