# Noncommutative Correction to the Entropy of BTZ Black Hole with GUP

Authors: M. A. Anacleto; F. A. Brito; B. R. Carvalho; E. Passos
Journal: Advances in High Energy Physics
 (2021)

---

## Abstract

We investigate the effect of noncommutativity and quantum corrections to the temperature and entropy of a BTZ black hole based on a Lorentzian distribution with the generalized uncertainty principle (GUP). To determine the Hawking radiation in the tunneling formalism, we apply the Hamilton-Jacobi method by using the Wentzel-Kramers-Brillouin (WKB) approach. In the present study, we have obtained logarithmic corrections to entropy due to the effect of noncommutativity and GUP. We also address the issue concerning stability of the noncommutative BTZ black hole by investigating its modified specific heat capacity.


---

## Body

## 1. Introduction

The study of three-dimensional gravity has been extensively explored in the literature [1]. It has become an excellent laboratory for a better understanding of the fundamentals of classical and quantum gravity and also to explore some ideas behind the AdS/CFT correspondence [2]. This special attention in three-dimensional gravity has been mainly due to the discovery of the black hole solution in 2+1 dimensions [3]. In addition, generalizations of the Bañados-Teitelboim-Zanelli (BTZ) black hole solution have also been constructed considering coupling with a dilaton/scalar field [4–6]. In recent years, the implementation of noncommutativity in black hole physics has been extensively explored (for a review see [7]). In [8], the authors have introduced a noncommutative Schwarzschild black hole solution in four dimensions. As shown in [8], one way to incorporate noncommutativity into general relativity is to modify the source of matter. Thus, noncommutativity is introduced by replacing the point-like source term with a Gaussian distribution—or otherwise by a Lorentzian distribution [9]. In addition, noncommutativity in a BTZ black hole has also been introduced in [10–14]. In [15], the gravitational Aharonov-Bohm effect due to a BTZ black hole in a noncommutative background has been analyzed. The process of massless scalar wave scattering by a noncommutative black hole via a Lorentzian smeared mass distribution has been explored in [16]. The thermodynamic properties of BTZ black holes in noncommutative spaces have been studied in [17–20].It is well known that string theories, loop quantum gravity, and noncommutative geometry presents important elements for the construction of a compatible theory of quantum gravity. Furthermore, these theories have a common feature, which is the appearance of a minimum length on the order of the Planck scale. This therefore leads to a modification of the Heisenberg uncertainty principle, which is called the generalized uncertainty principle (GUP) [21–23]. In recent years, several works have been devoted to investigating the effect of GUP on computing the Hawking radiation from black holes in 2+1 dimensions. In this sense, the Hamilton-Jacobi method via the WKB approach to calculate the imaginary part of the action is an effective way of investigating the Hawking radiation as a process of tunneling particles from a black hole [24–30]. In [31], the effect of GUP on the Hawking radiation from the BTZ black hole has been investigated using the modified Dirac equation. The Hawking radiation has been analyzed in [32], by considering the Martinez-Zanelli black hole in 2+1 dimensions [5] and using the Dirac equation modified by the GUP. By applying the quantum tunneling formalism, the Hawking radiation from a new type of black hole in 2+1 dimensions has also been studied in [33], and in [34], the Hawking radiation of a charged rotating BTZ black hole with GUP was explored. Moreover, in [35, 36], the entropy of the BTZ black hole with GUP has been determined, and in [37], by adopting a new principle of extended uncertainty, its effect on the thermodynamics of the black hole has been examined.The purpose of this paper is to investigate the effect of noncommutative and quantum corrections coming from GUP for the calculation of the temperature and entropy of a BTZ black hole based on a Lorentzian distribution, by considering the tunneling formalism framework through the Hamilton-Jacobi method. Thus, the Hawking radiation will be computed using the WKB approach. Therefore, we show that the entropy of the BTZ black hole presents logarithmic corrections due to both of the aforementioned effects.The paper is organized as follows. In Section2, we consider noncommutative corrections for the BTZ black hole metric implemented via the Lorentzian mass distribution. We also have applied the Hamilton-Jacobi approach to determine noncommutative corrections for the Hawking temperature and entropy. In Section 3, we consider GUP to compute quantum corrections to the Hawking temperature and entropy and also briefly comment on the correction of the specific heat capacity at a constant volume. In Section 4, we make our final considerations.


