Quasinormal Modes of Dirac Field in Generalized Nariai Spacetimes

Abstract

The exact electrically charged solutions to the Dirac equation in higher-dimensional generalized Nariai spacetimes are obtained. Using these solutions, the boundary conditions leading to quasinormal modes of the Dirac field are analyzed, and their correspondent quasinormal frequencies are analytically calculated.

1. Introduction

Quasinormal modes (QNMs) are eigenmodes of dissipative systems. For instance, if a spacetime with an event or cosmological horizon is perturbed from its equilibrium state, QNMs arise as damped oscillations with a spectrum of complex frequencies that do not depend on the details of the excitation. In fact, these frequencies depend just on the charges of the black hole, such as the mass, electric charge, and angular momentum [1, 2]. QNMs have been studied for a long time, and its interest has been renewed by the recent detection of gravitational waves, inasmuch as these are the modes that survive for a longer time when a background is perturbed and, therefore, these are the configurations that are generally measured by experiments [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Mathematically, this discrete spectrum of QNMs stems from the fact that certain boundary conditions must be imposed to the physical fields propagating in such background [30]. In this chapter, we consider a higher-dimensional generalization of the charged Nariai spacetime [31], namely, ${\mathit{dS}}_2\times S^2\times \dots \times S^2$, and investigate the dynamics of perturbations of the electrically charged Dirac field (spin 1/2). In such a geometry, the spinorial formalism [32, 33, 34] is used to show that the Dirac equation is separable [35] and can be reduced to a Schrödinger-like equation [36] whose potential is contained in the Rosen-Morse class of integrable potentials, which has the so-called Pöschl-Teller potential as a particular case [37, 38]. Finally, the boundary conditions leading to QNMs are analyzed, and the quasinormal frequencies (QNFs) are analytically obtained [5, 39].

2. Presenting the problem

In D dimensions, the dynamics of general relativity in spacetimes with a cosmological constant $\mathrm{\Lambda }$ is described by the Einstein-Hilbert action¹

$$S=\frac{1}{16\pi }\int d^{D}x\sqrt{\left|g\right|}\left[\mathcal{R}-\left(D-2\right)\mathrm{\Lambda }\right]+S_m,$$E1

where $\mathcal{R}$ is the Ricci scalar and $S_m$ stands for the action of all matter fields $\left\{{\mathbf{\Phi }}_i\right\}$ coupled to gravity appearing in the theory, which can be scalar, spinorial, vectorial, and so on. The least action principle allows to find the equations of motion for the fields $g_{\mathit{\mu \nu }}$ and ${\mathbf{\Phi }}_i$ which are given, respectively, by

$${\mathcal{R}}_{\mathit{\mu \nu }}-\frac{1}{2}\mathcal{R}{g}_{\mathit{\mu \nu }}+\frac{\left(D-2\right)}{2}\mathrm{\Lambda }{g}_{\mathit{\mu \nu }}=8\pi {\mathcal{T}}_{\mathit{\mu \nu }},\frac{\delta S_m}{\delta {\mathbf{\Phi }}_i}=0,$$E2

where ${\mathcal{T}}_{\mathit{\mu \nu }}$ is the symmetric stress-energy tensor associated to ${\mathbf{\Phi }}_i$ defined by the equation

$${\mathcal{T}}^{\mathit{\mu \nu }}=\frac{2}{\sqrt{\left|g\right|}}\frac{\delta S_m}{\delta g_{\mathit{\mu \nu }}}.$$E3

Since any symmetry has been imposed, the general solution of the system of Eq. (2) is some metric and fields in the background this metric

$${\mathit{ds}}^2=g_{\mathit{\mu \nu }}\left(x\right){\mathit{dx}}^{\mu }{\mathit{dx}}^{\nu },{\mathbf{\Phi }}_i={\mathbf{\Phi }}_i\left(x\right).$$E4

Now, let the pair $g_{\mathit{\mu \nu }}^{\left(0\right)}$ and ${\mathbf{\Phi }}_i^{\left(0\right)}$ be a solution for the equations of motion Eq. (2). Then, in order to study the perturbations around this particular solution, we write our fields as a sum of the unperturbed fields $g_{\mathit{\mu \nu }}^{\left(0\right)}$ and ${\mathbf{\Phi }}_i^{\left(0\right)}$ and the small perturbations $h_{\mathit{\mu \nu }}$ and ${\mathbf{\Psi }}_i$

$$g_{\mathit{\mu \nu }}=g_{\mathit{\mu \nu }}^{\left(0\right)}+h_{\mathit{\mu \nu }},{\mathbf{\Phi }}_i={\mathbf{\Phi }}_i^{\left(0\right)}+{\mathbf{\Psi }}_i,$$E5

where by “small” we mean that we neglect the quadratic and higher-order powers of the perturbation fields. Inserting the above equation into Eq. (2), we are left with a set of linear equations satisfied by the perturbed fields $h_{\mathit{\mu \nu }}$ and ${\mathbf{\Psi }}_i$. In general, these equations are coupled, namely, ${\mathbf{\Psi }}_i$ is a source for $h_{\mathit{\mu \nu }}$ and vice versa. However, in the special case in which ${\mathbf{\Phi }}_i^{\left(0\right)}=0$, the equations governing the perturbed fields ${\mathbf{\Psi }}_i$ can be decoupled from the metric perturbation $h_{\mathit{\mu \nu }}$ and vice versa. The reason why this happen is that when ${\mathbf{\Phi }}_i^{\left(0\right)}=0$, the stress-energy tensor ${\mathcal{T}}_{\mathit{\mu \nu }}$ can be set to zero at first order in the perturbation, since ${\mathcal{T}}_{\mathit{\mu \nu }}$ is typically quadratic or of higher order in the matter fields and, therefore, can be neglected. Therefore, investigating the linear dynamics of generic small perturbations of the matter fields with ${\mathcal{T}}_{\mathit{\mu \nu }}=0$ is equivalent to studying the test fields ${\mathbf{\Psi }}_i$ in the background $g_{\mathit{\mu \nu }}^{\left(0\right)}$.

