On Superstatistics and Black Hole Quasinormal Modes

On superstatistics and black hole quasinormal modes.

Aldo Mart´ınez-Merinoa , M. Sabidob a Departamento de Ciencias Naturales y Exactas, CU Valles, Universidad de Guadalajara. Carretera Guadalajara - Ameca Km. 45.5, C.P. 46600, Ameca, Jalisco, México. b Departamento de F´ısicade la Universidad de Guanajuato, A.P. E-143, C.P. 37150, León,Guanajuato, México.

Abstract

It is known that one can determine that lowest value of spin is jmin = 1, by using the quasinormal modes of black holes, the Bekenstein-Hawking entropy and Boltzmann-Gibbs statistics. In this paper, to determine jmin, we have used non extensive entropies that depend only on the probability (known as Obregon’s entropies and have been derived from superstatistics), as well as non extensive entropies that have free parameters . We find that jmin depends on the area and the non extensive parameter. In particular, for the non extensive entropies that only depend on the probability and find that the modification is only present for micro black holes. For classical black holes, the results are the same as for the Boltzmann-Gibbs statistics.

1. Introduction

Black holes are one of the most enigmatic and enticing objects of study and just recently, a direct veriﬁcation of their existence was provided by Event Horizon Telescope collaboration [1]. Despite the amount of research on the subject there are several unanswered questions. Quantization of black holes was proposed in the pioneering work of Bekenstein [2], he suggested that the surface gravity is proportional to its temperature and that the area of its event horizon is proportional to its entropy. In his remarkable work he conjectured that the horizon area of non-extremal black holes plays the role of a classic adiabatic invariant. He concluded that the horizon area should have a discrete spectrum with uniformly spaced eigenvalues

2 An = γ¯lpn, n = 1, 2, ... (1)

whereγ ¯ is a dimensionless constant. With the development of Loop Quantum Gravity (LQG), the correct spectrum of the area operator was obtained in [3, 4], being 2 p A( j) = 8πγlp j( j + 1), (2) where γ is the Immirzi parameter [5, 6]. The value of the Immirzi parameter determines the value of the minimal area, but it is a free parameter in Loop Quantum Gravity (LQG). As any fundamental constant in the theory we must arXiv:2106.15751v2 [gr-qc] 19 Aug 2021 ﬁnd a way to determine its value, and this is where the entropy comes into play. The entropy is a quantity related to spectrum and therefore seems to be the main candidate to determine the vale of γ. In [7], the author established a method to determine the value of γ by using the quasinormal modes of the Schwarzschild black hole. The method relates the area derived in LQG with the mass and area of the Schwarzschild black hole and uses Boltzmann-Gibbs statistics to ﬁnally derive that γ = ln√ 3 . 2π 2 In this work, we will follow [7] and use quasinormal modes together with Obregon’s entropies [8] to determine the minimum value jmin. These entropies are the only generalizations of the Boltzmann-Gibbs entropy that only depends

Email addresses: [email protected] (Aldo Mart´ınez-Merinoa), [email protected] (M. Sabidob)

Preprint submitted to Elsevier August 23, 2021 on the probability. This has been done before for Tsallis [9] and Barrow entropies [10], but in those cases the entropy has free parameters that are used to yield some given value of jmin. The paper is organized as follows. In section 2, we follow the approach in [7] using the non-extensive entropies that only depend on the probability. Section 3, we workout the case for other non-extensive entropies that, as for Tsallis, have free parameters. Finally, section 4 is devoted for ﬁnal remarks.

