## 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.

### Keywords

- quasinormal modes
- generalized Nariai spacetimes
- Dirac field
- boundary conditions

## 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, 42, 43, 44]. 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 [27]. In this chapter, we consider a higher-dimensional generalization of the charged Nariai spacetime [28], namely,

## 2. Presenting the problem

In *D* dimensions, the dynamics of general relativity in spacetimes with a cosmological constant ^{1}

where

where

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

Now, let the pair

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

In what follows, let us consider a specific matter field

where

The radii

where

This spacetime is a locally static solution of Einstein’s equation with a cosmological constant

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 *r* in de Sitter space assume values in the interval *x* defined by the equation

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

and the gauge field can be rewritten as

In particular, note that the tortoise coordinate maps the domain between two horizons,

The QNMs accounting for an important class of fields are associated to

From the mathematical of view, since we are assuming that the time dependence of

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 [42].

## 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 *q* and mass *m* propagating in such spacetime is a spinorial field obeying the following version of the Dirac equation:

where

with *d* and label the vector fields of an orthonormal frame

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

where *a* and *d*. The index *a* labels the first *d* vector fields of the orthonormal frame *d* vectors of the frame

Since the metric *g* is a covariantly constant tensor, it follows that the coefficients of the spin connection with all low indices

with

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

are the Hermitian Pauli matrices and

where ^{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

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

Indeed, in *d* degrees of freedom and can be written as

where each of the indices *d* terms, which is exactly the number of components of a spinorial field in

In the representation (Eq. (21)), the operator

where

is the Dirac operator on

where from the first to the second line we have changed the index

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. [36]. 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

where each index

The separation constant *L* in the above equation depends on the angular modes. In particular, in the special case of vanishing magnetic charges

as demonstrated in Appendix A of Ref. [36]. In our frame of vectors, the only components of the spin connection that are potentially nonvanishing are

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

Now, since the components of the metric are independents of the coordinate *t*, the vector

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:

In order to solve these equations, we should first decouple the fields

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

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

These are known as potentials of Rosen-Morse type, which are generalizations of the Pöschl-Teller potential [34, 35]. It is straightforward to see that this potential satisfies the following properties:

In many cases, the potential function *V* is regular at *V* can be equal to a constant different from zero. In fact, in our case, we find that

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 [33]. 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,

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

In particular, notice that *y* is defined on the domain

with the parameters *α* and *β* being constants conveniently chosen as

the functions

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

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

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

where *D* and *E* are arbitrary integration constants. Given the hypergeometric solution for

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* and *y* assumes the simpler form

Thus, taking into account the latter relation and using the fact that at

On the other hand, in order to apply the boundary conditions at

where

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

In particular, the following relations hold

Now we are ready to impose the boundary conditions. Obviously, without loss of generality, we can consider that the spin

Now, Figure 1 tells us that the field is assumed to move toward higher values of *x* at the boundary *x*. Then, since the time dependence of the field

Finally, to satisfy the QNM boundary condition near the boundary at *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,

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

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 *n* being a nonnegative integer. This, in its turn, leads to the same spectrum obtained for the component

## 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,

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 *R*_{1}. 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 [36].

where *mj* are integers, *R*_{1}, 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

## Notes

- The coefficient of Λ in S can be chosen of several manners. In particular, for any dimension D, in order to insure that the pure dS or pure AdS spacetimes are described by g tt = 1 − Λ / 3 r 2 , as occurs in the case D = 4 , this coefficient should be D − 1 D − 2 .
- In D = 2 d + 1 , besides the 2 d Dirac matrices Γ a and Γ a ˜ , we need to add one further matrix, which will be denoted by Γ d + 1 given by Γ d + 1 = σ 3 ⊗ σ 3 … ⊗ σ 3 ⏟ d times .