In what follows, let us consider a specific matter field $\mathbf{\Psi }$ propagating in a generalized version of the Nariai spacetime described in Ref. [31]. Here, $\mathbf{\Psi }$ is an electrically charged spinorial field of mass $m$ that obeys the Dirac equation minimally coupled to an electromagnetic field in such spacetime. In $D=2d$, this spacetime is formed from the direct product of the de Sitter space ${\mathit{dS}}_2$ with $\left(d-1\right)$ copies of the unit spheres $S^2$ possessing different radii $R_j$. Thus, the natural line element of the higher-dimensional version of the Nariai spacetime is given by

$${\mathit{ds}}^2=g_{\mathit{\mu \nu }}^0{\mathit{dx}}^{\mu }{\mathit{dx}}^{\nu }=-f\left(r\right){\mathit{dt}}^2+\frac{1}{f\left(r\right)}{\mathit{dr}}^2+\sum _{j=2}^d{R}_j^{2}d{\mathrm{\Omega }}_j^2,$$E6

where $f\left(r\right)$ is a function of the coordinate $r$ and $d{\mathrm{\Omega }}_j^2$ is the line element of the $j$th unit sphere $S^2$ as follows

$$f\left(r\right)=1-\frac{r^2}{R_1^2},d{\mathrm{\Omega }}_j^2=d{\theta }_j^2+{sin}^2{\theta }_{j}d{\varphi }_j^2.$$E7

The radii $R_1$ and $R_j$ are given by

$$R_1={\left[\mathrm{\Lambda }-\frac{1}{2}{Q}_1^2+\frac{Q}{2\left(D-2\right)}\right]}^{-1/2},R_j={\left[\mathrm{\Lambda }+\frac{1}{2}{Q}_j^2+\frac{Q}{2\left(D-2\right)}\right]}^{-1/2},$$E8

where $Q_1$ is an electric charge and $Q_j$ are magnetic charges, while $Q$ is defined by

$$Q=Q_1^2-\sum _{j=2}^d{Q}_j^2.$$E9

This spacetime is a locally static solution of Einstein’s equation with a cosmological constant $\mathrm{\Lambda }$ and electromagnetic field $ℱ=d\mathcal{A}$ whose gauge field $\mathcal{A}$ is given by

$$\mathcal{A}=Q_{1}r\mathit{dt}+\sum _{j=2}^d{Q}_j{R}_j^{2}cos{\theta }_{j}d{\varphi }_j.$$E10

The coordinates in the metric are also called static, because they do not depend explicitly on the time coordinate t. One may notice that, in this coordinate system, this background has a local Killing vector ${\partial }_t$ whose norm vanishes at $r=\pm R_1$. Indeed, $r=\pm R_1$ define closed null surfaces that surround the observer at all times, known as event horizons. The boundary conditions defining QNMs in our spacetime will be posed at these surfaces, as discussed in [39]. For this reason, the dependence of all the components of the field $\mathbf{\Psi }$ on the coordinates along the Killing vector ${\partial }_t$ is assumed to be of the form $e^{-\mathit{i\omega t}}$. Usually, the articles consider that the coordinate r in de Sitter space assume values in the interval $r\in \left(0,R_1\right)$ [40, 41, 42]. However, this is just justified for de Sitter with $D>2$, but not for $D=2$; see [39] for more details. By this reason, our domain of interest will be $r\in \left(-R_1,R_1\right)$. In such domain, it is useful to introduce the tortoise coordinate x defined by the equation

$$\mathit{dx}=\frac{1}{f\left(r\right)}\mathit{dr}\Rightarrow x=R_1\mathrm{arctanh}\left(\frac{r}{R_1}\right),$$E11

in terms of which the line element Eq. (6) becomes

$${\mathit{ds}}^2=\frac{1}{{cosh}^2\left(x/R_1\right)}\left(-{\mathit{dt}}^2+{\mathit{dx}}^2\right)+\sum _{j=2}^d{R}_j^{2}d{\mathrm{\Omega }}_j^2,$$E12

and the gauge field can be rewritten as

$$\mathcal{A}=Q_1{R}_{1}tanh\left(x/R_1\right)\mathit{dt}+\sum _{j=2}^d{Q}_j{R}_j^{2}cos{\theta }_{j}d{\varphi }_j.$$E13

In particular, note that the tortoise coordinate maps the domain between two horizons, $r\in \left(-R_1,R_1\right)$, into the interval $x\in \left(-\infty ,\infty \right)$.

The QNMs accounting for an important class of fields are associated to $\mathbf{\Psi }$ which are solutions to the equations of motion that satisfy specific boundary conditions imposed at the horizons of the spacetime in which the field is propagating; see [5, 6, 43, 44] for more details. In this chapter, we will use the boundary conditions as illustrated in Figure 1.

Figure 1.

Illustration of the boundary condition associated to QNMs in our spacetime. The wavy arrows represent the direction of the perturbation field at the boundaries $r=\pm R_1$, while the cones are the local light cones. Mathematically, the wavy arrow pointing to the right represents $e^{-\mathit{i\omega }\left(t-x\right)}$, while the wavy arrow pointing to the left represents $e^{-\mathit{i\omega }\left(t+x\right)}$. For more details, see Ref. [39].

