# Dirac quasinormal modes for a \(4\)-dimensional Lifshitz black hole

- First Online:

- Received:
- Accepted:

- 8 Citations
- 414 Downloads

## Abstract

We study the quasinormal modes of fermionic perturbations for an asymptotically Lifshitz black hole in four dimensions with dynamical exponent \(z=2\) and plane topology for the transverse section, and we find analytically and numerically the quasinormal modes for massless fermionic fields by using the improved asymptotic iteration method and the Horowitz–Hubeny method. The quasinormal frequencies are purely imaginary and negative, which guarantees the stability of these black holes under massless fermionic field perturbations. Remarkably, both numerical methods yield consistent results; i.e., both methods converge to the exact quasinormal frequencies; however, the improved asymptotic iteration method converges in a less number of iterations. Also, we find analytically the quasinormal modes for massive fermionic fields for the mode with lowest angular momentum. In this case, the quasinormal frequencies are purely imaginary and negative, which guarantees the stability of these black holes under fermionic field perturbations. Moreover, we show that the lowest quasinormal frequencies have real and imaginary parts for the mode with higher angular momentum by using the improved asymptotic iteration method.

## 1 Introduction

Lifshitz spacetimes have received great attention from the condensed matter point of view, i.e., the search for gravity duals of Lifshitz fixed points due to the AdS/CFT correspondence for condensed matter physics and quantum chromodynamics [1]. From the quantum field theory point of view, there are many invariant scale theories of interest when studying such critical points. Such theories exhibit the anisotropic scale invariance \(t\rightarrow \lambda ^zt\), \(x\rightarrow \lambda x\), with \(z\ne 1 \), where \(z\) is the relative scale dimension of time and space, and they are of particular interest in studies of critical exponent theory and phase transitions. Systems with such a behavior appear, for instance, in the description of strongly correlated electrons. The importance of possessing a tool to study strongly correlated condensed matter systems is beyond question, and consequently much attention has been focused on this area in recent years. Thermodynamically, it is difficult to compute conserved quantities for Lifshitz black holes; however, progress has been made on the computation of mass and related thermodynamic quantities by using the ADT method [2, 3] and the Euclidean action approach [4, 5]. Also, phase transitions between Lifshitz black holes and other configurations with different asymptotes have been studied in [6]. However, due to their different asymptotes these phases transitions do not occur.

An important property of black holes is their quasinormal modes (QNMs) and their quasinormal frequencies (QNFs) [7, 8, 9, 10, 11, 12]. The oscillation frequency of these modes is independent of the initial conditions and it only depends on the parameters of the black hole (mass, charge, and angular momentum) and the fundamental constants (Newton constant and cosmological constant) that describe a black hole, just like the parameters that define the test field. The study of the QNFs gives information as regards the stability of black holes under matter fields that evolve perturbatively in their exterior region, without back reacting on the metric. In general, the oscillation frequencies are complex, where the real part represents the oscillation frequency and the imaginary part describes the rate at which this oscillation is damped, with the stability of the black hole being guaranteed if the imaginary part is negative. The QNFs have been calculated by means of numerical and analytical techniques, and the Mashhoon method, the Chandrasekhar–Detweiler approach, the WKB method, the Frobenius method, the method of continued fractions, the Nollert method, and the asymptotic iteration method (AIM) are some remarkable numerical methods. For a review see [12] and the references therein. Generally, the Lifshitz black holes are stable under scalar perturbations, and the QNFs show the absence of a real part [5, 13, 14, 15, 16, 17]. In the context of black hole thermodynamics, the QNMs allow the quantum area spectrum of the black hole horizon to be studied [13] as well as the mass and the entropy spectrum.

