Why the Entropy of a Black Hole is A/4.

A black hole considered as a part of a thermodynamical system possesses the Bekenstein-Hawking entropy SH = AH/(4l 2 P), where AH is the area of a black hole surface and lP is the Planck length. Recent attempts to connect this entropy with dynamical degrees of freedom of a black hole generically did not provide the universal mechanism which allows one to obtain this exact value. We discuss the relation between the ’dynamical’ contribution to the entropy and SH , and show that the universality of SH is restored if one takes into account that the parameters of the internal dynamical degrees of freedom as well as their number depends on the black hole temperature. PACS numbers: 04.60.+n, 03.70.+k, 98.80.Hw Electronic address: [email protected] 1 According to the thermodynamical analogy in black hole physics, the entropy of a black hole in the Einstein theory of gravity equals SH = AH/(4l 2 P), where AH is the area of a black hole surface and lP = (h̄G/c ) is the Planck length [1,2]. The calculations in the framework of the Euclidean approach initiated by Gibbons and Hawking [3,4] relate this quantity with the tree-level contribution of the gravitational action, namely the action of the Euclidean black hole instanton. In this approach the entropy of a black hole has pure topologival origin, and it remains unclear whether there exist any real dynamical degrees of freedom which are responsible for it. The problem of the dynamical origin of the black hole entropy was intensively discussed recently (see e.g., [5–10,?]). The basic idea which was proposed is to relate the dynamical degrees of freedom of a black hole with its quantum excitations. This idea has different realizations. In particular, it was proposed to identify the dynamical degrees of freedom of a black hole with the states of all fields (including the gravitational one) which are located inside the black hole [6,11]. By averaging over states located outside the black hole one generates the density matrix of a black hole and can calculate the corresponding entropy S1. The main contribution to the entropy is given by inside modes of fields located in the very close vicinity of the horizon. It appears that the so defined S1 is divergent. It was argued that quantum fluctuations of the horizon may provide natural cut-off and make S1 finite. Simple estimations [6] of the cut-off parameter show that S1 ≈ SH . The generic feature of this as well as other ’dynamic’ approaches is that the entropy of a black hole arises at the one-loop level. The relation between the ’topological’ (tree-level) calculations and one-loop calculations based on the counting dynamical degrees of freedom of a black hole remains unclear. In particular, S1 depends on the number and characteristics of the fields, while SH does not. What is the general mechanism which provides the universal relation of the dynamically defined entropy with the universal thermodynamical value SH? The aim of this paper is to give a simple explanation of this puzzle. A black hole considered as a part of a thermodynamical system possesses a remarkable 2 property: its properties (size, gravitational field and so on) are determined only by one parameter (mass M), which in its turn in a state of a thermal equilibrium is directly connected with the temperature of the system. By varying the temperature one at the time inevitably changes all the internal parameters of the system. We show that namely this property together with scaling properties of the free energy results in the iniversality of the expression for SH . We illustrate the basic idea by considering a simple thermodynamical system described by the Hamiltonian