From the mathematical of view, since we are assuming that the time dependence of $\mathbf{\Psi }$ is $e^{-\mathit{i\omega t}}$, this boundary condition means that near the horizons $r=\pm R_1$, that is, as $x\to \pm \infty$, the radial component of the field $\mathbf{\Psi }$ should behave as $e^{-\mathit{i\omega }\left(t+x\right)}$ at $x\to \infty$, while it should go as $e^{-\mathit{i\omega }\left(t-x\right)}$ at $x\to -\infty$. The eigenfrequencies of this problem are complex, the reason why they are called QNFs. The real part of the QNFs is associated with the oscillation frequencies of the signal, while the imaginary part is related to its decay in time. This decay in time is closely related to the fact that the event horizon has a dissipative nature.

One interesting feature of this spacetime is that we can compute exactly the QNMs. The exactly solvable systems are usually limits of more realistic systems and allow us to study in detail some properties of a physical process and test some methods which can be used to analyze more complicated systems. Thus they are powerful tools in many research lines. Therefore we expect that the exactly computed QNFs for D-dimensional generalized Nariai spacetime may play an important role in future research [27].

3. Dirac equation in D-dimensional generalized Nariai spacetime

Let us present the construction of a solution to the Dirac equation minimally coupled to the electromagnetic field of D-dimensional generalized Nariai spacetime. A field of spin $1/2$ with electric charge q and mass m propagating in such spacetime is a spinorial field obeying the following version of the Dirac equation:

$${\mathrm{\Gamma }}^{\alpha }\left({\nabla }_{\alpha }-\mathit{iq}{\mathcal{A}}_{\alpha }\right)\mathbf{\Psi }=m\mathbf{\Psi },$$E14

where ${\mathcal{A}}_{\alpha }$ stands for the components of the background gauge field. In $D=2d$ dimensions, the Dirac matrices ${\mathrm{\Gamma }}^{\alpha }$ represent faithfully the Clifford algebra by $2^d\times 2^d$ matrices obeying the relation

$${\mathrm{\Gamma }}_{\alpha }{\mathrm{\Gamma }}_{\beta }+{\mathrm{\Gamma }}_{\beta }{\mathrm{\Gamma }}_{\alpha }=\mathit{g}\left({\mathit{e}}_{\alpha },{\mathit{e}}_{\beta }\right){\mathbb{I}}_d,$$E15

with ${\mathbb{I}}_d$ standing for the $2^d\times 2^d$ identity matrix. The index $\alpha ,\beta ,\gamma$ run from 1 to 2d and label the vector fields of an orthonormal frame $\left\{{\mathit{e}}_{\alpha }\right\}$. In order to solve the Dirac equation, we must introduce a suitable orthonormal frame of vector fields, which in the case of our background is given by

$$\begin{array}{c}{\mathit{e}}_1=-icosh\left(x/R_1\right){\partial }_t,{\mathit{e}}_j=\frac{1}{R_{j}sin{\theta }_j}{\partial }_{{\varphi }_j},\\ {\mathit{e}}_{\tilde{1}}=cosh\left(x/R_1\right){\partial }_x,{\mathit{e}}_{\tilde{j}}=\frac{1}{R_j}{\partial }_{{\theta }_j},\end{array}$$E16

where the index j ranges from 2 to d. In particular, note that

$$\mathit{g}\left({\mathit{e}}_{\alpha },{\mathit{e}}_{\beta }\right)={\delta }_{\mathit{\alpha \beta }}\leftrightarrow \left\{\begin{array}{cc}\mathit{g}\left({\mathit{e}}_a,{\mathit{e}}_b\right)& ={\delta }_{\mathit{ab}},\\ \mathit{g}\left({\mathit{e}}_a,{\mathit{e}}_{\tilde{b}}\right)& =0,\\ \mathit{g}\left({\mathit{e}}_{\tilde{a}},{\mathit{e}}_{\tilde{b}}\right)& ={\delta }_{\tilde{a}\tilde{b}},\end{array}\right.$$E17

where a and $\tilde{a}$ are indices that range from 1 to d. The index a labels the first d vector fields of the orthonormal frame $\left\{{\mathit{e}}_a\right\}$, while the index $\tilde{a}$ labels the remaining d vectors of the frame $\left\{{\mathit{e}}_a\right\}$. The derivatives of the frame vector fields determine the spin connection according to the following relation:

$${\nabla }_{\alpha }{\mathit{e}}_{\beta }={\omega }_{\mathit{\alpha \beta }}^{\gamma }{\mathit{e}}_{\gamma }.$$E18

Since the metric g is a covariantly constant tensor, it follows that the coefficients of the spin connection with all low indices ${\omega }_{\mathit{\alpha \beta \gamma }}={\omega }_{\mathit{\alpha \beta }}^{\epsilon }{\delta }_{\mathit{\epsilon \gamma }}$ are antisymmetric in their two last indices, ${\omega }_{\mathit{\alpha \beta \gamma }}=-{\omega }_{\mathit{\alpha \gamma \beta }}$. Note that the indices of the spin connection are raised and lowered with ${\delta }_{\mathit{\alpha \beta }}$ and ${\delta }^{\mathit{\alpha \beta }}$, respectively, so that frame indices can be raised and lowered unpunished. In particular, ${\omega }_{\alpha }^{\mathit{\beta \gamma }}={\omega }_{\alpha }^{\left[\mathit{\beta \gamma }\right]}$, where indices inside the square brackets are antisymmetrized. The covariant derivative of a spinorial field $\mathbf{\Psi }$ is, then, given by

$${\nabla }_{\alpha }\mathbf{\Psi }={\partial }_{\alpha }\mathbf{\Psi }-\frac{1}{4}{\omega }_{\alpha }^{\mathit{\beta \gamma }}{\mathrm{\Gamma }}_{\beta }{\mathrm{\Gamma }}_{\gamma }\mathbf{\Psi },$$E19

with ${\partial }_{\alpha }$ denoting the partial derivative along the vector field ${\mathit{e}}_{\alpha }$.

