M ar 1 99 7 Notes on the Generalised Second Law of

Several comments are given to previous proofs of the generalised second law of thermodynamics: black hole entropy plus ordinary matter entropy never decreases for a thermally closed system. Arguments in favour of its truism are given in the spirit of conventional thermo-dynamics. Thermodynamics (see [1, 2] for introduction) is one of the physical disciplines in which physical laws are governed by simplicity and generality (S&G). Due to the largeness of physical degrees of freedom, most macro-scopic systems are untraceable microscopically. Therefore, a systematic way, based on macroscopic S&G regardless of the microscopic details, is needed in order to extract the information we are interested, amongst which one of the most important is perhaps the equilibrium states. Thermodynam-ics suffices such task by employing extremising (maximising or minimising) principles [1]. Even though a microscopic model is introduced later to give thermodynamic quantities a statistical-mechanical interpretation, the major roles of thermodynamic quantities, and hence the extremising principles, are unquestionable. 1 Like all other microscopic physical laws, the status of second law of ther-modynamics (SLT) is more a postulate than a theorem [1, 2]: It is taken as one of the starting points for a long journey of searching statistical descriptions of a physical system. Therefore, it has to be checked up again and again throughout the journey. In other words, it can only be verified in a self-consistent way, or in a circular way by Callen's word [1]. And because SLT is an experience law applicable only at macroscopic scale, it cannot survive under the closest scrutiny from the viewpoint of microscopic unitary evolution. Even so, up to now, we have all of the reasons to believe that its S&G is unquestionable if we do not go beyond the border. On the other side of physics, we are used to thinking of space-time in geometric language after Einstein formulated general relativity, which is always regarded as a dynamic theory. Presumably, it would have surprised him very much, as we are, that, along with the development of black hole physics [3, 4], gravitational degrees of freedom can also be cast in the language of thermodynamics [5], as can be seen most transparently from the identification of the area of an event horizon with entropy [6, 7] and the formulation of four laws of black hole mechanics [8]. However, as being pointed out by Callen [1], thermodynamics by itself is not a theory; it is …