2. Black holes and non-extensive entropies that only depend on the probability

The generalization of the Boltzmann entropy usually has a big drawback, the loss of the extensive property. Despite of this, non-extensive entropies can be useful in the study of some more general phenomena. There is a large set of entropies that have been termed Superstaitstics, which are derived using different temperature distributions [11]. One of the more useful and better studied generalization is the Tsallis entropy [12]. One of the features of Tsallis entropy is the presence of a free parameter. This parameter takes different values depending on the physical system and when the parameter is one, the Tsallis entropy reduces to the usual Boltzmann entropy. In the context of superstatistics, the author in [8] starts with a Gamma distribution for the temperature to derive a non-extensive entropies depending only on the probability; these entropies are known as Obregon’s entropies. In contrast to Tsallis entropy, these entropies do not have free parameters. Furthermore, Obregon’s entropies (unlike Tsallis entropy) in the limit of small probabilities or, equivalently, a large number of states correspond to Boltzmann- Gibbs. It is worth mentioning that this entropy has been used in the context of entropic gravity [13], AdS/CFT [14]. For now, let us explicitly give the functional form of Obregon’s entropies, the first of them being1

XΩ pl S+ = 1 − pl , (3) l=1 P with the probabilities satisﬁying the usual constraint l pl = 1. There is also another entropy of the form

XΩ −pl S− = pl − 1 , (4) l=1

1 and a third one which is the sum of the previous entropies S ± = 2 (S + + S −). To relate the entropy to black holes, we make use of quasinormal modes. For a large imaginary part the frequency of the quasinormal modes [15] is ! ln 3 i 1 Mω = + n + , (5) n 8π 4 2 2 the value of the real part of Eq.(5) was previously proposed in [16]. The energy spectrum is ∆M = lpwn (where we are 1 ln 3 2 using units G = c = 1) and we have deﬁned wn = M Re(ωn) = 8πM . Considering that A = 16πM , when we introduce 2 a change in the mass ∆M we get a change in the area ∆A = 4lp ln 3 and, using the relation of the area from Eq.(2), we get next value of γ, ln 3 γ = p (6) 2π jmin( jmin + 1) The number of microstates of the conﬁgurations in a punctured sphere is

YN Ω = (2 jn + 1), (7) n=1

1Note that in the expressions we are dealing with are dimensionless quantities, thus we are already dividing the entropy by the Boltzmann constant, kB.

2 where jn is the spin label of such punctures, and N are the number of the punctures. In order to ﬁnd a lower bound for the number of states, we consider the minimum value of the spin labels for all of them at the same time, jmin, then the product (7) results in the relation N Ω = (2 jmin + 1) . (8) The number of punctures N is given by the ratio A/∆A, A N . = 2 (9) 4lp ln 3

Now we can write Ω in terms of jmin and the area of the event horizon,

A/4l2 ln 3 Ω = (2 jmin + 1) p . (10)

PΩ Furthermore, we assume that we have a considerable number of states Ω. From Shannon’s entropy S = − l=1 pl ln pl, and considering equipartition, we get Boltzmann’s entropy S = ln Ω. Moreover, using Eq.(7) and equating to the Hawking-Bekenstein entropy get that jmin = 1. Finally, from Eq.(6) we ﬁnd that the value of the Immirzi parameter is γ = ln√ 3 . 2π 2 As we already mentioned, we are interested to determine jmin for the non-extensive entropies. In order to achieve this, we start with the entropy in Eq.(3). Working in the microcanonical ensemble and assuming equipartition, the −1 probability of ﬁnding the system in a particular state is equal to the inverse of the number of states, p = pl = Ω . Then (3) takes the form Ω  ! 1   ! 1  X  1 Ω   1 Ω  S+ = 1 −  = Ω 1 −  . (11a)  Ω   Ω  l=1 With the same assumptions, we also rewrite Eq. (4), and also their sum (this case also represents a non extensive entropy that only depends on the probability)

Ω  !− 1   !− 1  X  1 Ω   1 Ω  S− =  − 1 = Ω  − 1 , (11b)  Ω   Ω  l=1 Ω  !− 1 ! 1   !− 1 ! 1  1 X  1 Ω 1 Ω   1 Ω 1 Ω  S± =  −  = Ω  −  . (11c) 2  Ω Ω   Ω Ω  l=1

