General laws of black-hole dynamics

A general definition of a black hole is given, and general ‘‘laws of black-hole dynamics’’ derived. The definition involves something similar to an apparent horizon, a trapping horizon, defined as a hypersurface foliated by marginal surfaces of one of four nondegenerate types, described as future or past, and outer or inner. If the boundary of an inextendible trapped region is suitably regular, then it is a (possibly degenerate) trapping horizon. The future outer trapping horizon provides the definition of a black hole. Outer marginal surfaces have spherical or planar topology. Trapping horizons are null only in the instantaneously stationary case, and otherwise outer trapping horizons are spatial and inner trapping horizons are Lorentzian. Future outer trapping horizons have nondecreasing area form, constant only in the null case: the ‘‘second law.’’ A definition of the trapping gravity of an outer trapping horizon is given, generalizing surface gravity. The total trapping gravity of a compact outer marginal surface has an upper bound, attained if and only if the trapping gravity is constant: the ‘‘zeroth law.’’ The variation of the area form along an outer trapping horizon is determined by the trapping gravity and an energy flux: the ‘‘first law.’’