## 2. Noncommutative Corrections to the BTZ Black Holes

Here, we introduce the noncomutativity by considering a Lorentzian mass distribution, given by [8, 9, 14, 16]:(1)ρθr=Mθ2πr2+θ3/2,where θ is the noncommutative parameter with dimension of length2 and M is the total mass diffused throughout the region of linear size θ. In this case, the smeared mass distribution function becomes [14](2)Mθ=∫0rρθr2πrdr=M1−θr2+θ=M−Mθr+Oθ3/2.By considering the above modified mass, the metric of the noncommutative BTZ black hole is given by(3)ds2=−Frdt2+Fr−1dr2+r2dϕ−J2r2dt2,where(4)Fr=−Mθ+r2l2+J24r2=−M+Mθr+r2l2+J24r2.Note that the metric obtained by a noncommutative correction is different from the metric in [1]. A term, Mθ/r, of the Schwarzschild type is generated due to the noncommutative correction. Our metric shows similarities with the metric obtained in [4, 5] and also with one of the classes of solutions found in [6] with a dilaton/scalar field.We shall now analyze the nonrotating caseJ=0, so the metric (3) becomes(5)ds2=−frdt2+fr−1dr2+r2dϕ2,where(6)fr=−M+Mθr+r2l2.The horizons are found by solving(7)fr=−M+Mθr+r2l2=0,which is equivalent to solving a cubic equation(8)r3−Ml2r+Ml2θ=0.The roots of this cubic equation are given by [38]:(9)r=2l2M3sin13sin−1323θl2M+ε2π3,ε∈0,±1.The three roots forε=1, 0, −1, up to the first order in θ, are given, respectively, by(10)r~h=rh−θ2+⋯,rc=θ+⋯,rv=−rh−θ2+⋯,where rh=l2M, r~h is the event horizon, rc the cosmological horizon, and rv the virtual (unphysical) horizon. From equation (7), we obtain the mass of the noncommutative black hole, up to the first order in θ, that is given by(11)M=r~h2l2+r~hθl2+⋯.In order to compute the Hawking temperature, we use the Klein-Gordon equation for a scalar fieldΦ in the curved space given by(12)1−g∂μ−ggμν∂ν−m2ℏ2Φ=0,where m is the mass of a scalar particle. In the sequel, we apply the WKB approximation(13)Φ=expiℏIt,r,xi,such that we obtain(14)gμν∂μI∂νI+m2=0.By applying the metric (5) in the above equation, we have(15)−1fr∂tI2+fr∂rI2+1r2∂ϕI2+m2=0.Now, we can write the solution of equation (15) as follows:(16)I=−Et+Wr+Jϕϕ,where(17)∂tI=−E,∂rI=dWrdr,∂ϕI=Jϕ,with Jϕ being a constant. By substituting (16) into equation (15) and solving for Wr, the classical action is written as follows:(18)I=−Et+∫drE2−frJϕ2/r2+m2fr+Jϕϕ.Next, in the regime near the event horizon of the noncommutative BTZ black hole,r⟶r~h, we can write fr≈κr−r~h, and so the spatial part of the action function reads(19)Wr=1κ∫drE2−κr−r~hJϕ2/r2+m2r−r~h=2πiκE,where κ is the surface gravity of the noncommutative BTZ black hole given by(20)κ=f′r~h=2r~hl2−Mθr~h2.The next step is to determine the probability of tunneling for a particle with energyE, and for this, we use the following expression:(21)Γ≃exp−2ImI=exp−4πEκ.In order to calculate the Hawking temperature of the noncommutative BTZ black hole, we can compare equation (21) with the Boltzmann factorexp−E/T~H , so we can find(22)T~H=κ4π=r~h2πl2−Mθ4πr~h2.Moreover, the above result can be rewritten in terms ofrh=l2M as follows:(23)T~H=rh−θ/22πl2−Mθ4πrh2=Th−θ4πl2−Mθ4πrh2.Therefore, the result above shows that the Hawking temperature is modified due to the presence of the noncommutative parameterθ. Note that when we take θ=0, we recover the temperature of the commutative BTZ black hole, which is