Thus, to determine jmin we must find Ω as a function of A, which is achieved by equating the Bekenstein-Hawking entropy to the entropies in Eqs.(11), which yields A p − pp, S , 2 = 1 for + (12a) 4lp A p p−p − , S , 2 = 1 for − (12b) 4lp A 1 p p−p − pp , S , 2 = for ± (12c) 4lp 2 retaining the variable p. We can solve these equations for p and get the solution for Ω, then substitute back in Eq.(10) to find jmin. In order to get an analytical expression, we consider Ω to be large and therefore perform an approximation for small p. The first order approximation reproduces the result obtained from Shannon’s entropy. By considering the next term in the expansion we look for deviations in the value of jmin. Concentrating for now on (12a), next to leading order we need to solve A 1 1 p −p p − p2 2 p − p3 3 p ..., 2 = ln ln ln + (13) 4lp 2 6

3 for p and use the solution in Eq.(7) to ﬁnd the value of jmin. We show in the Appendix how we solve p which lead us to express jmin in terms of the area, which results in

2 jmin + 1 = (14)   2    3   2   − A − 2A − 3A   1 A 4l2 1  A   A  4l2 1  A   A  A   4l2  exp ln 3 1 + e p +   4 − 3  e p +   10 − 24 + 9    e p + ...  .  2  2   2   2   2  2     2 4lp 12 4lp 4lp 48 4lp 4lp 4lp 

We observe that higher order terms are exponentially suppressed, then we have a good approximation using the ﬁrst three terms. Following the same procedure we calculate jmin using the entropy S−, and from Eq. (12b) we ﬁnd A 1 1 p −p p p2 2 p − p3 3 p ..., 2 = ln + ln ln + (15) 4lp 2 6 which, again, solving for p, we arrive at the equation for jmin for this case,

2 jmin + 1 = (16)   2    3   2   − A − 2A − 3A   1 A 4l2 1  A   A  4l2 1  A   A  A   4l2  exp ln 3 1 − e p +   4 − 3  e p −   10 − 24 + 9    e p + ...  .  2  2   2   2   2  2     2 4lp 12 4lp 4lp 48 4lp 4lp 4lp 

The last case, entropy S±, we proceed as before to get A 1 1 − p p2 3 p p4 5 p ..., 2 = ln + ln + ln + (17) 4lp 3! 5! from Eq. (12c), and ﬁnally, for jmin we have

  2 − 2A  1  A  2 2 j + 1 = exp ln 3 1 −   e 4lp (18) min   2   6 4lp  4    6   2  − 4A − 6A  1  A   A  4l2 1  A   A  A   4l2  +   27 − 20  e p −   2880 − 4860 + 1800    e p − ...  .  2   2   2   2  2    360 4lp 4lp 64800 4lp 4lp 4lp 

For this the entropy S ± the terms are rapidly suppressed, then using the ﬁrst two terms we have a very good approximation. Because the entropy only depends on the probability, the are no free parameters. Now we ask ourselves, if there is jmin , 1. The answer is yes, we can see in Fig.(1), that for small A (quantum black holes) we have a diﬀerent value for jmin. Even though we have values of jmin < 1 for S− and S±, we can not have a value for the area for which jmin = 1/2, which is one of the main results in [9]. Up to now, we do not have a group theoretic argument for the present minimum value of jmin for Obregon’s entropies. Consequently, we can ask ourselves, what is the value of the area where we obtain the minimum for jmin.

3. Non-extensive entropies with free parameters

The entropies studied in the previous section seem to be the only generalization to Shannon’s entropy that only depends on the probability. Nonetheless, there are several other generalizations that have free parameters besides Tsallis entropy. In this section we calculate jmin for these entropies.

