Mechanism of Generation of Black Hole Entropy in Sakharov's Induced Gravity

Abstract

The mechanism of generation of the Bekenstein-Hawking entropy $S^{BH}$ of a black hole in the Sakharov's induced gravity is proposed. It is suggested that the "physical" degrees of freedom, which explain the entropy $S^{BH}$, form only a finite subset of the standard Rindler-like modes defined outside the black hole horizon. The entropy $S_R$ of the Rindler modes, or entanglement entropy, is always ultraviolet divergent, while the entropy of the "physical" modes is finite and it coincides in the induced gravity with $S^{BH}$. The two entropies $S^{BH}$ and $S_R$ differ by a surface integral Q interpreted as a Noether charge of non-minimally coupled scalar constituents of the model. We demonstrate that energy E and Hamiltonian H of the fields localized in a part of space-time, restricted by the Killing horizon $\Sigma$, differ by the quantity $T_H Q$, where $T_H$ is the temperature of a black hole. The first law of the black hole thermodynamics enables one to relate the probability distribution of fluctuations of the black hole mass, caused by the quantum fluctuations of the fields, to the probability distribution of "physical" modes over energy E. The latter turns out to be different from the distribution of the Rindler modes. We show that the probability distribution of the "physical" degrees of freedom has a sharp peak at E=0 with the width proportional to the Planck mass. The logarithm of number of "physical" states at the peak coincides exactly with the black hole entropy $S^{BH}$. It enables us to argue that the energy distribution of the "physical" modes and distribution of the black hole mass are equivalent in the induced gravity. Finally it is shown that the Noether charge Q is related to the entropy of the low frequency modes propagating in the vicinity of the bifurcation surface $\Sigma$ of the horizon.