Spin-Coefficient Form of the New Laws of Black-Hole Dynamics

Abstract

General laws of black-hole dynamics, some of which are analogous to the laws of thermodynamics, have recently been found for a general definition of black hole in terms of a future outer trapping horizon, a hypersurface foliated by marginal surfaces of a certain type. This theory is translated here into spin-coefficient language. Second law: the area form of a future outer trapping horizon is generically increasing, otherwise constant. First law: the rate of change of the area form is given by an energy flux and the trapping gravity. Zeroth law: the total trapping gravity of a compact outer marginal surface has an upper bound, attained if and only if the trapping gravity is constant. Topology law: a compact future outer marginal surface has spherical topology. Signature law: an outer trapping horizon is generically spatial, otherwise null. Trapping law: spatial surfaces sufficiently close to a compact future outer marginal surface are trapped if they lie inside the trapping horizon. Confinement law: if the interior and exterior of a future outer trapping horizon are disjoint, an observer inside the horizon cannot get outside.