3.1. Tsallis-Cirto entropy The so-called Tsallis-Cirto entropy was proposed in [17], to solve thermodynamic inconsistencies for the Schwarzschild black hole. This entropy is deﬁned by the relation

δ S TC = (S BH) , (19) 4 ��

��

�� + �- �±

��

�

� � � � � ��

Figure 1: Plot of jmin for the diﬀerent non-extensive entropies that only depend on the probability. We can see that for Planck scale black holes jmin , 1, but for large ones the result that derived from Shannon’s entropy is recovered.

where if δ goes to 1 then we return to the Bekenstein-Hawking entropy. If we take the entropy in Eq.(19) equal to ln(Ω) with Ω given in Eq.(10), we get for jmin the result

 !δ−1   A  1  4l2  j = 3 p − 1 . (20) min 2  

For the Tsallis-Cirto entropy, we ﬁnd that the parameter δ can be given as a function of the black hole horizon area in

jmin 2.0

1.5

A= 16 lp2 1.0 A=32lp2

0.5

δ 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Figure 2: Plot of jmin for the for the Tsallis-Cirto entropy. We can see in this plot that we can ﬁnd a value of δ that gives jmin = 1/2.

2 order to attain jmin such value . The expression is

ln 2 ln ln 3 δ = A + 1. (21) ln 2 4lp

In Fig.(2) we show jmin as a function of δ for two given values of the area, where we see explicitly that we can ﬁnd a ∆ value of 1/2. We can relate the Tsallis-Cirto entropy to Barrow’s entropy by taking δ = 1 + 2 . This entropy is related to a fractal structure on the black hole surface and is of quantum origin and is encoded in the parameter ∆. When using the Barrow’s interpretation, one gets min > 1 for ∆ > 0. Furthermore, when considering Barrow’s entropy we can not determine the minimum value of min and the Immirzi parameter [10].

2 For the usual Tsallis entropy, the authors in [9] find the possibilities for which jmin can be equal to 1/2. 5 3.2. Modified Rényientropy By considering a non extensive Tsallis entropy to study the thermodynamics of the Schwarzschild back hole. The authors in [18], use the Renyi´ entropy. However, we consider the modified Rényientropy given by the relation 1 S = ln(1 + λS ), (22) MR λ BH where λ is a positive constant. In the limit λ → 0 we recover the Bekenstein-Hawking entropy. The relation for jmin is now given by the equation 1 h i j = 3 f (A;λ) − 1 , (23) min 2 where

2   4lp  A  f (A; λ) = ln 1 + λ  . (24)  2  λA 4lp

It is easy to verify that in the limit when λ goes to 0, jmin goes to 1, recovering the result for the Bekenstein-Hawking entropy. Looking for the value of λ which gives us jmin = 1/2, we arrive at the next transcendental equation

14 jmin 1.0 A= 16 lp2 12

0.8 A=32lp2 10

A=64lp2 2 8 0.6 A lp 4 6 0.4 4

0.2 2

0 λ 0.0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 14 λ

Figure 3: The ﬁrst graph is a plot of jmin derived from the modiﬁed Renyi entropy. We can see in this plot that to have jmin = 1/2 the value of λ is negative. Therefore, for this entropy jmin ≥ 1. The second plot is for the values of δ and A that give jmin = 1/2. The solid line corresponds to the numerical solution to Eq.(25), the dotted line for the approximation.

   λA  ln 2 λA ln 1 +  = . (25)  2  2 4lp ln 3 4lp

The case λ = 0, although a solution is excluded since for this value the entropy goes to Bekenstein-Hawking’s jmin = 1. 8lp2 For large λ we ﬁnd the approximate relation λ = 3A ln[3] .

3.3. Sharma-Mittal entropy The last entropy we consider is the Sharma-Mittal. This entropy was proposed to construct a new model for holographic dark energy [19]. Moreover, it is a generalization of both the Renyi´ and Tsallis entropy. In the gravitational context, this entropy is deﬁned by the relation 1 h i S = (1 + δS )R/δ − 1 . (26) SM R BH This entropy interpolates between the modiﬁed Renyi´ entropy (R → 0) and Bekenstein-Hawking entropy (R → δ). Thus, the value for the minimum spin is 1 h i j = 3g(A;R,δ) − 1 , (27) min 2 6 where 2  R/δ  4lp  δA   g(A; R, δ) =  + 1 − 1 . (28)  2   RA 4lp

The deﬁnition of this entropy in terms of these two parameters, permits the possibility of ﬁnding a region where the value of jmin be equal to 1/2. In the next plot, we show that for certain values of the area which pairs of such values gives jmin = 1/2.

