  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 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 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.

General Relativity and Quantum Cosmology