# Quasinormal modes and quantization of area/entropy for noncommutative BTZ black hole

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## Abstract

We investigate the quasinormal modes and area/entropy spectrum for the noncommutative BTZ black hole. The exact expressions for QNM frequencies are presented by expanding the noncommutative parameter in horizon radius. We find that the noncommutativity does not affect conformal weights (\(h_{L}, h_{R}\)), but it influences the thermal equilibrium. The intuitive expressions of the area/entropy spectrum are calculated in terms of Bohr–Sommerfeld quantization, and our results show that the noncommutativity leads to a nonuniform area/entropy spectrum. We also find that the coupling constant \(\xi \), which is the coupling between the scalar and the gravitational fields, shifts the QNM frequencies but not influences the structure of area/entorpy spectrum.

## 1 Introduction

In classical gravitational theory, perturbations of black holes or branes naturally lead to quasinormal modes (QNM) which are eigenmodes of dissipative systems [1]. It’s well known that QNM is a significant tool to study *AdS* / *CFT* duality [2, 3, 4, 5, 6]. In Refs. [7, 8, 9], the QNM frequencies determine the decay of the gravitational perturbations in the bulk and relaxation time of the two-point function of the thermal *CFT* at the boundary of BTZ black holes. The quasinormal modes provide a correspondence between the perturbation of the gravity in the bulk to that of a boundary CFT in BTZ space-time. While this result provides a proposal for the AdS/CFT duality, it would be significant to test its validity at the Planck scale, which is a natural regime for holography and quantum gravity. Noncommutative geometry is one of the candidates of the quantum gravity at the Planck scale [10]. In particular, string theory, loop gravity [11] and noncommutative geometry [10] suggest that the space-time might have some discrete structures at the Planck scale, thus the validity of *AdS* / *CFT* duality becomes very important. It is widely believed that the noncommutative space-time can be described by involving the quantum theory and general relativity together [12]. For instance, the noncommutative BTZ background can be described by \(\kappa \)-Minkowski algebra [13, 14, 15]. Therefore, the investigations on \(\kappa \)-Minkowski QNM in BTZ background turn out to be extremely important, which should be based on \(\kappa \)-Minkowski framework [16, 17].

In Ref. [18], the unitarity of scalar noncommutative field theories has been checked at the one loop level. They found that noncommutative field theories with only space noncommutativity (that is \(\theta ^{0i}= 0\)) are perturbatively unitary. And space-time noncommutative field theories (that is \(\theta ^{0i}\ne 0\)) do not have a unitary S-matrix and do not satisfy the cutting rules for Feynman diagrams. What’s more, there is a subtle point that two limits don’t necessarily commute, one is the process of putting the deformation parameter to zero, and the other is to perform the integral in the calculation of various Feynman diagrams (where also the UV/IR mixing enters) [19, 20]. An example of a finite theory for finite noncommutative parameter was also considered in Refs. [21, 22]. In other words, space noncommutative field theories describe a low energy limit of string theory in a background magnetic field. String theory can not be appropriately described by space-time noncommutative field theories (see [18]). Usually most of the analyses from string theory to noncommutative field theory are done with the assumption that the background flux is constant. However, when the background flux coming from the string theory is not constant, Aschieri et al. [23], with developing the metric aspects of non-associative differential geometry tailored to the parabolic phase space model of constant locally non-geometric closed string vacua, obtained explicit expressions for the torsion, curvature, Ricci tensor and Levi-Civita connection in non-associative Riemannian geometry on phase space, constructed R-flux corrections to the Ricci tensor on space-time, and commented on the potential implications of these structures in non-geometric string theory. By studying the noncommutative scalar and fermionic quasinormal modes in commutative BTZ black hole, Gupta et al. [24, 25] exploited a duality between a non-rotating and rotating BTZ black hole, the spin being proportional to the noncommutative deformation parameter. Therefore, their works showed how the Plank scale physics might influence the QNM frequency by setting the correct commutative limit and noncommutative corrections up to the first order in the deformation parameter. For example, they showed that the horizon temperatures in the dual *CFT* are modified due to noncommutative contributions through the *AdS* / *CFT* correspondence. Furthermore, they demonstrated the equivalence between the quasinormal and non-quasinormal modes for the noncommutative fermionic probes, which provides further evidence of holography in the noncommutative setting.