3.0

2.5

2.0

A=32 lp2

δ 1.5 A=64 lp2

A=128 lp2 1.0 A=256 lp2

0.5

0.0 0.0 0.5 1.0 1.5 2.0 R

Figure 4: Plot for the values of R and δ in order to have jmin for the the Sharma-Mittal entropy.

4. Final remarks

In this paper we have we have followed [7] and use quasinormal modes and non-extensive entropies to determine the minimum value jmin. These entropies are generalizations of the Boltzmann-Gibbs entropy that only depends on the probability. This has been done for the Tsallis entropy [9] and Barrows entropy [10], but in those cases the entropy has free parameters that are used to give the value of jmin. We worked out to classes of non-extensive entropies, does that only depend on the probability and those that have free parameters. With respect on non-extensive entropies that only depend on the probability, the only entropies that satisfy this requirement are the so called Obregon’s entropies. Assuming equipartition we find jmin for these entropies. For S − and S ± we see from Fig.(1) that the minimum value is less than one, but for large area, we recover the usual value. Therefore, we can conclude that the modification from using these entropies is only present for quantum black holes, 2 this is consistent with results obtained for fluids [20]. Moreover, for A > 8lp the result is the same as BG, therefore we conclude that that the effects of using non extensive entropies are only present on micro black holes. Of the non extensive entropies that have free parameters, the presence of the free parameters allows us to find jmin = 1/2, independent for particular values of the A. Furthermore, for these entropies for specific values of their respective parameter, they reproduce the Boltzmann-Gibbs results. Therefore, using this class of non-extensive entropies SO(3) and SU(2) spin networks are valid. It is worth mentioning, that we can have jmin > 1 and therefore these entropies generalize the value of jmin. In summary, the use of non extensive entropies modify jmin (at least for Obregon’s entropies) only for quantum black holes. Therefore, the possibility that non extensive statistics modifies the dynamics of quantum gravity systems exist. Of particular interest is the effects on cosmology, consequently the effects of no extensive entropies could change the dynamics of the very early Universe. This is work in progress and will be reported elsewhere.

7 Acknowledgements

This work is supported by CONACYT grants 257919, 258982. M. S. is supported by CIIC-032/2021. A. M-M. would like to thank O. Obregon´ for fruitful discussions, and acknowledges that this work began when he was part of Catedras´ Conacyt program.

Appendix

In order to have an expression for jmin in terms of the area of the black hole for the entropies (11), we need to solve the variable p in terms the area. In this Appendix we show how we accomplished this. Let us begin with Eq. (12a), which we rewrite it as ∞ A 1 X = (1 − pp) = − ln p − pn−1 lnn p, (29) 4l2 p p n=2 or A X∞ A X∞ ln p = − − pn−1 lnn p = − − lnn pe(n−1) ln p. (30) 4l2 4l2 p n=2 p n=2 A This equation has the form y = x + f (y) for y = ln p and x = − 2 ; observe that the function f (y) = 0 when y = 0. 4lp Applying the Lagrange inversion theorem [21], we can solve y in terms of x as a series using the formula

∞ !n−1 X 1 d y = x + { f (x)n}. (31) n! dx n=1 Using n = 3 in the formula, the result is

 2  3    4   2 − A − 2A − 3A A 1  A  4l2 1  A   A  4l2 1  A   A  A   4l2 ln p = − −   e p −   4 − 3  e p −   10 − 24 + 9    e p − ... (32) 2  2   2   2   2   2  2   4lp 2 4lp 12 4lp 4lp 48 4lp 4lp 4lp Observe that the terms become exponentially suppressed for higher order terms, so we can approximate with good accuracy up to the showed terms. We can proceed to apply the same equation for the entropies (11b) and (11c), with the results presented in the main text. Then, write down the expression of jmin in terms of the area of the black hole.

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