Th=rh/2πl2.At this point, we are prepared to go further. Let us now consider the noncommutative BTZ black hole in the rotating regimeJ≠0. Now the line element of equation (3) can be written in the form(24)ds2=−Frdt2+Fr−1dr2+r2dφ2,where(25)Fr=−M+Mθr+r2l2+J24r2,dφ=dϕ−J2r2dt.Thus, to find the horizons, we have to solve(26)Fr=−M+Mθr+r2l2+J24r2=0,which is equivalent to solving a quartic equation(27)r4−l2Mr2+l2J24+l2Mθr=0.We can now rewrite this equation as follows [38]:(28)r2−r+2r2−r−2+l2Mθr=0,that for θ=0, we have(29)r±2=l2M21±1−JMl2,where r+ is the outer event horizon and r− is the inner event horizon of the commutative BTZ black hole. Now, rearranging equation (28) in the form(30)r2=r±2−rh2θrr2−r∓2,where rh=l2M, we can solve it perturbatively. So, in the first approximation we get the event horizon:(31)r~+2≈r+2+rh2θr+r−2−r+2,or by keeping terms up to the first order in θ, we obtain(32)r~+=r++rh2θ2r−2−r+2+⋯,for the outer horizon. For the internal horizon, we have(33)r~−2≈r−2−rh2θr−r−2−r+2,so that for r~−, we find(34)r~−=r−−rh2θ2r−2−r+2+⋯.In order to determine the Hawking temperature for the case of the rotating black hole, we can follow the same steps as presented above, and so for the tunneling probability we have(35)Γ=exp−4πEκ¯,where the surface gravity is given by(36)κ¯=F′r~+=2r~+l21−l2J24r~+4−Mθr~+2.Again, by comparingΓ with the Boltzmann factor exp−E/TH, we obtain the Hawking temperature of the noncommutative rotating BTZ black hole:(37)TH=κ¯4π=F′r~+4π=2r~+4πl21−l2J24r~+4−Mθ4πr~+2.Forθ=0, we recover the result for the Hawking temperature of the rotating BTZ black hole which is given by(38)Th=r+2πl21−l2J24r+4.From equation (26), we obtain the mass of the noncommutative black hole, up to the first order in θ, that is given by(39)M=r~+2l2+J24r~+2+r~+θl2+θJ24r~+3+⋯.In order to analyze the entropy, we consider the following equation:(40)S=∫1TH∂M∂r~+dr~+,where(41)∂M∂r~+=2r~+l21−l2J24r~+4+θl21−3l2J24r~+4+⋯.The next step is to perform an expansion inTH−1 up to the first order in θ, so we can find(42)TH−1=4π2r~+l21−l2J24r~+4−11+rh2θ2r~+31−l2J24r~+4−1+⋯.Now, by replacing (41) and (42) in (40), we obtain(43)S^=4π∫1+rh2θ2r~+3+θ2r~+1−3l2J24r~+41+l2J24r~+4+⋯dr~+=4πr~++2πθlnr~+−πrh2θr~+2−πrh2l2J2θ12r~+6+2πl2J2θ8r~+4+3πl4J4θ64r~+8+S0+⋯,where S0 is an integration constant, and by rewriting in terms of r+, we have(44)S^=4πr++2πrh2θr−2−r+2+2πθlnr+−πrh2θr+2−πrh2l2J2θ12r+6+2πl2J2θ8r+4+3πl4J4θ64r+8+S0+⋯.Forθ=0 in (44), we have S=4πr+, which is the entropy of the commutative rotating BTZ black hole. On the other hand, for the case J=0, we have r+=rh, and the entropy becomes(45)S^=4πrh−3θ2+3πθ+2πθlnrh+S0+⋯.Note that we have obtained a logarithmic correction for the noncommutative BTZ black hole. Besides, our metric corresponds to that of Ref. [6] with the equivalence θ≡B and which is given for the nonrotating case (J=0) by(46)fBr=−M+MBr+r2l2,where B is a finite constant parameter introduced by a dilaton/scalar field. Hence, the horizon radius can be computed from equation (9) as above by taking the approximation B/l2M=BΛ/M≪1. So, we find(47)rhb=rh−B2+⋯.Thus, from equation (40), a logarithmic correction is obtained for entropy, given by(48)SB=4πrhb+2πBlnrhb+⋯,with ΔS≡SB−S=2πBlnrh+⋯ associated with small (thermal) fluctuations. This approach could also be considered in [2–5].


