Kinetic description of classical matter infalling in black holes

A popular aspect of black holes physics is the mathematical analogy between their laws, coming from general relativity and the laws of thermodynamics. The analogy is achieved by identifying a suitable set of observables, precisely: \emph{(a)} $E=M$ (being $E$ the thermodynamic free energy and $M$ the mass of the BH), \emph{(b)} $T=\alpha \kappa $ (with $T$ the absolute temperature, $\kappa $ the so-called surface gravity on event horizon and $% \alpha $ a suitable dimensional constant) and \emph{(c)} $S=(1/8\pi \alpha)A $ (where $S$ is the thermodynamic entropy of the black hole and $A$ the surface of the event horizon). However, despite numerous investigations and efforts spent on the subject, the theoretical foundations of such identifications between physical quantities belonging to apparently unrelated frameworks are not yet clear. The goal of this work is to provide the contribution to the black hole entropy, coming from matter in the black hole exterior. We propose a classical solution for the kinetic description of matter falling into a black hole, which permits to evaluate both the kinetic entropy and the entropy production rate of classical infalling matter at the event horizon. The formulation is based on a relativistic kinetic description for classical particles in the presence of an event horizon. An H-theorem is established which holds for arbitrary models of black holes and is valid also in the presence of contracting event horizons.