Black hole thermodynamic potentials for asymptotic observers

The generalized second law states the total entropy of any closed system as the Universe cannot decrease if we include black hole entropy. From the point of view of an asymptotic observer, a black hole can be described at late time as an open system at fixed temperature which can radiate energy and entropy to infinity. I argue that for massless free quantum fields propagating on a black hole background, we can define a black hole dynamical free energy using observables defined at future null infinity which decreases on the successive cross sections of I+. The proof of this spontaneous evolution law is similar to Wall's derivation of the generalized second law and relies on the monotonicity properties of the relative entropy. I discuss first the simpler case of the Schwarzschild background in which the gray body factors are neglected and show that in this case the free energy only depends on the Bondi mass, the Hawking temperature and the von Neumann entropy of the propagating quantum fields. Then I argue that taking into account the gray body factors adds a new term to the thermodynamic potential involving the number of particles detected at I+ conjugated to a chemical potential. Finally, I discuss the case of the Kerr black hole for which an angular momentum piece needs to be added to the free energy.