On the other hand, the QNMs determine how fast a thermal state in the boundary theory will reach thermal equilibrium according to the AdS/CFT correspondence [18], where the relaxation time of a thermal state of the boundary thermal theory is proportional to the inverse of the imaginary part of the QNFs of the dual gravity background, which was established due to the QNFs of the black hole being related to the poles of the retarded correlation function of the corresponding perturbations of the dual conformal field theory [19]. Fermions on the Lifshitz background have been studied in [20], by using the fermionic Green’s function in \(4\)-dimensional Lifshitz spacetime with \(z=2\), and also the authors considered a non-relativistic (mixed) boundary condition for fermions and showed that the spectrum has a flat band.

In this work, we will consider a matter distribution outside the horizon of the Lifshitz black hole in four dimensions with a plane transverse section and dynamical exponent \(z=2\). The matter is parameterized by a fermionic field, which we will perturb by assuming that there is no back reaction on the metric. We obtain analytically and numerically the QNFs for massless fermionic fields by using the improved AIM [21, 22] and the Horowitz–Hubeny method [23], and then we study their stability under fermionic perturbations. Also, we obtain analytically the QNFs of massive fermionic fields perturbations for the mode with lowest angular momentum and numerically the lowest QNF for the mode with higher angular momentum by using the improved AIM.

The paper is organized as follows. In Sect. 2 we give a brief review of the Lifshitz black holes considered in this work. In Sect. 3 we calculate the QNFs of fermionic perturbations for the \(4\)-dimensional Lifshitz black hole with plane topology and \(z=2\). Finally, our conclusions are in Sect. 4.

## 2 Lifshitz black hole

## 3 Fermionic quasinormal modes of a \(4\)-dimensional Lifshitz black hole

### 3.1 Case \(\kappa =0\)

### 3.2 Case \(\kappa \ne 0\)

In this section we will compute the QNFs for the case \(\kappa \ne 0\). We will obtain analytical solutions for massless fermions, then we will employ two numerical methods as mentioned previously. Firstly, we will use the improved AIM and then we will compute some QNFs with the Horowitz–Hubeny method, and finally we will compare the results obtained with both methods.

#### 3.2.1 Analytical solution

#### 3.2.2 Improved asymptotic iteration method

Improved AIM. Quasinormal frequencies for \(\kappa = 1, 2\) and \(3\), \(m=0\) and \(l=1\) (set 1)

\(\kappa \) | \(n\) | \(\omega \) | Exact | \(n\) | \(\omega \) | Exact |
---|---|---|---|---|---|---|

\(1\) | \(0\) | \(-0.75000i\) | \(-0.75000i\) | \(4\) | \(-2.55000i\) | \(-2.55000i\) |

\(1\) | \(-1.12500i\) | \(-1.12500i\) | \(5\) | \(-3.04167i\) | \(-3.04167i\) | |

\(2\) | \(-1.58333i\) | \(-1.58333i\) | \(6\) | \(-3.53571i\) | \(-3.53571i\) | |

\(3\) | \(-2.06250i\) | \(-2.06250i\) | \(7\) | \(-4.03125i\) | \(-4.03125i\) | |

\(2\) | \(0\) | \(-1.50000i\) | \(-1.50000i\) | \(4\) | \(-3.16667i\) | \(-3.16667i\) |

\(1\) | \(-1.83333i\) | \(-1.83333i\) | \(5\) | \(-3.64285i\) | \(-3.64286i\) | |

\(2\) | \(-2.25000i\) | \(-2.25000i\) | \(6\) | \(-4.12500i\) | \(-4.12500i\) | |

\(3\) | \(-2.70000i\) | \(-2.70000i\) | \(7\) | \(-4.61111i\) | \(-4.61111i\) | |

\(3\) | \(0\) | \(-2.12500i\) | \(-2.12500i\) | \(4\) | \(-2.95000i\) | \(-2.95000i\) |

\(1\) | \(-2.25000i\) | \(-2.25000i\) | \(5\) | \(-3.37500i\) | \(-3.37500i\) | |

\(2\) | \(-2.56250i\) | \(-2.56250i\) | \(6\) | \(-3.82143i\) | \(-3.82143i\) | |

