Barrow Fractal Entropy and the Black Hole Quasinormal Modes ∗ Everton M.C

Physics Letters B 807 (2020) 135602

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Physics Letters B

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Barrow fractal entropy and the black hole quasinormal modes ∗ Everton M.C. Abreu a,b,c, , Jorge Ananias Neto b a Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23890-971, Seropédica, RJ, Brazil b Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil c Programa de Pós-Graduação Interdisciplinar em Física Aplicada, Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil a r t i c l e i n f o a b s t r a c t

Article history: Using the Boltzmann-Gibbs statistical mechanics together with the quasinormal modes of the black holes Received 24 June 2020 and the Bekenstein-Hawking area entropy law, we can determine univocally the lowest possible value for Accepted 29 June 2020 the spin j, which is jmin = 1, in the framework of the√ Loop Quantum Gravity theory. Subsequently, the Available online 8 July 2020 value of Immirzi parameter is given by γ = ln 3/(2π 2). In this paper, we have demonstrated that if we Editor: N. Lambert use the Barrow formulation for the black hole entropy then the minimum value of the label j depends Keywords: on the value, which characterizes the fractal structure of the black hole surface called parameter, and = Loop Quantum Gravity may have values other than jmin 1. Barrow entropy © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license Immirzi parameter (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The quantum operator for area in Loop Quantum Gravity (LQG) In this paper we will use the quasinormal modes to obtain a has discrete spectra [1]. This result is without doubt very impor- new equation for the minimum value of the spin j that appears in tant and, certainly, can reveal many interesting physical insights. Eq. (1)as a consequence of the Barrow black hole entropy [8]. It is The Hilbert space of LQG is formed by spin networks which are important to mention that in Ref. [7]the author has considered the graphs with edges that carry labels such as j = 0, 1/2, 1, 3/2, .... Boltzmann-Gibbs statistics combined with the Bekenstein-Hawking The area of a given region of space is intersected or, as we can say area entropy law. Here we would like to comment that the article in the LQG language, is punctured by the spin network edge that [9] has generalized the Dreyer work [7]by using Tsallis’ statistics carries the label j. Then the surface can be represented by an area [10]. element written as [2–4] Recently, Barrow [8] analyzed the circumstance where quantum gravitational effects could cause about sophisticate, fractal = 2 + a( j) 8πlp γ j( j 1), (1) structure on the black hole surface. It changes its actual horizon area, which in turn leads us to a new black hole entropy relation, where lp is the Planck length and γ is the so-called Immirzi parameter [5]. Eq. (1)is, certainly, a relevant prediction of LQG namely, and, as we can see, the Immirzi parameter provides the size of 1+ a quantum of area in the Planck units. However, there is a dif- 2 = A ﬁculty concerned with the fact that the Immirzi parameter is, in S B , (2) A principle, undetermined. One way to determine the Immirzi pa- o rameter can be implemented with the help of quasinormal modes where A is the usual horizon area and Ao the Planck area. It is in the black holes theory. Quasinormal modes are the modes of important to mention that this extended entropy is different from energy dissipation that appear in the perturbation equations of the standard quantum corrected one [11–13]with logarithmic cor- the Schwarzschild geometry. These solutions were initially found rections [14], although it is a kind of Tsallis nonextensive entropy by Regge and Wheeler [6]. This procedure, as we will see, con- expression [10,15,16]. We can see clearly that this quantum grav- nects the relation between area and mass of a Schwarzschild black itational perturbation is encoded by the new exponent . There hole to the area induced by the spin network in the framework of are some characteristic values for . For example, when = 0 LQG [7]. we have the simplest construction. In this case we obtain the well known Bekenstein-Hawking entropy. On the other hand, when = 1we have the so-called maximal deformation. * Corresponding author. E-mail addresses: [email protected] (E.M.C. Abreu), jorge@ﬁsica.ufjf.br For a large imaginary part of the quasinormal frequency, which (J. Ananias Neto). will be denoted as ω, Nollert [17], Anderson [18] and Hod [19,20] https://doi.org/10.1016/j.physletb.2020.135602 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 E.M.C. Abreu, J. Ananias Neto / Physics Letters B 807 (2020) 135602 have obtained the following limiting of the quasinormal mode fre- quencies ln 3 i 1 Mωn = + n + , (3) 8π 4 2 where M is the mass of black hole and n is a non-negative integer. Here we are using the gravitational units G = c = 1. The real part of Eq. (3)is equal to ln 3 Re[Mωn]= . (4) 8π

Denoting the real part of the quasinormal modes as wn then from Eq. (4)we have = 2 = 2 Fig. 1. Values of jmin as a function of for areas A 16πlp (line) and A 32πlp ln 3 (dash). wn = . (5) 8π M A In addition, based on the considerations made by Hod [19]we can S = k ln W = k ln(2 j + 1). (15) B 2 min assume that the quantum of energy is 4lp ln 3

In order to solve this equation for jmin, we make S B equal to the M = hw¯ n . (6) + 2 1 = A Borrow black hole entropy, i.e., S B k 2 so that On the other hand, since 4lp

