Energy and entropy conservation for dynamical black holes

The Ashtekar-Krishnan energy-balance law for dynamical horizons, expressing the increase in mass-energy of a general black hole in terms of the infalling matter and gravitational radiation, is expressed in terms of general trapping horizons, allowing the inclusion of null (isolated) horizons as well as spatial (dynamical) horizons. This first law of black-hole dynamics is given in differential and integral forms, regular in the null limit. 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 for dynamical black holes which the stationary Killing vector plays for stationary black holes. Identifying an energy flux, vanishing if and only if the horizon is null, allows a division into energy-supply and work terms, as in the first law of thermodynamics. The energy supply can be expressed in terms of area increase and a newly defined surface gravity, yielding a Gibbs-like equation, with a similar form to the so-called first law for stationary black holes. A Clausius-like relation suggests a definition of geometric entropy flux. Taking entropy as $\mathrm{\text{area}}/4$ for dynamical black holes, it is shown that geometric entropy is conserved: The entropy of the black hole equals the geometric entropy supplied by the infalling matter and gravitational radiation. The area or entropy of a dynamical horizon increases by the so-called second law, not because entropy is produced, but because black holes classically are perfect absorbers.