  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.
