Fundamental thermodynamical equation of a self-gravitating system

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Abstract
The features of the fundamental thermodynamical relation (expressing entropy as function of state variables) that arise from the self-gravitating character of a system are analyzed. The models studied include not only a spherically symmetric hot matter shell with constant particle number but also a black hole characterized by a general thermal equation of state. These examples illustrate the formal structure of thermodynamics developed by Callen as applied to a gravitational configuration as well as the phenomenological manner in which Einstein equations largely determine the thermodynamical equations of state. We consider in detail the thermodynamics and quasistatic collapse of a self-gravitating shell. This includes a discussion of intrinsic stability for a one-parameter family of thermal equations of state and the interpretation of the Bekenstein bound. The entropy growth associated with a collapsing sequence of equilibrium states of a shell is computed under different boundary conditions in the quasistatic approximation and compared with black hole entropy. Although explicit expressions involve empirical coefficients, these are constrained by physical conditions of thermodynamical origin. The absence of a Gibbs-Duhem relation and the associated scaling laws for self-gravitating matter systems are presented. © 1996 The American Physical Society.