Perturbative semiclassical entropy of dynamical black holes

Abstract

We consider perturbative quantum gravity as a quantum field theory of linearized metric perturbation on an asymptotically flat spacetime with a bifurcate Killing horizon. We include the perturbative gravitational constraints into the algebra of observables restricted to the right half of the future horizon of the spacetime. We use the boundary charge, associated to the horizon Killing field, as an auxiliary "observer" degree of freedom. The observables "dressed" with the additional charge are invariant under the Killing symmetry and generate a Type-$\text{II}_{\infty}$ von Neumann factor. We compute the von Neumann entropy of the reduced density matrix of a classical-quantum coherent state constructed from the metric perturbations and the "observer wavefunction". This von Neumann entropy satisfies an analogue of the first law of thermodynamics. We further show that this entropy is related to Hollands-Wald-Zhang entropy of the (second order) perturbed dynamical black hole through the flux of perturbations through the horizon and future null infinity.