Our aim is to separate the Dirac Eq. (14). In order to accomplish this, it is necessary to use a suitable representation for the Dirac matrices. We recall that

$${\sigma }_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\sigma }_3=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],$$E20

are the Hermitian Pauli matrices and $\mathbb{I}$ denote the $2\times 2$ identity matrix. Using this notation, a convenient representation of the Dirac matrices is the following:

$$\begin{array}{l}{\mathrm{\Gamma }}_a=\underset{\left(a-1\right)\text{times}}{\underbrace{{\sigma }_3\otimes \dots \otimes }}{\sigma }_1\underset{\left(d-a\right)\text{times}}{\underbrace{\otimes \mathbb{I}\otimes \dots \otimes \mathbb{I}}},\\ {\mathrm{\Gamma }}_{\tilde{a}}=\underset{\left(a-1\right)\text{times}}{\underbrace{{\sigma }_3\otimes \dots \otimes }}{\sigma }_2\underset{\left(d-a\right)\text{times}}{\underbrace{\otimes \mathbb{I}\otimes \dots \otimes \mathbb{I}}},\end{array}$$E21

where $\mathbb{I}$ stands for the $2\times 2$ identity matrix. Indeed, we can easily check that the Clifford algebra given in Eq. (15) is properly satisfied by the above matrices.² In this case, spinorial fields are represented by the column vectors on which these matrices act. We can introduce a basis of this representation by the direct products of spinors ${\mathit{\xi }}^s$ given by

$${\mathit{\xi }}^{+}=\left[\begin{array}{cc}1& \\ 0& \end{array}\right],{\mathit{\xi }}^{-}=\left[\begin{array}{cc}0& \\ 1& \end{array}\right],$$E22

which, under the action of the Pauli matrices, satisfy concisely the relations

$${\sigma }_1{\mathit{\xi }}^s={\mathit{\xi }}^{-s},{\sigma }_2{\mathit{\xi }}^s=is{\mathit{\xi }}^{-s},{\sigma }_3{\mathit{\xi }}^s=s{\mathit{\xi }}^s.$$E23

Indeed, in $D=2d$ dimensions, a general spinor field has 2^d degrees of freedom and can be written as

$$\mathbf{\Psi }=\sum _{\left\{s\right\}}{\mathbf{\Psi }}^{s_1{s}_2\dots s_d}{\mathit{\xi }}^{s_1}\otimes {\mathit{\xi }}^{s_2}\otimes \dots \otimes {\mathit{\xi }}^{s_d},$$E24

where each of the indices $s_a$ can take the values “+1” and “−1.” Since every $s_a$ can take just two values, it follows that the sum over $\left\{s\right\}\equiv \left\{s_1,s_2,\dots ,s_d\right\}$ comprises 2^d terms, which is exactly the number of components of a spinorial field in $D=2d$ dimensions.

In the representation (Eq. (21)), the operator ${\mathrm{\Gamma }}^{\alpha }{\nabla }_{\alpha }$, called Dirac operator, is then represented by

$${\mathrm{\Gamma }}^{\alpha }{\nabla }_{\alpha }=\sum _{a=1}^d\left({\mathrm{\Gamma }}_a{\nabla }_a+{\mathrm{\Gamma }}_{\tilde{a}}{\nabla }_{\tilde{a}}\right)=\sum _{a=1}^d\underset{\left(a-1\right)\text{times}}{\underbrace{{\sigma }_3\otimes \dots \otimes }}{D}_a\underset{\left(d-a\right)\text{times}}{\underbrace{\otimes \mathbb{I}\otimes \dots \otimes \mathbb{I}}},$$E25

where

$$D_a={\sigma }_1{\nabla }_a+{\sigma }_2{\nabla }_{\tilde{a}},$$E26

is the Dirac operator on ${\mathrm{ℝ}}^2$ with coordinates $\left\{x^a,y^a\right\}$. The spinorial basis introduced previously is very convenient, since the action of the Dirac matrices on the spinor fields can be easily computed. Indeed, using Eqs. (21), (23), and (24), we eventually arrive at the following equation

$$\begin{array}{l}{\mathrm{\Gamma }}_a\mathbf{\Psi }=\sum _{\left\{s\right\}}\left(s_1{s}_2\dots s_{a-1}\right){\mathbf{\Psi }}^{s_1{s}_2\dots s_d}{\mathit{\xi }}^{s_1}\otimes {\mathit{\xi }}^{s_2}\otimes \dots \otimes {\mathit{\xi }}^{s_{a-1}}\otimes {\mathit{\xi }}^{-s_a}\otimes {\mathit{\xi }}^{s_{a+1}}\otimes \\ \dots \otimes {\mathit{\xi }}^{s_d}=\sum _{\left\{s\right\}}\left(s_1{s}_2\dots s_a\right){\mathbf{\Psi }}^{s_1{s}_2\dots s_{a-1}\left(-s_a\right)s_{a+1}\dots s_d}{\mathit{\xi }}^{s_1}\otimes {\mathit{\xi }}^{s_2}\otimes \\ \dots \otimes {\mathit{\xi }}^{s_{a-1}}\otimes {\mathit{\xi }}^{s_a}\otimes {\mathit{\xi }}^{s_{a+1}}\otimes \dots \otimes {\mathit{\xi }}^{s_d},\end{array}$$E27

where from the first to the second line we have changed the index $s_a$ to $-s_a$, which does not change the final result, since we are summing over all values of $s_a$, which comprise the same list of the values of $-s_a$. Moreover, we have used ${\left(s_a\right)}^2=1$. Analogously, we have:

$$\begin{array}{l}{\mathrm{\Gamma }}_{\tilde{a}}\mathbf{\Psi }=\sum _{\left\{s\right\}}\left(s_1{s}_2\dots s_{a-1}\right)\left({\mathit{\text{is}}}_a\right){\mathbf{\Psi }}^{s_1{s}_2\dots s_d}{\mathit{\xi }}^{s_1}\otimes {\mathit{\xi }}^{s_2}\otimes \dots \otimes {\mathit{\xi }}^{s_{a-1}}\otimes {\mathit{\xi }}^{-s_a}\otimes {\mathit{\xi }}^{s_{a+1}}\otimes \\ \dots \otimes {\mathit{\xi }}^{s_d}=-i\sum _{\left\{s\right\}}\left(s_1{s}_2\dots s_a\right)s_a{\mathbf{\Psi }}^{s_1{s}_2\dots s_{a-1}\left(-s_a\right)s_{a+1}\dots s_d}{\mathit{\xi }}^{s_1}\otimes {\mathit{\xi }}^{s_2}\otimes \\ \dots \otimes {\mathit{\xi }}^{s_{a-1}}\otimes {\mathit{\xi }}^{s_a}\otimes {\mathit{\xi }}^{s_{a+1}}\otimes \dots \otimes {\mathit{\xi }}^{s_d}.\end{array}$$E28

All that was seen above are necessary tools to attack our initial problem of separating the general Eq. (14). In order to solve such an equation, we need to separate the degrees of freedom of the field, which can be quite challenging in general. Fortunately, the spacetime considered here is the direct product of two-dimensional spaces of constant curvature, which is exactly the class of spaces studied in Ref. [39]. Indeed, in this latter paper, it is shown that the Dirac equation minimally coupled to an electromagnetic field is separable in such backgrounds. In particular, assuming that the components of the spinor field Eq. (24) can be decomposed in the form

$${\mathrm{\Psi }}^{s_1{s}_2\dots s_d}={\mathrm{\Psi }}_1^{s_1}\left(t,x\right){\mathrm{\Psi }}_2^{s_2}\left({\mathrm{\Phi }}_2,{\theta }_2\right)\dots {\mathrm{\Psi }}_d^{s_d}\left({\mathrm{\Phi }}_d,{\theta }_d\right),$$E29

where each index $s_a$ can take the values $s_a=\pm 1$, the fields ${\mathrm{\Psi }}_1^{s_1}\left(t,x\right)$ satisfy the following differential equation (the reader is invited to demonstrate the equation below or consult more details in [39]):

$$\left[{\partial }_{\tilde{1}}+\frac{{\omega }_{1\tilde{1}1}}{2}-\mathit{iq}{\mathcal{A}}_{\tilde{1}}-{\mathit{\text{is}}}_1\left({\partial }_1+\frac{{\omega }_{\tilde{1}1\tilde{1}}}{2}-\mathit{iq}{\mathcal{A}}_1\right)\right]{\mathrm{\Psi }}_1^{s_1}=\left(L-i{s}_{1}m\right){\mathrm{\Psi }}_1^{-s_1}.$$E30

The separation constant L in the above equation depends on the angular modes. In particular, in the special case of vanishing magnetic charges $Q_j$, it is determined by the eigenvalues ${\lambda }_j$ of the Dirac operator on unit sphere $S^2$ according to the following relation

$$L=\sqrt{{\lambda }_2^2+{\lambda }_3^2+\dots +{\lambda }_d^2},{\lambda }_j=\pm 1,\pm 2,\pm 3,\dots ,$$E31

as demonstrated in Appendix A of Ref. [39]. In our frame of vectors, the only components of the spin connection that are potentially nonvanishing are

$$\left\{\begin{array}{c}{\omega }_{1\tilde{1}1}=-{\omega }_{11\tilde{1}}=-\frac{1}{R_1}sinh\left(x/R_1\right),\\ {\omega }_{j\tilde{j}j}=-{\omega }_{\mathit{jj}\tilde{j}}=\frac{1}{R_j}cot{\theta }_j,\end{array}\right.$$E32

and the nonzero components of the gauge field can be written as

$${\mathcal{A}}_1=-i{Q}_1{R}_{1}sinh\left(x/R_1\right),{\mathcal{A}}_j=Q_j{R}_{j}cot{\theta }_j.$$E33

Now, since the components of the metric are independents of the coordinate t, the vector ${\partial }_t$ is a Killing vector for this metric. So, it is useful to assume the following time dependence for the field ${\mathrm{\Psi }}_1^{s_1}\left(t,x\right)$

$${\mathrm{\Psi }}_1^{s_1}\left(t,x\right)=e^{-\mathit{i\omega t}}{\psi }^{s_1}\left(x\right).$$E34

Inserting this field along with the gauge field Eq. (33), and taking into account the first relation of the Eq. (32) into the Eq. (30), we end up with the following coupled system of differential equations:

$$\left[\frac{d}{\mathit{dx}}+{\mathit{\text{is}}}_1\omega +\left(isq{Q}_1{R}_1-\frac{1}{2R_1}\right)tanh\left(x/R_1\right)\right]{\psi }^{s_1}=\frac{\left(L-{\mathit{\text{is}}}_{1}m\right)}{cosh\left(x/R_1\right)}{\psi }^{-s_1}.$$E35

