**Author details**

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

Likewise, imposing the boundary condition to the component *s*<sup>1</sup> ¼ � of the spinorial field, we find that we must set *E* ¼ 0 at Eq. (50) and then *c* � *a* ¼ �*n* or *c* � *b* ¼ �*n*, with *n* being a nonnegative integer. This, in its turn, leads to the same spectrum obtained for the component *s*<sup>1</sup> ¼ þ as expected, namely, Eq. (59).

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

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>m</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup>*Q*<sup>2</sup>

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 [39].

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>ℓ</sup>*<sup>j</sup>* <sup>ℓ</sup>*<sup>j</sup>* <sup>þ</sup> <sup>1</sup> � � *R*2 *j*

> � <sup>1</sup> 4*R*<sup>2</sup> 1

� � inside the square root appearing in the bosonic spectrum, it follows that for small enough *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 1*=*2

where ℓ*<sup>j</sup>* and *mj* are integers, ∣*mj*∣ ≤ ℓ*j*, and ℓ≥0. Due to the negative factor

1*R*2 <sup>1</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup>

it is interesting to note that the imaginary parts of the QNFs, which represent the

þ *i R*1

� <sup>1</sup> 4*R*<sup>2</sup> 1

� *i R*1 � *i R*1

*n* þ 1 2 � �

*n* þ 1 2 � �

*,*

*,*

(61)

*n* þ 1 2 � �

*,* (60)

angular mode of the field.

*Progress in Relativity*

**4. Conclusions**

Eq. (59), namely,

�1*<sup>=</sup>* <sup>4</sup>*R*<sup>2</sup> 1

**12**

*ω<sup>D</sup>* ¼ �

*ωKG* ¼ �

*ω<sup>M</sup>* ¼ �

q

*<sup>m</sup>*<sup>2</sup> <sup>þ</sup> <sup>P</sup>*<sup>d</sup>*

s

s

P*<sup>d</sup> j*¼2 *j*¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>ℓ</sup>*<sup>j</sup>* <sup>ℓ</sup>*<sup>j</sup>* <sup>þ</sup> <sup>1</sup> � � *R*2 *j*

Joás Venâncio<sup>1</sup> \* and Carlos Batista<sup>2</sup>


\*Address all correspondence to: joasvenancio@df.ufpe.br

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
