Geometry of Non-expanding Horizons and Their Neighborhoods

This is a contribution to MG9 session BHT4. Certain geometrically distinguished frame on a non-expanding horizon and in its space-time neighborhood, as well as the Bondi-like coordinates are constructed. The construction provides free degrees of freedom, invariants, and the existence conditions for a Killing vector field. The reported results come from the joint works with Ashtekar and Beetle [2]. In the quasi-local theory of black holes proposed recently by Ashtekar [1] a BH in equilibrium is described by a 3-dimensional null cylinder H generated in space-time by null geodesic curves intersecting orthogonally a spacelike, 2-dimensional closed surface S. The standard stationarity of space-time requirement is replaced by the assumption that the cylinder has zero expansion, that is H is a non-expanding horizon. This implies, upon the week and the dominant energy conditions, that the induced on H (degenerate) metric tensor q is Lie dragged by a null, geodesic flow tangent to H. The geometry induced on H consists of the metric tensor q and the induced covariant derivative D. It is enough for the mechanics of H [1]. The geometry of a non-expanding horizon is characterized by local degrees of freedom. They are an arbitrary 2-geometry of the null generators space S, the rotation scalar, and certain tangential ‘radiation’ evolving along the horizon.