In order to solve these equations, we should first decouple the fields ${\psi }^{s_1}$ and ${\psi }^{-s_1}$. Eliminating ${\psi }^{-s_1}$ we obtain a second-order equation for ${\psi }^{s_1}$. Indeed, we can prove that the fields ${\psi }^{s_1}$ satisfy the following second-order ordinary differential equation

$$\left[\frac{d^2}{{\mathit{dx}}^2}+{\omega }^2-V\left(x\right)\right]{\psi }^{s_1}=0,$$E36

which is a Schrödinger-like equation with V being a potential of the form

$$V\left(x\right)=A+Btanh\left(x/R_1\right)+\frac{C}{{cosh}^2\left(x/R_1\right)},$$E37

where the parameters A, B, and C are given by

$$\left\{\begin{array}{l}A=\frac{1}{4R_1^2}-q{Q}_1\left({\mathit{\text{is}}}_1+q{Q}_1{R}_1^2\right),\\ B=-\frac{\omega }{R_1}\left({\mathit{\text{is}}}_1+2q{Q}_1{R}_1^2\right),\\ C=m^2+L^2+\frac{1}{4R_1^2}+q^2{Q}_1^2{R}_1^2.\end{array}\right.$$E38

These are known as potentials of Rosen-Morse type, which are generalizations of the Pöschl-Teller potential [37, 38]. It is straightforward to see that this potential satisfies the following properties:

$$V\to \left\{\begin{array}{c}A+B\mathrm{at}x\to +\infty ,\\ A-B\mathrm{at}x\to -\infty .\\ \end{array}\right.$$E39

In many cases, the potential function V is regular at $r=0\left(x=0\right)$, in particular V can be equal to a constant different from zero. In fact, in our case, we find that

$$V\to A+C\mathrm{at}x\to 0,$$E40

which clearly is regular. So, we point out that for this potential both limits (Eqs. (39) and (40)) are finite, and thus there is no reason to demand for a regular solution in this point.

Thus, the problem of finding the QNMs is reduced to the searching of the corresponding spectrum of QNFs ω of Eq. (36). Most of the problems concerning the QNMs fall into Schrödinger-like equation with real potentials which vanish at both horizons [5], highlighting the fact that the solutions can be taken to be plane waves. However, clearly this is not the case. Although it is possible to make field redefinitions in order to make the potential real, we shall not do this here. For such procedure we refer the reader to [36]. Once an analytical form for the QNFs of Rosen-Morse type potential is not known, we must find an analytical exact solution of Eq. (36) and impose physically appropriate boundary conditions at the horizons, $x\to \pm \infty$, which define the QNFs in a unique way.

In order to solve Eq. (36), let us make the following change of variable

$$y=\frac{1}{2}+\frac{1}{2}tanh\left(x/R_1\right).$$E41

In particular, notice that y is defined on the domain $y\in \left(0,1\right)$ with the boundaries $x\to \pm \infty$ being given by $y=0$ and $y=1$. In addition to this change of independent variable, if we now set the Ansatz

$${\psi }^{s_1}\left(x\right)=y^{\alpha }{\left(1-y\right)}^{\beta }{H}^{s_1}\left(y\right),$$E42

with the parameters α and β being constants conveniently chosen as

$$\alpha =\frac{R_1}{2}\sqrt{A-B-{\omega }^2},\beta =-\frac{R_1}{2}\sqrt{A+B-{\omega }^2},$$E43

the functions $H^{s_1}$ must be solutions of the following differential equation

$$y\left(1-y\right)\frac{d^2{H}^{s_1}}{{\mathit{dy}}^2}+\left[2\alpha +1-\left(2+2\alpha +2\beta \right)y\right]\frac{{\mathit{dH}}^{s_1}}{\mathit{dy}}-\left[{\mathit{CR}}_1^2+\left(\alpha +\beta \right)\left(1+\alpha +\beta \right)\right]H^{s_1}.$$E44

This new variable as well as the Ansatz that we have been using are really interesting because in terms of these, it is immediate to see that the functions $H^{s_1}$ satisfy a hypergeometric equation. Indeed, comparing with the standard hypergeometric differential equation

$$y\left(1-y\right)\frac{d^2{H}^{s_1}}{{\mathit{dy}}^2}+\left[c-\left(1+a+b\right)y\right]\frac{{\mathit{dH}}^{s_1}}{\mathit{dy}}-\mathit{ab}{H}^{s_1}=0,$$E45

we find that the constants a, b, and c are given by

$$\left\{\begin{array}{cc}a& =\frac{1}{2}+\alpha +\beta +\sqrt{\frac{1}{4}-{\mathit{CR}}_1^2},\\ b& =\frac{1}{2}+\alpha +\beta -\sqrt{\frac{1}{4}-{\mathit{CR}}_1^2},\\ c& =2\alpha +1.\end{array}\right.$$E46

Such an equation admits two linearly independent solutions whose linear combination furnishes the following general solution:

$$H^{s_1}\left(y\right)=D_2{F}_1\left(a,b,c;y\right)+E{y^{\left(1-c\right)}}_2{F}_1\left(1+a+c,1+b+c,2-c;y\right),$$E47

where ${}_2{F}_1$ is the hypergeometric function and D and E are arbitrary integration constants. Given the hypergeometric solution for $H^{s_1}$ is known, one can immediately find the general solution for ${\psi }^{s_1}$. Indeed, from Eqs. (42), (46), and (47), we conclude that the solution of Eq. (36), which is regular at the origin, can be written as

$$\begin{array}{c}{\psi }^{s_1}={\left(1-y\right)}^{\frac{1}{2}\left(a+b-c\right)}[D{y^{\frac{1}{2}\left(c-1\right)}}_2{F}_1\left(a,b,c;y\right)\\ +E{y^{-\frac{1}{2}\left(c-1\right)}}_2{F}_1\left(1+a-c,1+b-c,2-c;y\right)].\end{array}$$E48