## 3. Quantum Correction to the Entropy

In this section, in order to derive quantum corrections to the Hawking temperature and entropy of the noncommutative BTZ black hole, we will apply tunneling formalism using the Hamilton-Jacobi method. So, we will adopt the following GUP [39], [40–49]:(49)ΔxΔp≥ℏ21−αlpℏΔp+α2lp2ℏ2Δp2,where α is a dimensionless positive parameter and lp is the Planck length.In sequence, without loss of generality, we will adopt the natural unitsG=c=kB=ℏ=lp=1, and by assuming that Δp~E and following the steps performed in [24], we can obtain the following relation for the corrected energy of the black hole:(50)Egup≥E1−α2Δx+α22Δx2+⋯.Thus, performing the same procedure as previously described, we have the following result for the probability of tunneling with corrected energyEgup given by(51)Γ≃exp−2ImI=exp−4πEgupa,where a is the surface gravity. Again, we compare with the Boltzmann factor and we obtain the corrected Hawking temperature of the noncommutative BTZ black hole(52)T≤T~H1−α2Δx+α22Δx2+⋯−1.Here, for simplicity, we will consider the caseJ=0. The temperature T~H is given by equation (22). Furthermore, since near the event horizon of the BTZ black hole the minimum uncertainty in our model is of the order of the radius of the horizon, so the corrected temperature due to the GUP is given by(53)Tgup≤T~H1−α4r~h+α28r~h2+⋯−1=2r~h4πl21−rh2θ2r~h31+α4r~h−α28r~h2+⋯.We can also write the result above in terms ofrh=lM as follows:(54)Tgup≤2rh4πl21−θ2rh21+α4rh1+θ2rh+⋯−α28rh21+θrh+⋯+⋯.Next, we will compute the entropy of the noncommutative BTZ black hole by using the following formula:(55)Sgup=∫1Tgup∂M∂r~hdr~h,where from equation (11), we have(56)∂M∂r~h=2r~hl21+θ2r~h.So, now we can obtain the corrected entropy(57)Sgup=4π∫1+θ2r~h1+rh2θ2r~h31−α4r~h+α28r~h2+⋯dr~h=4πr~h+2πθlnr~h−πrh2θr~h2−παlnr~h−πα22r~h+παθ2r~h−πα2θ8r~h2+πrh2αθ6r~h3−πrh2α2θ16r~h4+S0+⋯,or by expressing the result above in terms of the rh we have(58)Sgup=4πrh−3θ2+2πθlnrh+3πθ−παlnrh−πα22rh+2παθ3rh−3πα2θ16rh2+S0+⋯.Therefore, by analyzing the result, we have obtained corrections to the entropy due to the effects of GUP and also noncommutative correction. Note that due to the effect of noncommutativity and GUP, we have found logarithmic corrections for the entropy of the BTZ black hole. Forα=0, we have precisely the noncommutative correction to the entropy given by (45). In [50], the authors analyzed the thermodynamics of the charged rotating BTZ black hole, and logarithmic corrections were also obtained for entropy in the presence of the GUP and thermal fluctuations (for small variations in β=1/T). The logarithmic corrections to entropy become important for very small black holes and negligible for very large black holes. Further studies addressing these issues were also considered in Refs. [51–54]. In our case, logarithmic corrections are due to the presence of GUP and/or the noncommutativity of spacetime that mimic small thermal