In quantum gravity, the black hole area/entropy quantization is always connected to QNM frequency, which is originally predicted by Bekenstein who assumed the event horizon as an adiabatic invariant [26]. An important character of loop quantum gravity is that the spectrum of geometrical quantities is not continued but discrete [27, 28, 29]. According to the correspondence principle, Hod proposed that the real part of the highly damped QNM might be related to the fundamental quanta of mass and angular momentum [30]. Particularly, the area/entropy spectrum has been deeply studied in the non-rotating BTZ background [31]. They have taken advantage of Boher-Sommerfeld quantization for an adiabatic invariant and given an equally spaced mass spectrum, which is different from 3+1-dimension black hole. Bekenstein [26] argued that the QNM frequency \(\omega _{(E)}\), in high damping limit, should take the form of \(\omega _{(E)}=\sqrt{| \omega _{R} |^{2}+ | \omega _{I} |^{2}}\). Therefore, the QNM frequency \(\omega _{(E)}\) approximately equaled \(\omega _{R}\) when \(\omega _{I}\rightarrow 0\). And the frequency would become \(\omega _{E}=\omega _{I}\) in the case of \(n\rightarrow \infty \).

In this paper, motivated by that the noncommutative field causes a corrections to the QNM of BTZ black hole and slightly affects the area/entropy spectrums of BTZ black hole [24], we mainly focus our attentions on noncommutative BTZ black hole with the nonminimally coupling scalar field and massless Dirac field. We plan to calculate the exact expressions for QNM frequencies in different cases and investigate area/entropy spectrum with the existence of noncommutativity in terms of Bohr–Sommerfeld quantization. The paper is organized as follows: in Sec. 2, we briefly introduce the noncommutative BTZ black hole. In Sec. 3, we analytically calculate the QNM of this black hole with nonminimally coupling scalar field. In Sec. 4, we present the exact expressions of QNM frequencies of this black hole with massless Dirac field. In Sec. 5, the quantitative agreement in the noncommutative BTZ background is discussed. In Sec. 6, the area/entropy spectrum of this black hole is investigated in terms of the QNM presented. At last, a brief conclusion is given.

## 2 Noncommutative BTZ black hole

*U*(1) theory. The extension of the commutative \(SU(1,1)\times SU(1,1)\) Chern–Simons theory has to contain

*U*(1) elements. Thus, the noncommutative extension of \(SU(1,1)\times SU(1,1)\) Chern–Simons theory has to be \(U(1,1)\times U(1,1)\) noncommutative Chern–Simons theory. Therefore, one can regard the \(U(1,1)\times U(1,1)\) noncommutative Chern–Simons theory as a noncommutative extension of the three dimensional Einstein’s gravity. And the solution of the noncommutative Chern–Simons theory can be obtained from its commutative counter by using the Seiberg–Witten map [33]. The metric of the noncommutative BTZ black hole is given by Ref. [32]

*B*is the magnitude of the magnetic field, \(\theta \) is the noncommutative parameter. The details of the background magnetic field can be found in Refs. [33, 44, 45]. \(r_{+}\) and \(r_{-}\) are the radius of outer and inner horizons

*g*(

*r*) can be rewritten as

*g*(

*r*) would be considered as a correction term in the asymptotic region (\( r> > r_{h}\)), i.e. \(g(r)_{r>> r_{h}}\simeq f(r)_{r> > r_{h}}\).