In order to fix the integration constants D and E, we need to apply the appropriate boundary conditions. Inverting the Eq. (41) we find that, near the boundaries $x\to \pm \infty$, the relation between the coordinates x and y assumes the simpler form

$$\begin{array}{l}y\simeq e^{+2x/R_1}\mathrm{at}x\to -\infty ,\\ 1-y\simeq e^{-2x/R_1}\mathrm{at}x\to +\infty .\end{array}$$E49

Thus, taking into account the latter relation and using the fact that at $y=0\left(x\to -\infty \right)$ the hypergeometric function ${}_2{F}_1\left(a,b,c;0\right)=1$, one eventually obtains that near the boundary $x\to -\infty$ the field ${\psi }^{s_1}$ behaves as

$${\left.{\psi }^{s_1}\right|}_{x\to -\infty }\simeq D{e}^{\left(c-1\right)x/R_1}+E{e}^{-\left(c-1\right)x/R_1}.$$E50

On the other hand, in order to apply the boundary conditions at $y=1\left(x\to \infty \right)$, it is useful to write the hypergeometric functions as functions of $\left(1-y\right)$, so that they become united at the boundary. This can be done by rewriting the hypergeometric functions appearing in Eq. (48) by means of the following identity [45]:

$$\begin{array}{cc}{2}_{}F\left(a,b,c;y\right)& ={\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}}_{2}F\left(a,b,a+b-c+1;1-y\right)\\ & +\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{{\left(1-y\right)}^{\left(c-a-b\right)}}_{2}F\left(c-a,c-b,c-a-b+1;1-y\right),\end{array}$$E51

where $\mathrm{\Gamma }$ stands for the gamma function. Doing so, and using Eq. (49), we eventually arrive at the following behavior of the solution at $x\to +\infty$:

$$\begin{array}{c}{\left.{\psi }^{s_1}\right|}_{x\to +\infty }\simeq \left[D\frac{\mathrm{\Gamma }\left(c-a-b\right)\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}+E\frac{\mathrm{\Gamma }\left(c-a-b\right)\mathrm{\Gamma }\left(2-c\right)}{\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(1-b\right)}\right]e^{-\left(a+b-c\right)x/R_1}\\ +\left[D\frac{\mathrm{\Gamma }\left(a+b-c\right)\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}+E\frac{\mathrm{\Gamma }\left(a+b-c\right)\mathrm{\Gamma }\left(2-c\right)}{\mathrm{\Gamma }\left(a-c+1\right)\mathrm{\Gamma }\left(b-c+1\right)}\right]e^{\left(a+b-c\right)x/R_1}.\end{array}$$E52

Now, from parameters Eqs. (38) and (43), we find that the constants appearing in the hypergeometric equation can be written as

$$\begin{array}{l}a={\mathit{iR}}_1\sqrt{{\mu }^2+q^2{Q}_1^2{R}_1^2+L^2}+\left(1+s_1\right)\left(\frac{1}{4}-\mathit{i\omega }\frac{R_1}{2}\right)-i\left(1-s_1\right)\frac{{\mathit{qQ}}_1{R}_1^2}{2},\\ b=-{\mathit{iR}}_1\sqrt{{\mu }^2+q^2{Q}_1^2{R}_1^2+L^2}+\left(1+s_1\right)\left(\frac{1}{4}-\mathit{i\omega }\frac{R_1}{2}\right)-i\left(1-s_1\right)\frac{{\mathit{qQ}}_1{R}_1^2}{2},\\ c=\frac{1}{2}+{\mathit{\text{is}}}_1\left({\mathit{qQ}}_1{R}_1^2-\omega R_1\right).\end{array}$$E53

In particular, the following relations hold

$$\left(c-1\right)/R_1=-{\mathit{\text{is}}}_1\omega +{\mathit{\text{is}}}_1{\mathit{qQ}}_1{R}_1-\frac{1}{2R_1},$$E54

$$\left(a+b-c\right)/R_1=-\mathit{i\omega }-{\mathit{iqQ}}_1{R}_1+s_1\frac{1}{2R_1}.$$E55

Now we are ready to impose the boundary conditions. Obviously, without loss of generality, we can consider that the spin $s_1$ is already chosen and fixed at $s_1=+$ or $s_1=-$ since the QNFs should not depend on the choice of $s_1=\pm$. Let us impose, for instance, the boundary conditions for the component $s_1=+$ of the spinorial field. In this case, using the identity Eq. (54) along with the Eq. (34), we eventually arrive at the following behavior of the solution at $x\to \infty$:

$${\left.{\mathrm{\Psi }}_1^{+}\left(t,x\right)\right|}_{x\to -\infty }=D{e}^{-\mathit{i\omega }\left(t+x\right)}{e}^{\left({\mathit{iqQ}}_1{R}_1-\frac{1}{2R_1}\right)x}+E{e}^{-\mathit{i\omega }\left(t-x\right)}{e}^{-\left({\mathit{iqQ}}_1{R}_1-\frac{1}{2R_1}\right)x}.$$E56

