Advances in High Energy Physics

Volume 2019, Article ID 6570896, 6 pages

https://doi.org/10.1155/2019/6570896

## An Area Rescaling Ansatz and Black Hole Entropy from Loop Quantum Gravity

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, Mexico D.F. 04510, Mexico

Correspondence should be addressed to Abhishek Majhi; moc.liamg@ihjam.kehsihba

Received 12 December 2018; Accepted 25 February 2019; Published 1 April 2019

Academic Editor: Elias C. Vagenas

Copyright © 2019 Abhishek Majhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Considering the possibility of ‘renormalization’ of the gravitational constant on the horizon, leading to a dependence on the level of the associated Chern-Simons theory, a rescaled area spectrum is proposed for the nonrotating black hole horizon in loop quantum gravity. The statistical mechanical calculation leading to the entropy provides a unique choice of the rescaling function for which the Bekenstein-Hawking area law is yielded without the need to choose the Barbero-Immirzi parameter . is determined, rather than being chosen, by studying the limit in which the ‘renormalized’ gravitational constant on the horizon asymptotically approaches the ‘bare’ value. The possible physical dynamics behind the ‘renormalization’ is discussed.

#### 1. Introduction

Loop quantum gravity (LQG) provides a platform for the calculation of entropy for nonrotating (assumed henceforth) black holes from the first principles, albeit in the kinematic framework [1]. The main criticism of this approach has been the necessity to choose a particular value of the Barbero-Immirzi parameter , which is a dimensionless constant that characterizes the family of inequivalent kinematic quantization sectors of LQG, to obtain the Bekenstein-Hawking area law (BHAL) [2]. If the derivation is correct, then it is expected that one should get the BHAL without having to choose . As it appears, the full knowledge of the dynamics of LQG, the horizon degrees of freedom and the correct semiclassical limit of the theory are required to achieve this goal [2], which, unfortunately, does not seem to become available in near future. Nonetheless, the kinematic framework holds the potential to give us the hints towards the correct physical elements that give rise to the black hole entropy, which in turn may lead the path towards understanding the underlying dynamics. Here, I shall point out that there is a possibility of the involvement of a ‘renormalization’ of the gravitational constant* on the horizon* and incorporation of this effect in the entropy calculation leads us to the BHAL from LQG* without having to choose *. Since I shall base my arguments on analogy, all the words associated with renormalization will appear in quotes. I shall heuristically argue that the quantum field theoretic structure that effectively describes the horizon degrees of freedom suggests that there is a possibility for a* rescaled* area spectrum to be used for the black hole horizon in LQG due to the ‘renormalization’ of the gravitational constant on the horizon. Further, the calculation of entropy with this rescaled area spectrum provides us with the unique rescaling function that leads to the BHAL without having to choose . The value of is* determined*, irrespective of obtaining the BHAL, by studying how the ‘renormalized’ gravitational constant on the horizon should* asymptotically* approach its ‘bare’ value in a limit that can be viewed as the ‘fixed point’ of the ‘renormalization group flow’ on the horizon. The novelty of this, albeit heuristic, work lies in the fact that the value of is now determined by a physical consistency requirement rather than being chosen just to match a desired result. Further, it appears that a ‘screening’ effect possibly is the physical cause that underlies the ‘renormalization’ procedure.

#### 2. Motivation

The entire procedure of the entropy calculation for black holes in LQG consists of the following steps:

(1)* Horizon field dynamics*: the effective quantum field dynamics on the horizon (of topology ) is governed by a quantum Chern-Simons (CS) theory on a punctured 2-sphere and these punctures act as point-like sources coupled to the CS field strength [1]. The Hilbert space of this theory provides the state space of the horizon degrees of freedom that give rise to the entropy [1, 3, 4].

(2)* Spectrum of the source*: consider any arbitrary geometric 2-surface that is topologically . The quantum area of such a surface, in LQG, is given by (setting ) and any can take values like . are the quantum numbers carried by the intersection points (punctures) of the spin network edges with that 2-surface, being the total number of punctures [5]. This is the same area spectrum that is used during the entropy calculation for black holes, with a crucial modification due to the interplay between quantum geometry and the CS theory on the horizon, that can take values like , where , being the classical area of the black hole [1]. Hence, the contribution from an individual puncture (point-like source of the CS theory on the horizon) is with .

(3)* Statistical mechanics*: having the estimate of the microstate count from the first step and the area spectrum of the black hole from LQG in the second step, the statistical mechanics is applied to calculate the entropy.

