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. PACS: 65.40.Gr 04.70.Bw 04.70.Dy.
PACS numbers:The remarkable mathematical analogy between the laws of thermodynamics and black hole (BH) physics following from classical general relativity still escapes a complete and satisfactory interpretation. In particular it is not yet clear whether this analogy is merely formal or leads to an actual identification of physical quantities belonging to apparently unrelated framework. The analogous quantities are E ↔ M , T ↔ ακ and S ↔ (1/8πα)A, where A and κ are the area and the surface gravity of the BH, while α is a constant. A immediate hint to believe in the thermodynamical nature of BH comes from the first analogy which actually regards a unique physical quantity: the total energy. However, at the classical level there are obstacles to interpret the surface gravity as the BH temperature since a perfectly absorbing medium, discrete or continuum, which is by definition unable to emit anything, cannot have a temperature different from absolute zero. An reconciliation was partially achieved by in 1975 by Hawking [1], who showed, in terms of quantum particle pairs nucleation, the existence of a thermal flux of radiation emitted from the BH with a black body spectrum at temperature T = κ/2πk B (Hawking BH radiation model). The last analogy results the most intriguing, since the area A should essentially be the logarithm of the number of microscopic states compatible with the observed macroscopic state of the BH, if we identify it with the Boltzmann definition. In such a context, a further complication arise when one strictly refers to the internal microstates of the BH, since for the infinite red shift they are inaccessible to an external observer. An additional difficulty with the identification S ↔ (1/8πα)A, however, follows from the BH radiation model, since it predicts the existence of contracting BH for which the radius of the BH may actually decrease. To resolve this difficulty a modified constitutive equation for the entropy was postulated [2,3], in order to include the contribution of the matter in the BH exterior, by setting(S ′ denoting the so-called Bekenstein entropy) where S is the entropy carried by the matter outside the BH andidentifies the contribution of the BH. As a consequence a generalized second lawwas proposed [2, 3] which can be viewed as nothing more than the ordinary second law of thermodynamics applied to a system containing a BH. From this point of view one notices that, by assumption and in contrast to the first term S, S bh cannot be interpreted, in a proper ...