fluctuations by properly identifying the corresponding parameter to values normally found in thermal fluctuations as well discussed in Refs [50–54].At this point, we will compute the Helmholtz free energy, which can be determined by using the following relationship:(59)Fgup=−∫SgupdTgup.So, from equations (53) and (57), we get(60)Fgup=−12πl2∫1+rh2θr~h3+α28r~h2+3rh2αθ8r~h4−rh2α2θ4r~h5Sgupdr~h=−r~h2l2+3rh2θ2l2r~h+5rh2αθ12l2r~h2−29rh2α2θ96l2r~h3−θr~hl2lnr~h+θr~hl2+α2θlnr~h8l2r~h+α2θ16l2r~h+αr~h2l2lnr~h−αr~h2l2−rh2α2θlnr~h16l2r~h3−αθlnr~h4l2+F0+⋯,where F0 is an integration constant.Forα=0 (in the absence of the GUP), the Helmholtz free energy becomes(61)Fθ=−r~hl2r~h−θ+3rh2θ2l2r~h−θr~hl2lnr~h+F0+⋯,or rewriting in terms of rh, we have(62)Fθ=−1l2rh−θ2rh−3θ2+3rhθ2l2−θrhl2lnrh+F0+⋯.The correction of the specific heat capacity is given by(63)Cgup=∂M∂Tgup=∂M∂r~h∂Tgup∂r~h−1=4πr~h1+θ2r~h1−θr~h−α28r~h2−3αθ8r~h2+α2θ4r~h3+⋯.Now ifα=0 in equation (63), we have(64)Cθ=4πr~h1+θ2r~h1−θr~h+⋯,which in terms of rh, becomes(65)Cθ=4πrh1+θ2rh1−3θ2rh+⋯.Forθ=0, we have C=4πrh, which is the specific heat for the commutative BTZ black hole. Note that the specific heat vanishes at the point rh=3θ/2 (or r~h=θ in equation (64)). In this case, we have a minimum radius(66)rθmin=l2Mθmin=3θ2,and then the noncommutative black hole reaches a minimum mass given by(67)Mθmin=9θ4l2.Thus, this result indicates that the black hole ceases to evaporate completely and becomes a remnant. Next, we obtain the temperature of the remnant of the black hole by replacing therh⟶rθmin in (23):(68)Tθrem≈T~H3=rθmin6πl2=θ4πl2.Furthermore, from equations (45) and (62) for rh⟶rθmin=3θ/2, we find(69)Srem≈3πθ+2πθln3θ2+S0≈0,S0=−3πθ−2πθln3θ2,Frem≈0+Oθ+F0,F0=0.Hence, we have that the entropy and the Helmholtz free energy are zero for the remnant of the noncommutative BTZ black hole.Now, to analyze the effect of the GUP, we consider the case whereθ=0 and α≠0. Thus, from equation (63), we have the following contribution to specific heat:(70)Cα=4πrh1−α28rh2+⋯,and the specific heat vanishes at the point rh=rαmin=α/22. Hence, the BTZ black hole with GUP becomes a remnant with a minimum mass, Mαmin=α2/8l2, and a temperature given by(71)Tαrem=Tgupθ=02=α8πl2.Moreover, the entropy and the Helmholtz free energy are zero for the remnant of the BTZ black hole with GUP with(72)Sαrem≈−παln2α4+S0≈0,S0=παln2α4,Fαrem≈−1+2α28l2−2α2ln2α4+F0≈0,F0=1+2α28l2+2α2ln2α4.Forθ≠0 and α≠0, we can write equation (63) as follows:(73)Cgup=4πr~h1+θ2r~h1−rm+r~h1−rm−r~h+⋯,where(74)rm±=θ2±12θ+αα+3θ2.Note that forr~h=rm+ (or rh=θ/2+rm+), the specific heat vanishes. The results obtained previously are recovered when θ≠0 and α=0 (or θ=0 and α≠0). For the condition of forming a remnant of a noncommutative BTZ black hole, we can write the following approximations for the minimum radius:(75)rmin=θ2+rm+=r1min=3θ2+3α8+Oα2,r2min=α22+32+88θ+Oθ.By applying the minimum radiusrmin, the specific heat, entropy, and