# Fermionic quasinormal modes for two-dimensional Hořava–Lifshitz black holes

## Abstract

To obtain fermionic quasinormal modes, the Dirac equation for two types of black holes is investigated. It is shown that two different geometries lead to distinctive types of quasinormal modes, while the boundary conditions imposed on the solutions in both cases are identical. For the first type of black hole, the quasinormal modes have continuous spectrum with negative imaginary part that provides the stability of perturbations. For the second type of the black hole, the quasinormal modes have a discrete spectrum and are completely imaginary.

## 1 Introduction

The investigation of gravitational perturbations of the Schwarzschild geometry started several decades ago [1, 2, 3]. That idea was applied for the examination of perturbations of other types of black holes caused by the fields of a different nature, for example scalar field or Dirac field. All this work gave birth to the method which is known nowadays as the quasinormal mode method (QNM) [4, 5, 6]. This method allows one to get important information as regards the stability of black holes against perturbations of different types which evolve in the exterior region of the black holes. We also note that in most cases the influence of the external fields is considered perturbatively and the backreaction of the field on the black hole’s metric is not taken into account. The quasinormal modes and their quasinormal frequencies are useful for different branches of investigations in general relativity. In particular, QN modes are important in gauge–string duality theories (AdS-CFT) [7] because they define the relaxation times of dual field theories [8]. The relation between QN modes and retarded correlators of dual field theories was also established [9, 10, 11, 12]. Another possibility is due to Hod’s conjecture as regards the quantization of a black hole’s area [13, 14, 15]. The connection between QN modes and Hawking radiation is also considered [16, 17]. The progress in experimental astrophysics and the discovery of gravitational waves opened a new perspective for application of the QNM method for the estimation of different parameters of compact sources of gravitational field or the verification of some conjectures of general relativity [5].

Another area of active research is related to two different disciplines, namely quantum mechanics and general relativity. A well-known and still open problem is the reconciliation of the principles of these theories. It might give some hints about the underlying theory of quantum gravity. For example, nonrenormalizability is a crucial problem when one tries to quantize general relativity in the way possible for other gauge fields. To overcome this difficulty, it was supposed that the general relativity should be treated as an effective theory and, in order to have a gravitation theory suitable for quantization, the principles of general relativity should be elaborated. One of the approaches that leads to power-countable UV-renormalizability is the so-called Hořava–Lifshitz (HL) theory [18, 19, 20]. General relativity can be recovered as an infrared limit of the Hořava–Lifshitz theory. Because of its attractive and promising features, the Hořava–Lifshitz approach has gained considerable interest in recent years. In particular, black hole solutions were found and their properties were investigated [21, 22, 23, 24, 25, 26, 27, 28, 29]. The quasinormal modes for HL black holes were studied in Refs. [30, 31, 32, 33, 34].

The examination of the evolution of the fields in a background of lower-dimensional black holes is an interesting and important problem. Firstly, because of the simplicity of those problems in comparison with higher-dimensional cases, analytical computations can be made and, as a result, for many kinds of black holes exact QN frequencies can be calculated. The second important point is the fact that lower-dimensional black holes and the fields evolving in their backgrounds give some hints or reveal some aspects of higher-dimensional cases. The other important moment which stimulates the interest to lower-dimensional black holes is the fact that they might be suitable models for analogue gravity [35].

Our paper is organized as follows: in Sect. 2 we briefly review some \(1+1\)-dimensional black hole solutions in HL gravity. In Sect. 3 the Dirac equations for fermion fields in specific black hole backgrounds are written. In the Sect. 4 we investigate fermionic QNMs for chosen BH metrics. The last section contains some concluding remarks.

