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 dynamical, spherically symmetric black holes from relative between coherent states quantum matter, generalising results by Hollands and Ishibashi on a Schwarzschild background. We consider state suitably chosen asymptotically vacuum scalar field theory propagating over dynamical hole. use conservation associated Kodama vector in spacetimes, flat spacetimes found Longo, Casini, Grillo, Pontiello. back-reaction matter metric region $\mathscr O$ outside From with field, we obtain equation form $(S + A/4)' =\Phi$, where $S$ $A$ apparent horizon area, $\Phi$ flux radiated at infinity. prime denotes derivative along outgoing light-rays.