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 on the relative entropy on a Schwarzschild background. 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 results on the entropy of coherent states in flat spacetimes found by Longo, and Casini, Grillo, and Pontiello. We consider the back-reaction of the quantum matter on the metric in a region $\mathscr O$ outside the black hole. From the conservation law associated with the Kodama vector field, we obtain an equation in the form $(S + A/4)' =\Phi$, where $S$ is the relative entropy between coherent states of the scalar field, $A$ is the apparent horizon area, and $\Phi$ is the flux radiated at infinity. The prime denotes a derivative along the outgoing light-rays.