2 A = 16π M , (7) A 2 2 ln 3 e 4lp − 1 such an increase in the mass of the black holes induces a corre- jmin = . (16) 2 sponding heighten in the horizon area given by From Eq. (16)we can observe that in the limit → 0we recover = = 2 the result obtained by Dreyer that is j = 1. In Fig. 1 we have A 32π MM 4ln3lp , (8) min plotted the minimum value for the spin j, Eq. (16), as a func- ¯ = 2 where we have used that h lp in gravitational units. At the same tion of . The parameter varies in the physical interval which time, by using the area result of LQG, Eq. (1), we have is 0 ≤ ≤ 1. We choose, for example, two different values for A = 2 = 2 which are A 16πlp and A 32πlp . So, we can see in Fig. 1 = = 2 + A a( jmin) 8πlpγ jmin( jmin 1). (9) that the minimum value for the spin j can have several values which are positive half-integers, i.e., jmin = 1, 3/2, 2, 5/2, ..., which Comparing Eq. (9)with (8)we can derive an expression for the are represented by the intersection of curves with grid horizon- Immirzi parameter γ given by tal points, and not only jmin = 1as obtained by Dreyer when the ln 3 usual Bekenstein-Hawking area entropy law is used. γ = √ . (10) An expression for the Immirzi parameter as a function of is 2π j ( j + 1) min min obtained from Eq. (10)as In order to derive the dependence of the minimum spin j, we ln 3 will consider the Boltzmann-Gibbs statistics. The number of con- γ = , (17) figurations (microstates) in a punctured surface is given by A 2 2 2 ln 3 π e 4lp − 1 N ln 3 W = (2 jn + 1), (11) which recovers the result obtained by Dreyer [7], γ = √ , in the 2π 2 n=1 limit → 0. Our result, Eq. (17), indicates to be correct because we have imposed that the Boltzmann-Gibbs entropy is equal to where the term (2 jn + 1) in Eq. (11)is the multiplicity of the state j. Statistically, the most important configurations that contribute the Barrow black holes entropy. The strategy of reproducing black holes entropy, which results in the obtainment of the Immirzi pa- into Eq. (11) come from jn = jmin, where jmin is the minimum rameter, is a standard procedure in LQG. label. Then, from Eq. (11), we have that To conclude, in this work we have investigated the behav- N ior of the lowest possible spin, jmin, in the framework of LQG W = (2 jmin + 1) . (12) when we consider Barrow entropy. Our result, which is inside From Eqs. (1) and (9)the number N of punctures in a surface of Eq. (16), shows that the minimum spin number depends on the 2 → area A is given by ratio A/4lp and the Barrow parameter. In the limit 0, where Bekenstein-Hawking framework must be recovered, we have A A = N = = . (13) reobtained the result jmin 1. It is important to mention that a( jmin) A Dreyer [7], considering only the Bekenstein-Hawking entropy, has = From (8)we can write Eq. (13)as obtained that the lowest possible spin is jmin 1. We show values of the parameter > 0lead to jmin > 1. Then, Barrow A black hole entropy, which is a consequence of the quantum gravi- N = , (14) 2 tational effects and presents a fractal structure on the BH surface, 4lp ln 3 can certainly generalize the value of the lowest spin possible jmin. = 2 where we have used in Eq. (2) that A0 4lp . Using Eqs. (12) and The Immirzi parameter is obtained with the aid of Eqs. (10) and (14)in the Boltzmann-Gibbs entropy we obtain (16) and the result is Eq. (17). As expected, in the limit → 0, E.M.C. Abreu, J. Ananias Neto / Physics Letters B 807 (2020) 135602 3 where Bekenstein-Hawking must be recovered, we have obtained [2] C. Rovelli, L. Smolin, Nucl. Phys. B 442 (1995) 593. γ = ln√ 3 . Therefore, our result indicates that if we consider Bar- [3] A. Ashtekar, J. Lewandowski, Class. Quantum Gravity 14 (1997) A55. 2π 2 [4] M. Sadiq, Phys. Lett. B 741 (2015) 280. row black hole entropy then the minimum possible value of the [5] G. Immirzi, Nucl. Phys. B, Proc. Suppl. 57 (1997) 65. spin networks and the Immirzi parameter are not univocally de- [6] T. Regge, J.A. Wheeler, Phys. Rev. 108 (1957) 1063. termined. [7] O. Dreyer, Phys. Rev. Lett. 90 (2003) 081301. [8] J.D. Barrow, The area of a rough black hole, arXiv:2004 .09444 [gr-qc]. [9] Everton M.C. Abreu, Jorge Ananias Neto, Edésio M. Barboza Jr., Bráulio M. Declaration of competing interest Soares, Phys. Lett. B 798 (2019) 135011. [10] C. Tsallis, J. Stat. Phys. 52 (1988) 479. The authors declare that they have no known competing finan- [11] E.N. Saridakis, Barrow holographic dark energy, arXiv:2005 .04115 [gr-qc]. cial interests or personal relationships that could have appeared to [12] E.N. Saridakis, S. Basilakos, The generalized second law of thermodynamics influence the work reported in this paper. with Barrow entropy, arXiv:2005 .08258 [gr-qc]. [13] Everton M.C. Abreu, Jorge Ananias Neto, Edésio M. Barboza Jr, Europhys. Lett. 130 (2020) 40005. Acknowledgements [14] R.K. Kaul, P. Majumdar, Phys. Rev. Lett. 84 (2000) 5255; S. Carlip, Class. Quantum Gravity 17 (2000) 4175. The authors thank CNPq (Conselho Nacional de Desenvolvi- [15] G. Wilk, Z. Wlodarczyk, Phys. Rev. Lett. 84 (2000) 2770. [16] C. Tsallis, L.J.L. Cirto, Eur. Phys. J. C 73 (2013) 2487. mento Científico e Tecnológico), Brazilian scientific support federal [17] H.-P. Nollert, Phys. Rev. D 47 (1993) 5253. agency, for partial financial support, Grants numbers 406894/2018- [18] N. Andersson, Class. Quantum Gravity 10 (1993) L61. 3 and 302155/2015-5 (E.M.C.A.) and 303140/2017-8 (J.A.N.). [19] S. Hod, Phys. Rev. Lett. 81 (1998) 4293. [20] Y. Ling, H. Zhang, Phys. Rev. D 68 (2003) 101501(R). References

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