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.

Figure 1

A dual-null foliation: the commuting evolution vectors $u_{\pm }=\partial /\partial x^{\pm }$ generate the transverse surfaces $S$, while their null normal projections $l_{\pm }$ generally do not commute.

Figure 2

Gravitational collapse: Penrose diagram of typical black-hole formation satisfying cosmic censorship, indicating the trapping horizon (blue bold line), null infinity (straight lines), and the center $R=0$ (curved line), which is regular outside the trapping horizon and singular and spatial inside. Advanced time $x^{+}$ runs diagonally up-rightwards and retarded time $x^{-}$ runs diagonally up-leftwards. Flow lines of the area-radius vector $(dR{)}^{♯}$ (green short dashes) and the Killing-like vector $\chi$ (magenta long dashes) are also indicated. Note the causal switch-over between $\chi$ and $(dR{)}^{♯}$ at the trapping horizon.