Kavli Affiliate: Robert M. Wald

| First 5 Authors: Stefan Hollands, Robert M. Wald, Victor G. Zhang, ,

| Summary:

We propose 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. In general relativity, our formula for the entropy of a dynamical black

hole is $1/4$ of the horizon area plus a term involving the integral of the

expansion of the null generators of the horizon, which we show is $1/4$ of the

area of the apparent horizon to leading order. Our formula for entropy in a

general theory of gravity obeys a "local physical process version" of the first

law of black hole thermodynamics. 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 second

law is obeyed at leading order if and only if the "modified canonical energy

flux" is positive (as is the case in general relativity but presumably would

not hold in general theories). We obtain a general relationship between our

formula for the entropy of a dynamical black hole and a formula proposed

independently by Dong and by Wall. We then consider the generalized second law

in semiclassical gravity for first order perturbations of a stationary black

hole. We show that the validity of the quantum null energy condition (QNEC) on

a Killing horizon is equivalent to the generalized second law using our notion

of black hole entropy but using a modified notion of von Neumann entropy for

matter. On the other hand, the generalized second law for the Dong-Wall entropy

is equivalent to an integrated version of QNEC, using the unmodified von

Neumann entropy for the entropy of matter.

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