title: Energy conservation for dynamical black holes.
abstract: An energy conservation law is described, expressing the increase in mass-energy of a general black hole in terms of the energy densities of the infalling matter and gravitational radiation. This first law of black-hole dynamics describes how a black hole grows and is regular in the limit where it ceases to grow. An effective gravitational-radiation energy tensor is obtained, providing measures of both ingoing and outgoing, transverse and longitudinal gravitational radiation on and near a black hole. Corresponding energy-tensor forms of the first law involve a preferred time vector which plays the role of a stationary Killing vector. Identifying an energy flux, vanishing if and only if the horizon is null, allows a division into energy supply and work terms. The energy supply can be expressed in terms of area increase and a newly defined surface gravity, yielding a Gibbs-like equation.
authors.affiliation: Department of Physics, National Central University, Jhongli, Taoyuan 320, Taiwan. sean_a_hayward@yahoo.co.uk
authors.forename: Sean A
authors.initials: SA
authors.lastname: Hayward