A Study of the Entropy of Dynamical Black Holes

This dissertation proposes and studies 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 any dimension. By requiring that a "local physical process version" of the first law of black hole thermodynamics hold for perturbations of a stationary black hole, we modify the usual Noether charge formula by a dynamical correction term which contributes nontrivially in nonstationary eras. It is shown that with the inclusion of this new term, our entropy obeys the second law of black hole thermodynamics for first order perturbations sourced by external matter that satisfies the null energy condition. In addition, it immediately follows that our entropy formula in general theories of gravity increases only as the external matter crosses the horizon instead of in anticipation of it crossing, the latter being a peculiar property of the area prescription for black hole entropy in general relativity given by the Bekenstein-Hawking formula. One can thus understand our new dynamical correction term as "correcting for" the growth of the event horizon due to matter yet to fall into the black hole. 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 (as is the case in general relativity but presumably would not hold in more general theories of gravity). In general relativity, our formula for the entropy of a dynamical black hole differs from the standard Bekenstein-Hawking formula $A/4$ by a term involving the integral of the expansion of the null generators of the horizon. We show that, to leading perturbative order, our dynamical entropy in general relativity is equal to $1/4$ of the area of the apparent horizon. We can therefore also interpret the first and second laws we proved for general relativity as applying to apparent horizons. Interestingly, our formula for the entropy of a dynamical black hole differs from a formula proposed independently by Dong and by Wall, and we obtain the general relationship between their formula and ours. We also explore a potential definition of apparent horizons in general theories of gravity, formulated in relation to our entropy formula and the Dong-Wall formula. Finally, we 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.