\(3\) | \(-2.75000i\) | \(-2.75000i\) | \(7\) | \(-4.28124i\) | \(-4.28125i\) |

Improved AIM. Quasinormal frequencies for \(\kappa = 1, 2\) and \(3\), \(m=0\) and \(l=1\) (set 2)

\(\kappa \) | \(n\) | \(\omega \) | Exact | \(n\) | \(\omega \) | Exact |
---|---|---|---|---|---|---|

\(1\) | \(0\) | \(-0.75000i\) | \(-0.75000i\) | \(4\) | \(-2.30556i\) | \(-2.30556i\) |

\(1\) | \(-0.91667i\) | \(-0.91667i\) | \(5\) | \(-2.79545i\) | \(-2.79545i\) | |

\(2\) | \(-1.35000i\) | \(-1.35000i\) | \(6\) | \(-3.28846i\) | \(-3.28846i\) | |

\(3\) | \(-1.82143i\) | \(-1.82143i\) | \(7\) | \(-3.78333i\) | \(-3.78333i\) | |

\(2\) | \(0\) | \(-1.41667i\) | \(-1.41667i\) | \(4\) | \(-2.47222i\) | \(-2.47222i\) |

\(1\) | \(-1.65000i\) | \(-1.65000i\) | \(5\) | \(-2.93214i\) | \(-2.93182i\) | |

\(2\) | \(-2.03571i\) | \(-2.03571i\) | \(6\) | \(-3.40385i\) | \(-3.40385i\) | |

\(3\) | \(-2.25000i\) | \(-2.25000i\) | \(7\) | \(-3.88333i\) | \(-3.88333i\) | |

\(3\) | \(0\) | \(-2.15000i\) | \(-2.15000i\) | \(4\) | \(-3.15909i\) | \(-3.15909i\) |

\(1\) | \(-2.25000i\) | \(-2.25000i\) | \(5\) | \(-3.59615i\) | \(-3.59615i\) | |

\(2\) | \(-2.39286i\) | \(-2.39286i\) | \(6\) | \(-4.05000i\) | \(-4.05000i\) | |

\(3\) | \(-2.75000i\) | \(-2.75000i\) | \(7\) | \(-4.51471i\) | \(-4.51471i\) |

Improved AIM. Lowest quasinormal frequencies for \(\kappa = 1\), \(m=0.5, 1.0, 1.5, 2.0\) and \(2.5\), and \(l=1\)

\(m\) | \(\omega \) |
---|---|

\(0.5\) | \(0.08970-0.76051i\) |

\(1.0\) | \(0.10195-0.77728i\) |

\(1.5\) | \(0.09063-0.83925i\) |

\(2.0\) | \(0.07884-0.92493i\) |

\(2.5\) | \(0.06907-1.02306i\) |

#### 3.2.3 Horowitz–Hubeny method

Horowitz–Hubeny method. Quasinormal frequencies for \(\kappa = 1, 2\) and \(3\), \(m=0\) and \(l=1\) (set 1)

\(\kappa \) | \(n\) | \(\omega \) | Exact | \(n\) | \(\omega \) | Exact |
---|---|---|---|---|---|---|

\(1\) | \(0\) | \(-\) | \(-0.75000i\) | \(4\) | \(-2.54992i\) | \(-2.55000i\) |

\(1\) | \(-1.12497i\) | \(-1.12500i\) | \(5\) | \(-3.04157i\) | \(-3.04167i\) | |

\(2\) | \(-1.58328i\) | \(-1.58333i\) | \(6\) | \(-3.53561i\) | \(-3.53571i\) | |

\(3\) | \(-2.06243i\) | \(-2.06250i\) | \(7\) | \(-4.03114i\) | \(-4.03125i\) | |

\(2\) | \(0\) | \(-1.50285i\) | \(-1.50000i\) | \(4\) | \(-3.16662i\) | \(-3.16667i\) |

