Generic weak isolated horizons

Isolated horizons [1], much like Killing horizons [2], were introduced in order to deal with the more practical problems involving black holes. Both formulations are local, in contrast with the global nature of event horizons [3]. However, isolated horizons, unlike Killing horizons, do not require a Killing vector field in their neighbourhood. Thus, isolated horizons are characterized by a weaker set of boundary conditions that give rise to both the zeroth and the first laws of black hole mechanics. Nevertheless, the isolated horizon boundary conditions, as developed in a series of papers [1, 4, 5, 6, 7, 8], consider a restricted equivalence class of null normal vector fields [ cl], where l1 , l a 2 belong to the class if and only if l1 = cl a 2 for some positive definite constant c. Even though it has been emphasized that the most natural equivalence class of null normals at the horizon is [ ξl], where ξ is an arbitrary positive function on the horizon, one restricts oneself, somewhat artificially perhaps, to the constant equivalence class. The purpose of this letter is to generalize the framework such that now the equivalence class of null normals is [ ξl ], where ξ is a given class of functions to be specified below, and to show how from this generalized setup both the zeroth and the first laws of black hole mechanics can be derived. This clearly makes isolated horizons applicable to a wider varieties of problems, both from analytical and numerical viewpoints. Another unexpected gain is that this generalized class of null normals include both non-extremal and extremal global solutions explicitly, namely there exists a choice of ξ for which the surface gravity associated with the vector field ξl vanishes, even though it is nonzero for other null normals in the class. In contrast, Killing horizons make essential use of the bifurcation two spheres in proving the zeroth and the first law [10], thereby distinguish extremal and non-extremal solutions in an explicit way.