Entropy for spherically symmetric, dynamical black holes from the relative entropy between coherent states of a scalar quantum field

Abstract

The goal of this paper is to prove an area law for the entropy of dynamical, spherically symmetric black holes from the relative entropy between coherent states of the quantum matter, generalising the results by Hollands and Ishibashi (2019 Class. Quantum Grav. 36 195001). We consider the relative entropy between a coherent state and a suitably chosen asymptotically vacuum state for a scalar quantum field theory propagating over a dynamical black hole. We use the conservation law associated to the Kodama vector field in spherically symmetric spacetimes, and the recent results on the relative entropy between coherent states found in Longo (2019 Lett. Math. Phys. 109 2587) and Casini et al (2019 Phys. Rev. D 99 125020). We consider the back-reaction of the quantum matter on the metric. From the conservation law associated with the Kodama vector field, we obtain an equation for the derivative along the outgoing lightrays of the generalised entropy, defined as the sum of the relative entropy between coherent states of the scalar field and one quarter of the horizon area.

Kodama-like vector fields in axisymmetric spacetimes

“…As mentioned before, the Kodama symmetry in spherically symmetric spacetimes has been a very useful tool in the field of semi-classical gravity, e.g. to study the back reaction of quantum effects on stationary black hole spacetimes, or to investigate quantum field theory in spherically symmetric dynamical backgrounds, see [11,12]. It would be fascinating to study, to which extent our results enable a first generalization of well-established results in spherical symmetry to axisymmetric spacetimes.…”

“…Especially in the field of semi-classical gravity and black hole evaporation, the Kodama vector has proven to be very useful. For instance, recent works use the existence and the properties of the Kodama vector field to compute the relative entropy between two coherent states of a quantum field theory in spherically symmetric dynamical spacetimes, see [11,12].…”

Modular Hamiltonian for de Sitter diamonds

“…For some recent studies of relative entropy in quantum field theory, we refer the reader to refs. [40][41][42][43][44]. In fact, the relative entanglement entropy can be used instead of the von Neumann entropy to derive thermodynamic laws [45,46], and both the relative entropy and its rigorous formulation using modular theory can be employed in the holographic correspondence [47][48][49][50] instead of the divergent entanglement entropy.…”