\(1\) | \(-1.83338i\) | \(-1.83333i\) | \(5\) | \(-3.64279i\) | \(-3.64286i\) | |

\(2\) | \(-2.25000i\) | \(-2.25000i\) | \(6\) | \(-4.12493i\) | \(-4.12500i\) | |

\(3\) | \(-2.69997i\) | \(-2.70000i\) | \(7\) | \(-4.61107i\) | \(-4.61111i\) | |

\(3\) | \(0\) | \(-2.12503i\) | \(-2.12500i\) | \(4\) | \(-2.95042i\) | \(-2.95000i\) |

\(1\) | \(-2.25000i\) | \(-2.25000i\) | \(5\) | \(-3.37514i\) | \(-3.37500i\) | |

\(2\) | \(-2.56218i\) | \(-2.56250i\) | \(6\) | \(-3.82149i\) | \(-3.82143i\) | |

\(3\) | \(-2.75000i\) | \(-2.75000i\) | \(7\) | \(-4.28128i\) | \(-4.28125i\) |

Horowitz–Hubeny method. Quasinormal frequencies for \(\kappa = 1, 2\) and \(3\), \(m=0\) and \(l=1\) (set 2)

\(\kappa \) | \(n\) | \(\omega \) | Exact | \(n\) | \(\omega \) | Exact |
---|---|---|---|---|---|---|

\(1\) | \(0\) | \(-0.75000i\) | \(-0.75000i\) | \(4\) | \(-2.30560i\) | \(-2.30556i\) |

\(1\) | \(-0.91678i\) | \(-0.91667i\) | \(5\) | \(-2.79549i\) | \(-2.79545i\) | |

\(2\) | \(-1.35007i\) | \(-1.35000i\) | \(6\) | \(-3.28849i\) | \(-3.28846i\) | |

\(3\) | \(-1.82148i\) | \(-1.82143i\) | \(7\) | \(-3.78336\) | \(-3.78333i\) | |

\(2\) | \(0\) | \(-1.41669i\) | \(-1.41667i\) | \(4\) | \(-2.47309i\) | \(-2.47222i\) |

\(1\) | \(-1.64992i\) | \(-1.65000i\) | \(5\) | \(-2.93218i\) | \(-2.93182i\) | |

\(2\) | \(-2.03511i\) | \(-2.03571i\) | \(6\) | \(-3.40409i\) | \(-3.40385i\) | |

\(3\) | \(-2.25000i\) | \(-2.25000i\) | \(7\) | \(-3.88353\) | \(-3.88333i\) | |

\(3\) | \(0\) | \(-2.14993i\) | \(-2.15000i\) | \(4\) | \(-3.15893i\) | \(-3.15909i\) |

\(1\) | \(-2.25000i\) | \(-2.25000i\) | \(5\) | \(-3.59577i\) | \(-3.59615i\) | |

\(2\) | \(-2.39311i\) | \(-2.39286i\) | \(6\) | \(-4.04918i\) | \(-4.05000i\) | |

\(3\) | \(-2.75000i\) | \(-2.75000i\) | \(7\) | \(-4.51185i\) | \(-4.51471i\) |

## 4 Conclusions

In this work we have calculated the QNFs of massless fermionic perturbations for the \(4\)-dimensional Lifshitz black hole with a plane topology and dynamical exponent \(z=2\). It is well known that the boundary conditions depend on the asymptotic behavior of spacetime. For asymptotically AdS spacetimes the potential diverges and thus the field must be null at infinity (Dirichlet boundary conditions) or the flux must vanish at infinity, which is known as Neumann boundary conditions. Here, as the black hole is asymptotically Lifshitz and the potential diverges at the boundary, we have considered that the fermionic fields will be null at infinity (Dirichlet boundary conditions) and that there are only ingoing modes at the horizon, and we have obtained analytical and numerical results using the improved AIM and the Horowit–Hubeny method, and we have found that the QNFs for the massless fermionic field are purely imaginary and negative, which ensures the stability of the black hole under massless fermionic perturbations. Remarkably, both numerical methods yield consistent results; i.e., both methods converge to the exact QNFs; however, the improved AIM converges in a fewer number of iterations.

