# Dirac quasinormal modes of new type black holes in new massive gravity

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## Abstract

We study a new type of black holes in three-dimensional new massive gravity and we calculate analytically the quasinormal modes for fermionic perturbations for some special cases. Then we show that for these cases black holes of the new type are stable under fermionic field perturbations.

## Keywords

Black Hole Event Horizon Massive Gravity Quasinormal Mode Extremal Black Hole## 1 Introduction

In recent years, there has been a remarkable interest in the study of three-dimensional models of gravity. Apart from the BTZ black hole [1], which is a solution to the Einstein equations with a negative cosmological constant, much attention was given to topologically massive gravity (TMG), which is a generalization of three-dimensional GR that amounts to augment the Einstein–Hilbert action adding a Chern–Simons gravitational term [2, 3, 4]. Here, the propagating degree of freedom is a massive graviton. TMG also admits the BTZ (and other) black holes as exact solutions. The renewed interest on TMG relies on the possibility of constructing a chiral theory of gravity at a special point of the space of parameters [5].

On the other hand, Bergshoeff, Hohm, and Townsend (BHT) introduced another three-dimensional massive gravity theory, which is known as new massive gravity (NMG), being the action with the standard Einstein–Hilbert term with a specific combination of the scalar curvature square term and the Ricci tensor square one [6, 7, 8, 9, 10], and it is equivalent at the linearized level to the (unitary) Fierz–Pauli action for a massive spin-2 field [6]. The model in three dimensions is indeed unitary at tree level, but the corresponding model in higher dimensions is not, due to the appearance of non-unitary massless spin-2 modes [11]. Also, NMG admits warped AdS black holes [12], AdS waves [13, 14], asymptotically Lifshitz black holes [15], gravitational solitons, kinks, and wormholes [16]; for further aspects of the BHT theory see [17, 18, 19, 20, 21, 22]. Besides, asymptotically AdS and Lifshitz black holes in NMG dressed by a (non-)minimally coupled scalar field have been studied recently in [23]. It is worth to mention that TMG and NMG share common features, however, there are different aspects, one of them is the existence of the new type of black holes, for a specific combination of parameters in the NMG Lagrangian, which was discovered by BHT, and which are known as new type black holes.

The particular motivation of this work is to calculate the quasinormal modes (QNMs) for fermionic field perturbations in the background of the new type black holes in three-dimensional new massive gravity and study the stability of these black holes under fermionic perturbations. The QNMs and their quasinormal frequencies (QNFs) are an important property of black holes and have a long history [24, 25, 26, 27, 28, 29]. It is well known that the presence of event horizons dampens the vibration modes of a matter field that evolves perturbatively in the exterior region. In this way, the system is intrinsically dissipative, i.e., there is no temporary symmetry. In general, the oscillation frequencies are complex, therefore the system is not Hermitian. Nevertheless, the oscillation frequency of these modes is independent of the initial conditions and it only depends on the parameters (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. In three-dimensional spacetime, the QNMs of the BTZ black hole have been studied in [30, 31, 32], and the QNMs for scalar field perturbations in the background of the new type black holes in NMG was studied in [33].

The QNMs give information as regards the stability of black holes under matter fields that evolve perturbatively in the exterior region of them, without backreacting on the metric. Also, the QNMs determine how fast a thermal state in the boundary theory will reach thermal equilibrium according to the AdS/CFT correspondence [34], where the relaxation time of a thermal state of the boundary thermal theory is proportional to the inverse of the imaginary part of the QNMs of the dual gravity background [35]. In the context of black hole thermodynamics, the QNMs allow the quantum area spectrum of the black hole horizon to be studied, as well as the mass and the entropy spectrum. In this regard, Bekenstein [36] was the first to propose the idea that in quantum gravity the area of black hole horizon is quantized, leading to a discrete spectrum which is evenly spaced. Then Hod [37] conjectured that the asymptotic QNF is related to the quantized black hole area, by identifying the vibrational frequency with the real part of the QNFs. However, it is not universal for every black hole background. Then Kunstatter [38] propose that the black hole spectrum can be obtained by imposing the Bohr–Sommerfeld quantization condition to an adiabatic invariant quantity involving the energy and the vibrational frequency. Furthermore, Maggiore [39] argued that in the large damping limit the identification of the vibrational frequency with the imaginary part of the QNF could lead to the Bekenstein universal bound. Then the consequences of these proposals were studied in several spacetimes; for instance, see [33], where the authors comment on the AdS/CFT correspondence and entropy/area spectrum for the new type of black holes. Besides, in [40, 41, 42, 43] the authors discuss a connection between Hawking radiation and black hole quasinormal modes, which is important in the route to quantize gravity, because one can naturally interpret black hole quasinormal modes in terms of quantum levels.

