Microscopic derivation of the Bekenstein-Hawking entropy for Schwarzschild black holes

Abstract

In this paper, we successfully derive the Bekenstein-Hawking entropy for Schwarzschild black holes in various dimensions by using a non-trivial phase space. It is appealing to notice that the thermodynamics of a Schwarzschild black hole actually behaves like that of a 1-dimensional quantum mechanical system. Our result suggests that black hole should be viewed as a system with the equation of state P = ρ, and it also suggests that a holographic stage should exist in the early universe.

Running of effective dimension and cosmological entropy in early universe

“…The fact that black holes have temperature and entropy implies that there must be some kinds of microscopic degrees of freedom behind it. In order to provide a statistical interpretation to the thermodynamic behaviors (1), we consider the Schwarzschild black hole as composed of microscopic particles that we call QG particles for convenience [2]. In fact such ideas are not new, for example, the charged AdS black holes have been suggested to be consisting of "molecules" with attractive or repulsive interactions [8,9].…”

“…Now turn to the system consisting of QG particles. According to [2], in order to account for the thermodynamics (1), we should take…”

“…with size L s = 9V /G. 2 Obviously this means that the black hole can be described as a 1 + 1 dimensional quantum system at least from the thermodynamic viewpoint, despite the fact that it actually lives in a 3 + 1 dimensional spacetime. The derivation can be generalized to higher dimensions, and the necessity of w = 1 for getting the exact Bekenstein-Hawking entropy has been emphasized in [2].…”

“…In [1] Bekenstein and Mayo revealed a secret behind this kind of thermodynamics, that is, black holes are effectively 1 + 1 dimensional as far as entropy flow is concerned. Recently, in [2] Xiao showed more directly that the thermodynamics (1) is 1 + 1 dimensional in essence. Quantum gravitational (QG) particles with non-trivial phase a e-mail: xiaoyong@hbu.edu.cn (corresponding author) space have been introduced there in order to provide a microscopic explanation to (1), with the equation of state (EoS) w ≡ P/ρ = 1 derived as a byproduct.…”

“…The paper is organized as follows. First we review the thermodynamics (1) can be microscopically explained by introducing the so-called QG particles and it is effectively 1 + 1 dimensional [2]. Next we study the universe filled with a single kind of constituents for simplicity and derive the concerned properties including the entropy inside particle horizon and effective dimension.…”