## 3 Scalar QNM in the noncommutative BTZ background

*R*(

*z*) takes the form as

*R*(

*z*) into the Eq. (23)

## 4 Fermionic QNM in the noncommutative BTZ background

*z*

## 5 Quantitative agreement in the noncommutative BTZ background

*CFT*). They have simulated the QNM for high-dimension

*AdS*-Schwarzschild black holes. Further numerical simulations of quasi-normal modes in asymptotically

*AdS*space-times have been presented in Refs. [50, 51, 52, 53, 54, 55]. Birmingham et al. [8] have showed that there is a precise quantitative agreement between QNM frequencies and the location of the poles of the retarded correlation function describing the linear response on the conformal field theory in the BTZ background. The dual conformal field theory on the boundary is \(1+1\) dimensional, the conformal symmetry being generated by two copies of the Virasoro algebra acting separately on left- and right-moving sectors. Consequently, the conformal field theory splits into two independent sectors at thermal equilibrium with temperatures

*s*propagating in \(AdS_{3}\) corresponds to conformal weights (\(h_{L}, h_{R}\)) in the dual conformal field theory [6]

*m*of the field. In particular, \(\Delta =1+\sqrt{1+m^{2}}\) is for scalar field, \(\Delta =1+\left| m \right| \) is for fermionic field.

*CFT*which coincides precisely with the QNM of the BTZ black hole was presented (more details in Ref. [8])

## 6 Quantization of area/entropy

*E*and \(\Delta \omega _{E}\) are the energy and vibrational frequency. It is widely believed that the oscillating black hole radiates or absorbs elementary quanta of energy/mass when it undergoes a transition between two quantum states in its QNM spectrum. However, its QNM frequencies are complex quantities which lead to a serious problem, i.e. which part of them carries the genuine information about the energy. In our paper, we replace the energy

*E*with mass

*M*and obtain vibrational frequency between the two adjacent states of the black hole by following the same methods in Ref. [57]. The radius of event horizon has been given in Eq. (16) and the area of this black hole has the form

*I*for this noncommutative BTZ black hole is

*I*around \(\theta =0\), we obtain

## 7 Conclusion

In summary, we have investigated the quasinormal modes and area/entropy spectrum for the noncommutative BTZ black hole. The analytical expressions for the scalar and feimionic QNM have been given in noncommutative BTZ background. Our results of QNM frequencies could reduce to the formula in Refs. [8, 24, 25] when the noncommutative parameters and the coupling constant disappear. We have found that the coupling constant \(\xi \) shifts the QNM frequencies but does not affect the structure of the area/entropy spectrum. We also have found that the noncommutativity does not affect conformal weights (\(h_{L}, h_{R}\)), but influences the thermal equilibrium, which suggests that the quantitative agreement also exists between the QNM of the noncommutative BTZ black hole and the poles of the retarded correlation function of the corresponding perturbations of the dual conformal field theory. According to Bohr–Sommerfeld quantization, we have presented the intuitive expressions of the area/entropy spectrum of the noncommutative BTZ black hole with the nonminimally coupling scalar field. Clearly, the area/entropy spectrum are not equidistant because of the existence of noncommutativity. Particularly, it is noncommutativity of this black hole that brings in this additional richness and complexity into the pattern of the area/entropy spectrum, leading to its nonuniform.

## Notes

### Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant nos. U1731107.