Now, let me focus on the second step. It implies, in principle, the quantum area of an arbitrary geometric 2-surface of topology can be infinite, irrespective of the classical area of the surface. So, it is expected on physical grounds that this should* not* be the case when the concerned 2-surface is that of a physical object and the value of should acquire an upper cut-off provided by the underlying physics associated with the surface of the physical object. This is exactly what happens for the black hole horizon. The value of acquires an upper bound , where the is the level of the CS theory associated with the horizon; i.e., the first step plays a crucial role. Therefore, the theory governing the physics associated with the horizon naturally provides this upper bound. This is a result which is already manifested from the LQG kinematics and the effective horizon theory. However, the lack of knowledge about the full dynamics of a quantum black hole in LQG leaves room for some physics, associated with the horizon degrees of freedom contributing to the entropy of a black hole, which may be missing in the kinematics. As I shall argue, the information that is already available from the kinematics (the first step), indeed, hints towards such a possibility.

The field equations on a black hole horizon are that of a CS theory coupled to sources: where is the curvature of the CS gauge fields on the horizon and are the sources from the bulk. In the quantum theory, the source is nonzero only at the punctures. Effectively, the theory on the horizon is a quantum CS theory coupled to point-like sources on a 2-sphere. The spectrum associated with a single source is calculated for an arbitrary 2-sphere where there is no coupling with any field strength and it is given by So, these punctures on an arbitrary 2-sphere are like ‘free’ excitations and the spectrum in (2) can be regarded as ‘bare’ spectrum. However, these ‘free’ excitations get coupled to the CS field strength in case the 2-sphere is a cross-section of a black hole horizon.

Now, in quantum field theory (QFT), the physical parameters like mass, charge, etc. associated with free particles get renormalized due to their coupling with fields, consequently affecting the mode spectrum. Analogously, in the present scenario, there is a possibility of in (2) (since and always appear as a product in the kinematics of LQG, one should consider the ‘renormalization’ of rather than alone [6]), which can be viewed as the “mode spectrum” for the sources [7], getting ‘renormalized’ due to the coupling with the CS field strength. This ‘renormalized’ , say , should depend on , which is the cut-off for the allowed values of that appears naturally in the theory on the horizon resulting from its gauge invariance [1]. Since the physical process involved with this ‘renormalization’ is associated with the quantum theory* on the horizon,* this can only affect the microscopic physics localized on the horizon.

Although this heuristic ‘renormalization’ argument is only at the level of an analogy made from a QFT viewpoint, the possibility of the scenario cannot be completely ruled out unless one gets to know the full dynamics of the theory.

#### 3. Rescaled Area Spectrum: An Ansatz

As I have just argued, (the ‘renormalized’ ), which enters the area spectrum of the black hole horizon, can only depend on and on the value of because there are no other quantities intrinsic to the theory on the horizon. If is the change in the value of , then Since and are both dimensionless, simply on dimensional grounds, . Further, as , must tend to zero because the sources get more weakly coupled to the CS field strength and the ‘renormalized’ spectrum should approach towards the ‘bare’ spectrum (this argument will be discussed more elaborately later).

Based on these arguments I propose that the area contribution from a single puncture with quantum number , for a black hole horizon in LQG, is given by where is the ‘renormalized’ gravitational constant on the horizon and has the following properties:(i)Since the ‘bare’ spectrum needs to be positive definite, one has . It is required that so that the ‘renormalized’ spectrum is positive definite too.(ii), i.e., as the sources get more weakly coupled to the CS field strength, asymptotically approaches .

Hence, I consider a* rescaled* area spectrum for the black hole horizon in LQG. As I shall show, the statistical mechanical calculation provides a unique choice of the function that leads to the BHAL and satisfies property (i). Satisfaction of property (ii) by , which is a physical consistency requirement, will determine . It is crucial to note that property (i) and property (ii) are independent of each other.

#### 4. Entropy

I shall consider here black holes with classical area . Quantum area of a cross-section of a black hole horizon, with spin configuration :where and number of punctures with quantum number . Since ranges from to , hence . Also, since is positive definite, the definition of suggests that . So, the quantum theory of the horizon is valid for . Now, I shall implement the method of most probable distribution [8, 9] to calculate the microcanonical entropy of the black hole in the area ensemble. One can find the calculation (but, with the ‘bare’ spectrum) in [10]. So, I shall provide the main steps and results here to avoid an unnecessary repeat.

The microstate count for a spin configuration for which : where and satisfies the following constraint (considering ): Then one finds the most probable configuration (MPC) by solving the following equation: where is the Lagrange multiplier. This yields the distribution for the MPC to be where and the entropy comes out to be

Now, a sum over on both sides of (10) leads to the following consistency condition: Equation (12), in principle, should lead to a solution for as a function of . To avoid the mathematical complication of finding this solution analytically, I plot (using Mathematica) the function where , , , and are some numbers. The plot (yellow coloured in Figure 1) fits to the curve obtained by plotting versus from (12), up to a ‘very good’ approximation, for I do not provide here a mathematical estimate of ‘how good’ a fit it is. This is just an ‘optical’ fit obtained by numerical experiments.