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

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.