The Entropy of Dynamical Black Holes and the Second Law

Recently, we have proposed a new formula for the entropy of a dynamical black hole---valid to leading order for perturbations off of a stationary black hole background---in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in $n$ dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. For general relativity, our formula gives the entropy of a dynamical black hole as its area minus an integral involving the expansion of the null generators of the horizon. Our formula for entropy in a general theory of gravity is obtained from the requirement that a ``local physical process version'' of the first law of black hole thermodynamics holds for perturbations of a stationary black hole. In this talk, we will show that for first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the leading order change in entropy occurs at second order in perturbation theory, and the second law is obeyed at leading order if and only if the ``modified canonical energy flux'' is positive. This is the case in general relativity but presumably would not be true in more general theories of gravity.