## References

- 1.E. Berti, V. Cardoso, A.O. Starinets, Class. Quantum Gravity
**26**, 163001 (2009)ADSCrossRefGoogle Scholar - 2.D. Birmingham, Phys. Rev. D
**64**, 064024 (2001)ADSMathSciNetCrossRefGoogle Scholar - 3.T. Regge, J.A. Wheeler, Phys. Rev.
**108**, 1063 (1957)ADSMathSciNetCrossRefGoogle Scholar - 4.C.V. Vishveshwara, Nature
**227**, 936 (1970)ADSCrossRefGoogle Scholar - 5.R.A. Konoplya, A. Zhidenko, Rev. Mod. Phys.
**83**, 793 (2011)ADSCrossRefGoogle Scholar - 6.O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, Phys. Rep.
**323**, 183 (2000)ADSMathSciNetCrossRefGoogle Scholar - 7.G.T. Horowitz, V.E. Hubeny, Phys. Rev. D
**62**, 024027 (2000)ADSMathSciNetCrossRefGoogle Scholar - 8.D. Birmingham, I. Sachs, S.N. Solodukhin, Phys. Rev. Lett.
**88**, 151301 (2002)ADSMathSciNetCrossRefGoogle Scholar - 9.D. Birmingham, I. Sachs, S.N. Solodukhin, Phys. Rev. D
**67**, 104026 (2003)ADSMathSciNetCrossRefGoogle Scholar - 10.A. Connes,
*Noncommutative Geometry*(Academic Press, San Diego, 1994)zbMATHGoogle Scholar - 11.C. Rovelli, L. Smolin, Phys. Rev. D
**52**, 10 (1995)CrossRefGoogle Scholar - 12.S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Phys.
**172**, 187 (1995)ADSCrossRefGoogle Scholar - 13.B.P. Dolan, K.S. Gupta, A. Stern, Class. Quantum Gravity
**24**, 1647 (2007)ADSCrossRefGoogle Scholar - 14.B.P. Dolan, K.S. Gupta, A. Stern, J. Phys. Conf. Ser.
**174**, 012023 (2009). (IOP Publishing)Google Scholar - 15.T. Ohl, A. Schenkel, JHEP
**0910**, 052 (2009)ADSCrossRefGoogle Scholar - 16.J. Lukierski, H. Ruegg, A. Nowicki, V.N. Tolstoi, Phys. Lett. B
**264**, 331 (1991)ADSMathSciNetCrossRefGoogle Scholar - 17.J. Lukierski, H. Ruegg, Phys. Lett. B
**329**, 189 (1994)ADSMathSciNetCrossRefGoogle Scholar - 18.J. Gomis, T. Mehen, Nucl. Phys. B
**591**, 265 (2000)ADSCrossRefGoogle Scholar - 19.A.P. Balachandran, T.R. Govindarajan, G. Mangano, A. Pinzul, B.A. Qureshi, S. Vaidya, Phys. Rev. D
**75**, 045009 (2007)ADSCrossRefGoogle Scholar - 20.A.P. Balachandran, P. Padmanabhan, A.R. de Queiroz, Phys. Rev. D
**84**, 065020 (2011)ADSCrossRefGoogle Scholar - 21.T. Juric, T. Poulain, J.C. Wallet, JHEP
**1605**, 146 (2016)ADSCrossRefGoogle Scholar - 22.A. Gere, T. Juric, J.C. Wallet, JHEP
**1512**, 045 (2015)ADSGoogle Scholar - 23.P. Aschieri, M. Dimitrijevic Ciric, R.J. Szabo, JHEP
**1802**, 036 (2018)ADSCrossRefGoogle Scholar - 24.K.S. Gupta, E. Harikumar, T. Juri, JHEP
**1509**, 25 (2015)CrossRefGoogle Scholar - 25.K.S. Gupta, T. Juri, A. Samsarov, JHEP
**1706**, 107 (2017)ADSCrossRefGoogle Scholar - 26.J.D. Bekenstein, Lett. Nuovo Cimento
**11**, 467 (1974)ADSCrossRefGoogle Scholar - 27.C. Rovelli, L. Smolin, Nucl. Phys. B
**442**, 593 (1995)ADSCrossRefGoogle Scholar - 28.C. Rovelli, L. Smolin, Nucl. Phys. B
**456**, 734 (1995)Google Scholar - 29.A. Ashtekar, J. Lewandowski, Class. Quantum Gravity
**14**, 55 (1997)ADSCrossRefGoogle Scholar - 30.S. Hod, Phys. Rev. Lett.
