The quantum of area $\Delta A = 8\pi l_P^{2}$ and a statistical interpretation of black hole entropy

Abstract

In contrast to alternative values, the quantum of area $\Delta A = 8\pi l_P^{2}$ does not follow from the usual statistical interpretation of black hole entropy; on the contrary, a statistical interpretation follows from it. This interpretation is based on the two concepts: nonadditivity of black hole entropy and Landau quantization. Using nonadditivity a microcanonical distribution for a black hole is found and it is shown that the statistical weight of black hole should be proportional to its area. By analogy with conventional Landau quantization, it is shown that quantization of black hole is nothing but the Landau quantization. The Landau levels of black hole and their degeneracy are found. The degree of degeneracy is equal to the number of ways to distribute a patch of area $8\pi l_P^{2}$ over the horizon. Taking into account these results, it is argued that the black hole entropy should be of the form $S_{bh} =2\pi\cdot\Delta\Gamma $, where the number of microstates is $\Delta\Gamma = A/8\pi l_P^{2}$. The nature of the degrees of freedom responsible for black hole entropy is elucidated. The applications of the new interpretation are presented. The effect of noncommuting coordinates is discussed.