Se p 20 04 Thermodynamics of a collapsing shell in an expanding

We describe the quasi-static collapse of a radiating, spherical shell of matter in de Sitter space-time using a thermodynamical formalism. It is found that the specific heat at constant area and other thermodynamical quantities exhibit singularities related to phase transitions during the collapse. Because of the paradigm of inflation, de Sitter space-time has recently become the subject of great research interest. One aspect of research has been the conjectured dS-CFT correspondence [1] which is the analogue of Maldacena's AdS-CFT conjecture [2] for the case of a space-time with a positive cosmological constant. As in the latter case, the " gravitational " (dS) side of the correspondence is expected to yield the macroscopic description of the system, whereas the CFT (conformal field theory) should be able to describe the microphysics. A well known example for the AdS-CFT is given by the microscopical description of the thermodynamical laws of black holes [3]. Furthermore, applications of the de Sitter space Bousso bound [4], related to the Bekenstein bound for flat space-times, have also drawn much interest. The dS-CFT correspondence has not yet reached the degree of comprehension of the AdS-CFT, since the corresponding CFT has not been completely identified. Therefore, one usually proceeds by studying simplified " gravitational " models in order to obtain general features that the relevant CFT is required to possess. One such model is given by a thin spherical shell of matter in a de Sitter space-time, since it may represent a collection of D-branes forming the so called D-Sitter space-time of Ref. [5], or can be used as a tool for the study of the entropy bound in a very simplified physical environment [6]. Finally, one may also consider the " operational " approach to black hole entropy, as illustrated in Ref. [7], which leads one to the conclusion that the entropy of a black hole is the same as that stored in a shell of matter sitting just outside its Schwartzschild radius after it has collapsed from infinity. According to the AdS-CFT correspondence, a black hole is represented by a thermal state and its formation by the approach to such a state. A thermodynamical description of the gravitational collapse should thus be useful in the de Sitter case as well.