The Entropy of Dynamical Black Holes

We study the entropy of black holes in arbitrary classical diffeomorphism-invariant Lagrangian theories of gravity in $n$ dimensions. We propose a new formula for the entropy of a dynamical black hole, valid to second order in perturbation theory off of a stationary black hole background. In stationary eras, this formula agrees with the Noether charge formula proposed by one of us, but in nonstationary eras this formula introduces a nontrivial correction term to the Noether charge formula. In particular, in general relativity, our formula gives the entropy of a dynamical black hole as the area minus the integral of the expansion of the null generators of the horizon. Using our formula for the entropy of a black hole, we prove the ``physical process version'' of the first law of black hole thermodynamics for arbitrary perturbations of a stationary black hole. We also show that this entropy formula obeys a ``second law'' of black hole thermodynamics at leading order. We conclude with a derivation of a relation between our formula with those proposed recently by Wall and Dong.