The Entropy of Dynamical Black Holes and Apparent Horizons

Recently, we 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. 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. In stationary eras, this formula agrees with the usual area formula, but in nonstationary eras, we obtain a nontrivial correction term. In this talk, I will show that our new formula in general relativity agrees with the apparent horizon area to leading order. I will also discuss the generalization of the apparent horizon to general theories of gravity in the context of our formula for dynamical black hole entropy.