Also, we have found analytically the QNFs for massive fermionic fields for the mode with lowest angular momentum, being the QNFs purely imaginary and negative, which guarantees the stability of these black holes under fermionic fields perturbations. Interestingly, in this case we obtain two sets of Dirac QNFs that cover all the range of mass (positive and negative) of the fermionic field in analogy with Neumann boundary condition which yields two sets of modes in the BTZ black hole. On the other hand, we have shown that the lowest QNFs for massive fermionic fields, for the mode with higher angular momentum, have real and imaginary parts, by using the improved AIM.

## Notes

### Acknowledgments

P.G. would like to thank Felipe Leyton for valuable discussions and comments on numerical methods. This work was funded by the Comisión Nacional de Investigación Científica y Tecnológica through FONDECYT Grant 11121148 (YV, MC) and also partially funded by Dirección de investigación, Universidad de La Frontera (MC). The authors also thank partial support by NLHCP (ECM-02) at CMCC UFRO. P.G. and Y.V. acknowledge the hospitality of the Universidad de La Frontera where part of this work was undertaken.

### References

- 1.S. Kachru, X. Liu, M. Mulligan, Phys. Rev. D
**78**, 106005 (2008). [arXiv:0808.1725 [hep-th]] - 2.D.O. Devecioglu, O. Sarioglu, Phys. Rev. D
**83**, 021503 (2011). [arXiv:1010.1711 [hep-th]] - 3.D.O. Devecioglu, O. Sarioglu, Phys. Rev. D
**83**, 124041 (2011). [arXiv:1103.1993 [hep-th]] - 4.H.A. Gonzalez, D. Tempo, R. Troncoso, JHEP
**1111**, 066 (2011). [arXiv:1107.3647 [hep-th]] - 5.Y.S. Myung, T. Moon, Phys. Rev. D
**86**, 024006 (2012). [arXiv:1204.2116 [hep-th]] - 6.Y.S. Myung, Eur. Phys. J. C
**72**, 2116 (2012). [arXiv:1203.1367 [hep-th]] - 7.T. Regge, J.A. Wheeler, Phys. Rev.
**108**, 1063 (1957)CrossRefMATHMathSciNetADSGoogle Scholar - 8.F.J. Zerilli, Phys. Rev. D
**2**, 2141 (1970)CrossRefMATHMathSciNetADSGoogle Scholar - 9.F.J. Zerilli, Phys. Rev. Lett.
**24**, 737 (1970)CrossRefADSGoogle Scholar - 10.K.D. Kokkotas, B.G. Schmidt, Living Rev. Rel.
**2**, 2 (1999). [gr-qc/9909058] - 11.H.-P. Nollert, Class. Quant. Grav.
**16**, R159 (1999)Google Scholar - 12.R.A. Konoplya, A. Zhidenko, Rev. Mod. Phys.
**83**, 793 (2011). [arXiv:1102.4014 [gr-qc]] - 13.B. Cuadros-Melgar, J. de Oliveira, C.E. Pellicer, Phys. Rev. D
**85**, 024014 (2012). [arXiv:1110.4856 [hep-th]] - 14.P.A. Gonzalez, J. Saavedra, Y. Vasquez, Int. J. Mod. Phys. D
**21**, 1250054 (2012). [arXiv:1201.4521 [gr-qc]] - 15.P.A. Gonzalez, F. Moncada, Y. Vasquez, Eur. Phys. J. C
**72**, 2255 (2012). [arXiv:1205.0582 [gr-qc]] - 16.R. Becar, P.A. Gonzalez, Y. Vasquez, Int. J. Mod. Phys. D
**22**, 1350007 (2013). [arXiv:1210.7561 [gr-qc]] - 17.A. Giacomini, G. Giribet, M. Leston, J. Oliva, S. Ray, Phys. Rev. D
**85**, 124001 (2012). [arXiv:1203.0582 [hep-th]] - 18.J.M. Maldacena, Adv. Theor. Math. Phys.