The paper is organized as follows. In Sect. 2 we give a brief review of the new type of black holes in three-dimensional New Massive Gravity. In Sect. 3 we calculate the exact QNMs of fermionic perturbations for the new type black holes. Finally, our conclusions are presented in Sect. 4.

## 2 New type black holes

## 3 Dirac quasinormal modes of new type black holes

### 3.1 Null angular momentum

#### 3.1.1 Asymptotically \(AdS\) new type black holes

#### 3.1.2 Asymptotically \(dS\) new type black holes

### 3.2 Massless Dirac QNMs of asymptotically \(dS\) and asymptotically locally flat new type black holes

In this section we compute the QNMs for asymptotically \(dS\) (\(A<0\)) and asymptotically locally flat (\(A=0\)) new type black holes for massless fermionic perturbations (\(m=0\)).

#### 3.2.1 Asymptotically \(dS\) new type black holes

#### 3.2.2 Asymptotically locally flat new type black holes

### 3.3 Massless Dirac QNMs of extremal new type black holes

## 4 Conclusions

In this work we have calculated analytically the QNMs of fermionic perturbations for some special cases for a new type of black holes, which are solutions of three-dimensional NMG and also are solutions of conformal gravity in three dimensions. The first case that we have analyzed concerns massive fermionic fields perturbations without angular momentum (\(\kappa =0\)) in the black hole background, and we have found the QNFs for an asymptotically \(AdS\) (\(dS\)) new type of black holes. For asymptotically \(AdS\) new type black holes the QNFs are purely imaginary and negative, which ensures the stability of the black hole under fermionic perturbations. For an asymptotically \(dS\) new type black holes the QNFs have a real and imaginary part, and the imaginary part is negative, which ensures the stability of the black hole under fermionic perturbations. It is worth mentioning that in these cases the Dirac equation can be written as a Riemann differential equation, as in [46]. Other cases, where is possible to find the QNFs analytically, are for asymptotically \(dS\) and asymptotically locally flat new types of black holes for massless fermionic field perturbations; in these cases we have found that the QNFs have a real and imaginary part, and the imaginary part is negative, which ensures the stability of the black hole under fermionic perturbations.

Finally, we have analyzed fermionic field perturbations in the extremal new type black holes for some special cases, and we found that there are not quasinormal modes as occurs, for instance, in [46, 47], where the authors have showed the absence of QNMs in the extremal BTZ black hole and the extremal four-dimensional Lifshitz Black Hole in Conformal Gravity. However, it was shown that it is possible to construct the QNMs of three-dimensional extremal black holes in an algebraic way as the descendants of the highest weight modes [48], with hidden conformal symmetry being an intrinsic property of the extremal black hole. Also, it is worth to mention that the absence of QNMs for extremal black holes does not always occur, for instance see [49], where the authors have showed the presence of QNMs for the extremal BTZ black holes in TMG.

## Notes

### Acknowledgments

We thank to Julio Oliva for useful comments. This work was funded by the Comisión Nacional de Investigación Científica y Tecnológica through FONDECYT Grant 11121148 (Y.V.). P. A. G. acknowledges the hospitality of the Universidad de La Serena where part of this work was undertaken.

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