The wave function of a black hole and the dynamical origin of entropy

Recently it was proposed to explain the dynamical origin of the entropy of a black hole by identifying its dynamical degrees of freedom with the physical modes propagating in the black hole interior. The present paper contains the further development of this approach. The no-boundary proposal (analogous to the cosmological no-boundary wave function) is put forward for the pure quantum state of a black hole. This is a functional on the configuration space of physical fields (including the gravitational one) inhabiting the three-dimensional space of the Einstein-Rosen bridge topology. For linearized field perturbations on the Schwarzschild-Kruskal background this no-boundary wave function coincides with the Hartle-Hawking vacuum state. The invariant definition of interior and exterior modes is proposed and the duality existing between them is discussed. The density matrix ρH describing the internal state of a black hole is obtained by averaging over the exterior modes. The ``dynamical'' entropy, determined by -TrρHlnρH, is calculated. It is shown that the one-loop contribution to the ``dynamical'' entropy calculated for a given black hole background is divergent. The notion of an instant horizon is proposed, which separates the interior from the exterior of the black hole. It is argued that quantum fluctuations of the instant horizon inherent in the proposed formalism may give the necessary cutoff and provide a black hole with finite dynamical entropy. The relation between the ``dynamical'' entropy and the standard ''dynamical' entropy and the standard Bekenstein-Hawking (``thermodynamical'') entropy is briefly discussed.