**2**, 231 (1998). [hep-th/9711200]MATHMathSciNetADSGoogle Scholar - 19.D. Birmingham, I. Sachs, S.N. Solodukhin, Phys. Rev. Lett.
**88**, 151301 (2002). [hep-th/0112055]CrossRefMathSciNetADSGoogle Scholar - 20.M. Alishahiha, M.R. Mohammadi, Mozaffar, A. Mollabashi. Phys. Rev. D
**86**, 026002 (2012). [arXiv:1201.1764 [hep-th]] - 21.H.T. Cho, A.S. Cornell, J. Doukas, W. Naylor, Class. Quant. Grav.
**27**, 155004 (2010). [arXiv:0912.2740 [gr-qc]] - 22.H.T. Cho, A.S. Cornell, J. Doukas, T.R. Huang, W. Naylor, Adv. Math. Phys. 2012 (2012) 281705. [arXiv:1111.5024 [gr-qc]]
- 23.G.T. Horowitz, V.E. Hubeny, Phys. Rev. D
**62**, 024027 (2000). [hep-th/9909056]CrossRefMathSciNetADSGoogle Scholar - 24.S.A. Hartnoll, J. Polchinski, E. Silverstein, D. Tong, JHEP
**1004**, 120 (2010). [arXiv:0912.1061 [hep-th]] - 25.G.T. Horowitz, B. Way, Phys. Rev. D
**85**, 046008 (2012). [arXiv:1111.1243 [hep-th]] - 26.E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, Phys. Rev. D
**80**, 104029 (2009). [arXiv:0909.1347 [hep-th]] - 27.R.-G. Cai, Y. Liu, Y.-W. Sun, JHEP
**0910**, 080 (2009). [arXiv:0909.2807 [hep-th]] - 28.E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, JHEP
**1004**, 030 (2010). [arXiv:1001.2361 [hep-th]] - 29.M.H. Dehghani, R.B. Mann, JHEP
**1007**, 019 (2010). [arXiv:1004.4397 [hep-th]] - 30.R.B. Mann, JHEP
**0906**, 075 (2009). [arXiv:0905.1136 [hep-th]] - 31.G. Bertoldi, B.A. Burrington, A. Peet, Phys. Rev. D
**80**, 126003 (2009). [arXiv:0905.3183 [hep-th]] - 32.K. Balasubramanian, J. McGreevy, Phys. Rev. D
**80**, 104039 (2009). [arXiv:0909.0263 [hep-th]] - 33.M. Abramowitz, A. Stegun,
*Handbook of Mathematical Functions*(Dover publications, New York, 1970)Google Scholar - 34.P. Breitenlohner, D.Z. Freedman, Phys. Lett. B
**115**, 197 (1982)CrossRefMathSciNetADSGoogle Scholar - 35.P. Breitenlohner, D.Z. Freedman, Ann. Phys.
**144**, 249 (1982)CrossRefMATHMathSciNetADSGoogle Scholar - 36.H. Ciftci, R.L. Hall, N. Saad, J. Phys. A
**36**(47), 1180711816 (2003)CrossRefMathSciNetADSGoogle Scholar - 37.H. Ciftci, R.L. Hall, N. Saad, Phys. Lett. A
**340**, 388 (2005)CrossRefMATHADSGoogle Scholar - 38.T. Barakat, Int. J. Mod. Phys. A
**21**, 4127 (2006)CrossRefMATHADSGoogle Scholar - 39.R.-G. Cai, Z.-Y. Nie, B. Wang, H.-Q. Zhang, arXiv:1005.1233 [gr-qc]
- 40.J.-l. Jing, gr-qc/0502010

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP^{3} / License Version CC BY 4.0.