Helmholtz free energy are null, and by (54) the temperature is given by(76)Tαθrem=Tguprh=rmin=rmin2πl2−θ2πl21+α4rmin+α8rmin2θ−α−α2θ8rmin3+⋯.We can obtain approximate expressions for the temperature of the remnant of the noncommutative BTZ black hole by expanding it inα and θ. So, by applying the minimum radii r1min and r2min, the temperatures are given, respectively, by(77)T1rem=Tguprmin=r1min=θ4πl2+35α144πl2+Oα2,T2rem=Tguprmin=r2min=α8πl2+2θ4πl2+θπl2+Oθ.Then, by doingα=0 in T1rem, we obtain the result of (68), and for θ=0 in T2rem, we recover the result of (71).Now in order to verify the stability of the black hole, we show in Figures1 and 2 the specific heat behavior. In Figure 1, we show that the specific heat is positive for θ=0.001 and α=0.1, and so the noncommutative BTZ black hole with GUP is stable. In addition, we observed that the specific heat vanishes to a critical radius. Furthermore, for θ=0.03 and α=0.1, one achieves two points where the specific heat vanishes, with an unphysical region in between.Figure 1
Specific heat capacity (equation (63)). For θ=α=0, we have the result for the specific heat of the commutative BTZ black hole. We also show the result when θ≠0 and α≠0.Figure 2
Specific heat capacity. (a) Forθ≠0 and α=0, we have the result for the specific heat of the noncommutative BTZ black hole. (b) For θ=0 and α≠0, we have the result for the specific heat of the commutative BTZ black hole with GUP.
(a)(b)In Figure2(a), we verify the behavior of the specific heat for θ≠0 and α=0, and in Figure 2(b) for θ=0 and α≠0. Note that the specific heat vanishes before entering into an unphysical zone. The BTZ black hole decreases its size until it achieves a critical radius where it ceases to evaporate and becomes a remnant of the noncommutative BTZ black hole.


## 4. Conclusions

In summary, we have considered the metric of a noncommutative BTZ black hole implemented via the Lorentzian mass distribution. Thus, applying the Hamilton-Jacobi approach and the WKB approximation, we have obtained noncommutative corrections to the Hawking temperature and entropy. In addition, we have found a logarithmic correction to the entropy of the BTZ black hole due to the effect of noncommutativity. We also have verified the stability of the BTZ black hole by calculating the specific heat capacity and have shown that the noncommutative BTZ black hole becomes a remnant with a minimum massMθmin=9θ/4l2. Therefore, the contribution of the noncommutative corrections introduces a GUP effect. We also investigated the effect of GUP by calculating the Hawking temperature and entropy of the noncommutative BTZ black hole. Due to the effect of noncommutativity and GUP, we have found logarithmic corrections for the entropy of the BTZ black hole, in the form of Sgup~S+c1+c2lnS+⋯, where the “species” ci=−α,θ are essentially related to each corresponding parameter of correction.



---

*Source: 6633684-2021-01-12.xml*