High-order corrections to the entropy and area of quantum black holes

Abstract

The celebrated area-entropy formula for black holes has provided the most important clue in the search for the elusive theory of quantum gravity. We explore the possibility that the (linear) areaentropy relation acquires some smaller corrections. Using the Boltzmann-Einstein formula, we rule out the possibility for a power-law correction, and provide severe constraints on the coefficient of a possible log-area correction. We argue that a non-zero logarithmic correction to the area-entropy relation, would also imply a modification of the area-mass relation for quantum black holes.The necessity in a quantum theory of gravity was already recognized in the 1930s. However, despite the flurry of research we still lack a complete theory of quantum gravity. It is believed that black holes may play a major role in our attempts to shed some light on the nature of a quantum theory of gravity (such as the role played by atoms in the early development of quantum mechanics).In particular, the area-entropy relation S BH = A/4ℓ2 P[1] for black holes has served as a valuable element of guidance for the quantum-gravity research. The intuition that has led Bekenstein to this discovery is actually based on very simple ingredients. In particular, to elucidate the relation between area and entropy, it is instructive to use a semiclassical version of Christodoulou's reversible processes [2,3], in which a particle is absorbed by a black hole. Bekenstein [1,4] has shown that the Heisenberg quantum uncertainty principle imposes a lower bound on the increase in black hole surface areawhere γ is a dimensionless constant of order unity, and ℓ P = G c 3 1/2h 1/2 is the Planck length. Remarkably, this bound is universal in the sense that it is independent of the black-hole parameters [5]. The universality of the fundamental lower bound is clearly a strong evidence in favor of a uniformly spaced area spectrum for quantum black holes (see Ref. [7]). Hence, one concludes that the quantization condition of the black-hole surface area should be of the formFurthermore, using the fact that the minimum increase in black-hole surface area should correspond to a minimum increase of its entropy (in order to compensate for the loss of the particle's entropy), one arrives to the proportionality between black-hole surface area and entropy S BH = ηA/ℓ 2 P . It should be recognized however that the precise values of the proportionality constants γ and η cannot be inferred from this simple line of reasoning. The very nature of Heisenberg quantum uncertainty principle allows only an order-of-magnitude estimate of the minimal increase in black-hole surface area. As a consequence, the proportionality constant η was fixed only few years later by Hawking, who determined the characteristic temperature of black holes [8].Mukhanov and Bekenstein [9,10,7] have suggested an independent argument in order to determine the value of the coefficient γ. In the spirit of Boltzmann-Einstein formula in statistical physics, they relate g n ≡ exp[S BH (n)] to the number of micro...

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

“…By way of some simple manipulations, we can re-express the GUP in the following 4 Let us point out two recent studies that have endeavored to at least put restrictions on the value of the logarithmic prefactor [14,15]. Nevertheless, the former was specific to loop quantum gravity whereas the latter used a premise that is believed to be contradictory with the same theory.…”

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

“…Supposing that the value of b cannot be fixed, one might ask if there is any means for at least constraining this prefactor. As it so happens, this very question was recently addressed in a paper by Hod [13]. 2 The premise of this work was to take seriously the statistical interpretation of the entropy in the context of Bekenstein's notion of a quantum black hole area spectrum [15].…”

“…3 Given this line of reasoning, the degeneracy of the nth area eigenvalue (which translates into the exponential of the entropy) should be constrained as follows [13]:…”

“…More explicitly, A n was written as an expansion in terms of n, and S bh was expanded in terms of A; where both expansions include a linear term, logarithmic term, a constant, as well as any number of sub-leading power-law terms (with the latter inclusion turning out to be irrelevant). To comply with the constraint equation (4), it was then shown that the entropy expansion must necessarily reduce to the form [13] …”

“…Before proceeding on with the main discussion, let us point out that Hod's paper [13] considers the quantum corrections at the most fundamental level; thus implying that the predictions therein refer strictly to the microcanonical corrections. So, under the premise of a comparison with [13], one might ask if it is fair to include canonical corrections in our calculation.…”