## 2 \(1+1\)-dimensional black holes solutions in Hořava theory

- The first case: for \(C_1=-M\), \(C_2=-\frac{1}{2}\) and \(A=B=C=0\) we have \(V_{\varphi }(\varphi )=0\) (or \(V(\varphi )=const\)) and one arrives at the solution:It should be noted that similar solution was obtained in the context of the ordinary \(1+1\)-dimensional gravity [38].$$\begin{aligned} \mathrm{d}s^2=-(2M|x|-1)\mathrm{d}t^2+\frac{1}{(2M|x|-1)}\mathrm{d}x^2. \end{aligned}$$(4)
- The second case: the constants are chosen in the following way: \(A=\Lambda \), \(B=C=0\), \(C_1=-M\) and \(C_2=-\epsilon /2\). For this case we have \(V_{\varphi }(\varphi )=\Lambda \), which leads to linear dependence for the scalar potential \(V(\varphi )=\Lambda \varphi \). The solution takes the formThe latter metric can be rewritten in a bit different form after some kind of transformation of coordinates [34]:$$\begin{aligned} \mathrm{d}s^2= & {} -\left( \left( \Lambda /\eta \right) ^2x^2+2Mx-\epsilon \right) \mathrm{d}t^2\nonumber \\&+\,\frac{1}{\left( \left( \Lambda /\eta \right) ^2x^2+2Mx-\epsilon \right) }\mathrm{d}x^2. \end{aligned}$$(5)Having used the above transformation we arrive at a new representation of the metric (5):$$\begin{aligned} u=\sqrt{\frac{\Lambda }{\eta }}x+\sqrt{\frac{\eta }{\Lambda }}M. \end{aligned}$$(6)and here \(u_+=\sqrt{(\eta /\Lambda )M^2+\epsilon }\) and \(l=\root 4 \of {\Lambda /\eta }\). It is worth of note that in the new coordinate system the horizon of the black hole is located at the point \(u=u_+\).$$\begin{aligned} \mathrm{d}s^2=-(u^2-u^2_+)\mathrm{d}t^2+\frac{l^2}{(u^2-u^2_+)}\mathrm{d}u^2 \end{aligned}$$(7)
- The third case: the so-called Schwarzschild-like solution. In this case one imposes the requirement that \(A=C=C_1=0\), \(B=-2M\), \(C_2=1/2\) and \(\eta =1\). As a result the metric would look as follows:We also note that in this case the potential can be written in explicit form [37]. So the metric takes the Schwarzschild-like form$$\begin{aligned} N^2(x)=1-\frac{2M}{x}, \quad \varphi (x)=\frac{1}{2}\ln \left( 1-\frac{2M}{x}\right) . \end{aligned}$$(8)$$\begin{aligned} \mathrm{d}s^2=-\left( 1-\frac{2M}{x}\right) \mathrm{d}t^2+\frac{1}{\left( 1-\frac{2M}{x}\right) }\mathrm{d}x^2. \end{aligned}$$(9)
- The fourth case is the so-called Reissner–Nordström-like case. The constants should be chosen as follows: \(A=C_1=0\), \(B=-2M\), \(C=3Q^2\) and \(C_2=1/2\). So we obtainAs a result the metric takes the Reissner–Nordström-like form$$\begin{aligned} N^2(x)=1-\frac{2M}{x}+\frac{Q^2}{x^4}, \quad \varphi (x)=\frac{1}{2}\ln \left( 1-\frac{2M}{x}+\frac{Q^2}{x^4}\right) .\nonumber \\ \end{aligned}$$(10)We note that in contrast to the previous cases here, it is not possible to find explicit form for the scalar potential.$$\begin{aligned} \mathrm{d}s^2=-\left( 1-\frac{2M}{x}+\frac{Q^2}{x^2}\right) \mathrm{d}t^2+\frac{1}{\left( 1-\frac{2M}{x}+\frac{Q^2}{x^2}\right) }\mathrm{d}x^2.\nonumber \\ \end{aligned}$$(11)

## 3 Dirac equation

*m*is the mass of fermionic field \(\psi \). The covariant derivative is defined as follows:

### 3.1 Dirac equation for the first kind of the metric

### 3.2 Dirac equation for the second kind of the metric

## 4 Quasinormal modes

In this section, the Dirac equations for two cases of the metric will be examined again separately. We will find the quasinormal modes and then compare the results. It should be remarked that the quasinormal modes for scalar perturbations in the same black hole background were considered in Ref. [34].

### 4.1 Quasinormal modes for the metric of the first kind

*x*by the following relation:

### 4.2 Quasinormal modes for the second kind of the metric

*z*is the interval: \(-1\leqslant z\leqslant 1\) and since we consider the motion of the particle outside the black hole our domain will be as follows: \(0\leqslant z\leqslant 1\). Having used the transformation (46) we rewrite the equation (29) in the form

*F*(

*z*) can be written

*a*,

*b*and

*c*are given by the relations

*a*,

*b*,

*c*(57) and the above general solution of the hypergeometric equation (58) immediately leads to the solution of the hypergeometric equation (49), which takes the form

*a*,

*b*and

*c*can be represented in the form

*n*. Now we conclude that the black hole metric (5) is stable under the influence of the fermionic perturbations. To consider QN modes for the lower component of the wavefunction, we use Eq. (28) which can be rewritten in the form

## 5 Concluding remarks

We studied fermionic quasinormal frequencies for two types of \(1+1\) dimensional HL black holes. The first type of black hole’s solution is similar to the corresponding solution, which can be found in the framework of the standard GR. We impose boundary conditions on the solutions of the Dirac equations to obtain quasinormal modes, namely we demand that in the vicinity of the horizon the wavefunction should behave as an ingoing wave and it has to vanish at infinity because the background geometry is not asymptotically flat. The solutions which satisfy the conditions mentioned above were found and the corresponding frequencies of the wavefunctions are complex and the imaginary parts of the frequencies are bounded from above. The obtained continuous spectrum for the quasinormal modes is in agreement with the results of Ref. [39]. As has been mentioned above a continuous QNF spectrum might appear for other types of black hole geometries [40, 41]. We also note that our analysis leads to the conclusion that the upper and lower components of the Dirac wavefunction have the same spectra.

The second type of black hole solution is defined in the presence of a dilatonic field. For this type of black hole’s geometry we impose the same boundary conditions on the solution of the wave equation. The solution which fulfills the imposed boundary condition has a purely imaginary discrete spectrum. It should be noted that the fermion field of arbitrary mass is stable in that geometry and as we mentioned before, for scalar field perturbations it might be unstable for large masses of the field [34].

## Notes

### Acknowledgements

This work was partly supported by Project FF-30F (No. 0116U001539) from the Ministry of Education and Science of Ukraine and Grant No. 0116U005055 of the State Fund For Fundamental Research of Ukraine.

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