ar X iv : 0 71 1 . 23 30 v 2 [ gr - q c ] 1 6 N ov 2 00 7 Black Hole Evaporation and Generalized 2 nd Law with Nonequilibrium Thermodynamics

In general, when a black hole evaporates, there arises a net energy flow from black hole into its outside environment due to Hawking radiation and energy accretion onto black hole. The existence of energy flow means that the thermodynamic state of the whole system, which consists of a black hole and its environment, is in a nonequilibrium state. To know the detail of evaporation process, the nonequilibrium effects of energy flow should be taken into account. The nonequilibrium nature of black hole evaporation is a challenging topic including issues of not only black hole physics but also nonequilibrium physics. Using the nonequilibrium thermodynamics which has been formulated recently, this report shows: (1) the self-gravitational effect of black hole which appears as its negative heat capacity guarantees the validity of generalized 2nd law without entropy production inside the outside environment, (2) the nonequilibrium effect of energy flow tends to shorten the evaporation time (life time) of black hole, and consequently specific nonequilibrium phenomena are suggested. Finally a future direction of this study is commented. This report summarizes three rather long papers [8, 9, 10] without loading readers too much into the detail of discussions and analyses. 1 Challenge to nonequilibrium nature of black hole evaporation The black hole evaporation is one of interesting phenomena in black hole physics [1]. A direct treatment of time evolution of the evaporation process suffers from mathematical and conceptual difficulties; the mathematical one will be seen in the dynamical Einstein equation in which the source of gravity may be a quantum expectation value of stress-energy tensor of Hawking radiation, and the conceptual one will be seen in the definition of dynamical black hole horizon. Therefore an approach based on the black hole thermodynamics [1, 2, 3] is useful. Exactly speaking, dynamical evolution of any system is a nonequilibrium process. If and only if thermodynamic state of the system under consideration passes near equilibrium states during its evolution, its dynamics can be treated by an approximate method, the so-called quasi-static process. In this approximation, it is assumed that the thermodynamic state of the system evolves on a path lying in the state space which consists of only thermal equilibrium states, and the time evolution is described by a succession of different equilibrium states. However once the system comes far from equilibrium, the quasi-static approximation breaks down. In that case a nonequilibrium thermodynamic approach is necessary. For dissipative systems, the heat flow inside the system can quantify the degree of nonequilibrium nature [4, 5, 6]. For the black hole evaporation, when its horizon scale is larger than Planck size, it is relevant to describe the black hole itself by equilibrium solutions of Einstein equation, Schwarzschild, ReissnerNortström and Kerr black holes, because the evaporation proceeds extremely slowly and those equilibrium This report is a short summary of three papers [8, 9, 10]. A full combined and rearranged version will be published as an invited contribution chapter (titled BH evaporation as a nonequilibrium process) in an edited book Classical and Quantum Gravity Research Progress (tentative title), Nova Science Publisher. Based on proceedings and talks given at APCTP Jeju Meeting on Gravitation and Cosmology (Gallery House, Jeju, Korea, 2007), Dynamics and Thermodynamics of Black Holes and Naked Singularities II (Politecnico di Milano, Milan, Italy, 2007) and 16th General Relativity and Gravitation (Niigata Prefectural Civic Center, Niigata, Japan, 2006)