Now, Figure 1 tells us that the field is assumed to move toward higher values of x at the boundary $x\to -\infty$, while at the boundary $x\to -\infty$ it should move toward lower values of x. Then, since the time dependence of the field ${\mathrm{\Psi }}_1^{+}$ is of the type $e^{-\mathit{i\omega t}}$, this means that ${\mathrm{\Psi }}_1^{+}$ should behave as $e^{-\mathit{i\omega }\left(t-x\right)}$ at $x\to -\infty$, while it should go as $e^{-\mathit{i\omega }\left(t+x\right)}$ at $x\to +\infty$. Thus, from Eq. (55), we conclude that we must set $D=0$. In such a case, from Eq. (52), the field ${\mathrm{\Psi }}_1^{+}$ becomes

$$\begin{array}{cc}{\left.{\mathrm{\Psi }}_1^{+}\right|}_{x\to +\infty }& \simeq E\left[\frac{\mathrm{\Gamma }\left(c-a-b\right)\mathrm{\Gamma }\left(2-c\right)}{\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(1-b\right)}\right]e^{-\mathit{i\omega }\left(t-x\right)}{e}^{\left({\mathit{iqQ}}_1{R}_1-\frac{1}{2R_1}\right)x}\\ & +E\left[\frac{\mathrm{\Gamma }\left(a+b-c\right)\mathrm{\Gamma }\left(2-c\right)}{\mathrm{\Gamma }\left(a-c+1\right)\mathrm{\Gamma }\left(b-c+1\right)}\right]e^{-\mathit{i\omega }\left(t+x\right)}{e}^{\left(-{\mathit{iqQ}}_1{R}_1+\frac{1}{2R_1}\right)x}.\end{array}$$E57

Finally, to satisfy the QNM boundary condition near the boundary at $x\to \infty$, we must eliminate the term $e^{-\mathit{i\omega }\left(t-x\right)}$ of the above equation. Since E cannot be zero (as otherwise the field would vanish identically), we need the combination of the gamma functions to be zero. Now, once the gamma function has no zeros, the way to achieve this is to let the gamma functions in the denominator diverge, $\mathrm{\Gamma }\left(1-a\right)=\infty$ or $\mathrm{\Gamma }\left(1-b\right)=\infty$. Since the gamma functions diverge only at nonpositive integers, we are led to the following constraint:

$$1-a=-n\text{or}1-b=-n,\text{where}n\in \left\{\mathrm{0,1,2},\dots \right\}.$$E58

Using the Eq. (53), we find that these constraints translate to

$$\omega =\pm \sqrt{m^2+q^2{Q}_1^2{R}_1^2+L^2}+\frac{i}{R_1}\left(n+\frac{1}{2}\right),$$E59

which are the QNFs of the Dirac field propagating in D-dimensional generalized Nariai spacetimes. The real part of a QNF is associated with the oscillation frequency, while the imaginary part is related to its decay rate. At this point, it is worth recalling that L is a separation constant of the Dirac equation that is related to the angular mode of the field.

Likewise, imposing the boundary condition to the component $s_1=-$ of the spinorial field, we find that we must set $E=0$ at Eq. (50) and then $c-a=-n$ or $c-b=-n$, with n being a nonnegative integer. This, in its turn, leads to the same spectrum obtained for the component $s_1=+$ as expected, namely, Eq. (59).

4. Conclusions

In this chapter we have investigated the perturbations on a spinorial field propagating in a generalized version of the charged Nariai spacetime. Besides the separability of the degrees of freedom of these perturbations, one interesting feature of this background is that the perturbations can be analytically integrated. They all obey a Schrödinger-like equation with an integrable potential that is contained in the Rosen-Morse class of integrable potentials. Such an equation admits two linearly independent solutions given in terms of standard hypergeometric functions. This is a valuable property, since even the perturbation potential associated to the humble Schwarzschild background is nonintegrable, despite the fact that it is separable. We have also investigated the QNMs associated to this spinorial field. Analyzing the Eq. (59), namely,

$${\omega }_D=\pm \sqrt{m^2+q^2{Q}_1^2{R}_1^2+L^2}+\frac{i}{R_1}\left(n+\frac{1}{2}\right),$$E60

it is interesting to note that the imaginary parts of the QNFs, which represent the decay rates, do not depend on any details of the perturbation; rather, they only depend on the charges of the gravitational background through the dependence on R₁. On the other hand, the real parts of the QNFs depend on the mass of the field and on the angular mode of the perturbations. Another fact worth pointing out is that the fermionic field always has a real part in its QNFs spectrum, meaning that it always oscillates. This is not reasonable. Indeed, for Klein-Gordon and Maxwell perturbations in the D-dimensional Nariai spacetime, their QNFs are equal to [39].

$$\begin{array}{l}{\omega }_{\mathit{KG}}=\pm \sqrt{m^2+{\sum }_{j=2}^d\frac{{\mathrm{\ell }}_j\left({\mathrm{\ell }}_j+1\right)}{R_j^2}-\frac{1}{4R_1^2}}-\frac{i}{R_1}\left(n+\frac{1}{2}\right),\\ {\omega }_M=\pm \sqrt{{\sum }_{j=2}^d\frac{{\mathrm{\ell }}_j\left({\mathrm{\ell }}_j+1\right)}{R_j^2}-\frac{1}{4R_1^2}}-\frac{i}{R_1}\left(n+\frac{1}{2}\right),\end{array}$$E61

where ${\mathrm{\ell }}_j$ and m_j are integers, $\mid m_j\mid \le {\mathrm{\ell }}_j$, and $\mathrm{\ell }\ge 0$. Due to the negative factor $-1/\left(4R_1^2\right)$ inside the square root appearing in the bosonic spectrum, it follows that for small enough R₁, along with small enough mass and angular momentum, the argument of the square root can be negative, so that this term becomes imaginary.

To finish, we believe that a good exercise is to calculate the QNFs of the gravitational field in D-dimensional generalized charged Nariai spacetime. Research on the latter problem is still ongoing and, due to the great number of degrees of freedom in the gravitational field, shall be considered in a future work. The next interesting step is the investigation of superradiance phenomena for the spin $1/2$ field. Although bosonic fields like scalar, electromagnetic, and gravitational fields can exhibit superradiant behavior in four-dimensional Kerr spacetime [46], curiously, this is not the case for the Dirac field [36]. Thus, it would be interesting to investigate whether an analogous thing happens in the background considered here [47].