Universal Canonical Black Hole Entropy

Abstract

Non-rotating black holes in three and four dimensions are shown to possess a canonical entropy obeying the Bekenstein-Hawking area law together with a leading correction (for large horizon areas) given by the logarithm of the area with a universal finite negative coefficient, provided one assumes that the quantum black hole mass spectrum has a power law relation with the quantum area spectrum found in Non-perturbative Canonical Quantum General Relativity. The thermal instability associated with asymptotically flat black holes appears in the appropriate domain for the index characterising this power law relation, where the canonical entropy (free energy) is seen to turn complex.The microcanonical entropy (S MC ) of generic stationary non-rotating macroscopic four dimensional black holes, modelled as isolated horizons [1], has been shown to obey, for large fixed horizon areas, the Bekenstein-Hawking Area Law (BHAL) [2] together with an infinite series of corrections, each term of which is finite and calculable [3]. The leading correction is logarithmic in area (BH entropy) with a coefficient (calculated in [3]) that has been argued to be universal [4], [5]. These results are obtained within Non-perturbative Canonical Quantum General Relativity (NCQGR), and crucially depend on the discrete spectrum of geometrical observables, especially the area [6].It is well-known in elementary statistical mechanics [7] that, unlike the canonical entropy, the definition of the microcanonical entropy in fact is not unique. One can define it as the logarithm of the degeneracy of quantum microstates characterizing the system, as has been employed in the calculations cited above [2], [3]. Alternatively, one can define it as the logarithm of the density of these microstates as a function of the energy. While both these definitions yield the same area law, they donot lead to the same logarithmic corrections. The difference actually depends on the relation between the energy (mass) of the black hole and the horizon area. This difference plays an important role when the NCQGR result is used to compute the effect of thermal fluctuations on the canonical entropy of black holes [8],In [8], the precise distinction and relationship between corrections to the BHAL arising due to quantum spacetime fluctuations, and due to thermal fluctuations within a canonical framework (admittedly heuristic), has been delineated. The logarithmic corrections due to quantum spacetime fluctuations are interpreted as 'finite size effects'. 1 Thermal fluctuations then generate additional log-area corrections which are computed in [8] for generic non-rotating antide Sitter black holes. Fortunately, the thermal fluctuation contributions to the canonical entropy can be expressed entirely in terms of the microcanonical entropy which has been computed within NCQGR as already mentioned. Unfortunately, however, the extra logarithmic contribution to the canonical entropy owing to the difference in the two definitions of the microcanonical entropy has been missed out in [8...

Universal canonical entropy for gravitating systems

“…Once again, the answer is in the affirmative with some caveats. The result found in [7], at least for the leading log area corrections, turns out to be universal in the sense that, just like the BHAL, it holds for all black holes independent of their parameters.…”

A comment on black hole entropy or does nature abhor a logarithm?

“…In this regard, let us first turn to some relevant papers by Gour and the current author [26], and Chatterjee and Majumdar [27]. 8 In these treatments, the black hole is assumed (as noted above) to be in a state of thermal equilibrium with its surroundings; for descriptive purposes, one might envision a black hole enclosed in a "reflective box".…”

When conceptual worlds collide: The generalized uncertainty principle and the Bekenstein-Hawking entropy

“…Using the GUP as our primary input, we present a perturbative calculation of the quantum-corrected entropy, which can readily be extended to any desired order. Note that, here, we consider strictly the microcanonical corrections, 6 as the canonical corrections have been dealt with elsewhere (e.g., [11][12][13]30]). The paper concludes with a discussion that emphasizes the relevance of our outcome.…”