**81**, 4293 (1998)ADSMathSciNetCrossRefGoogle Scholar - 31.M.R. Setare, Class. Quantum Gravity
**21**, 1453 (2004)ADSCrossRefGoogle Scholar - 32.E. Chang-Young, D. Lee, Y. Lee, Class. Quantum Gravity
**26**, 185001 (2009)Google Scholar - 33.N. Seiberg, E. Witten, JHEP
**9909**, 032 (1999)ADSCrossRefGoogle Scholar - 34.A.H. Chamseddine, Phys. Lett. B
**504**, 33 (2001)ADSMathSciNetCrossRefGoogle Scholar - 35.P. Aschieri, C. Blohmann, M. Dimitrijevi et al., Class. Quantum Gravity
**22**, 3511 (2005)ADSCrossRefGoogle Scholar - 36.P. Aschieri, M. Dimitrijević, F. Meyer et al., Class. Quantum Gravity
**23**, 1883 (2006)ADSCrossRefGoogle Scholar - 37.M. Chaichian, M. Oksanen, A. Tureanu et al., Phys. Rev. D
**79**, 044016 (2009)ADSMathSciNetCrossRefGoogle Scholar - 38.M. Chaichian, P.P. Kulish, K. Nishijima, A. Tureanu et al., Phys. Lett. B
**604**, 98 (2004)ADSMathSciNetCrossRefGoogle Scholar - 39.M. Chaichian, P. Presnajder, A. Tureanu, Phys. Rev. Lett.
**94**, 151602 (2005)ADSCrossRefGoogle Scholar - 40.A. Achucarro, P.K. Townsend, Phys. Lett. B
**180**, 89 (1986)ADSMathSciNetCrossRefGoogle Scholar - 41.E. Witten, Nucl. Phys. B
**311**, 46 (1988)ADSCrossRefGoogle Scholar - 42.M. Banados, O. Chandia, N. Grandi et al., Phys. Rev. D
**64**, 084012 (2001)ADSMathSciNetCrossRefGoogle Scholar - 43.S. Cacciatori, D. Klemm, L. Martucci et al., Phys. Lett. B
**536**, 101 (2002)ADSMathSciNetCrossRefGoogle Scholar - 44.A. Connes, M.R. Douglas, A. Schwarz, JHEP
**9802**, 003 (1998)ADSCrossRefGoogle Scholar - 45.M.R. Douglas, C. Hull, JHEP
**9802**, 008 (1998)ADSCrossRefGoogle Scholar - 46.H.C. Kim, M.I. Park, C. Rim, JHEP
**0810**, 060 (2008)ADSCrossRefGoogle Scholar - 47.G. Panotopoulos, Á. Rincón, Phys. Lett. B
**772**, 523 (2017)ADSCrossRefGoogle Scholar - 48.M. Abramowitz, I.A. Stegun,
*Handbook of Mathematical Functions*(Dover, New York, 1970)zbMATHGoogle Scholar - 49.D.R. Brill, J.A. Wheeler, Rev. Mod. Phys.
**29**, 465 (1957)ADSMathSciNetCrossRefGoogle Scholar - 50.J.S. Chan, R.B. Mann, Phys. Rev. D
**55**, 7546 (1997)ADSCrossRefGoogle Scholar - 51.B. Wang, C.Y. Lin, E. Abdalla, Phys. Lett. B
**481**, 79 (2000)ADSMathSciNetCrossRefGoogle Scholar - 52.T.R. Govindarajan, V. Suneeta, Class. Quantum Gravity
**18**, 265 (2001)ADSCrossRefGoogle Scholar - 53.V. Cardoso, J.P.S. Lemos, Phys. Rev. D
**63**, 124015 (2001)ADSMathSciNetCrossRefGoogle Scholar - 54.V. Cardoso, J.P.S. Lemos, Phys. Rev. D
**64**, 084017 (2001)ADSCrossRefGoogle Scholar - 55.B. Wang, E. Abdalla, R.B. Mann, Phys. Rev. D
**65**, 084006 (2002)ADSMathSciNetCrossRefGoogle Scholar - 56.G. Kunstatter, Phys. Rev. Lett.
**90**, 161301 (2003)ADSMathSciNetCrossRefGoogle Scholar - 57.Y.X. Liu, S.W. Wei, R. Li, JHEP
**0903**, 076 (2009)ADSCrossRefGoogle Scholar

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