Open access peer-reviewed chapter

Quasinormal Modes of Dirac Field in Generalized Nariai Spacetimes

Written By

Joás Venâncio and Carlos Batista

Reviewed: August 14th, 2019 Published: November 11th, 2019

DOI: 10.5772/intechopen.89179

From the Edited Volume

Progress in Relativity

Edited by Calin Gheorghe Buzea, Maricel Agop and Leo Butler

Chapter metrics overview

603 Chapter Downloads

View Full Metrics


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.


  • 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, 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, dS2×S2××S2, 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 Λ is described by the Einstein-Hilbert action1


where R is the Ricci scalar and Sm stands for the action of all matter fields Φi 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 gμν and Φi which are given, respectively, by


where Tμν is the symmetric stress-energy tensor associated to Φi defined by the equation


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 gμν0 and Φi0 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 gμν0 and Φi0 and the small perturbations hμν and Ψi


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 hμν and Ψi. In general, these equations are coupled, namely, Ψi is a source for hμν and vice versa. However, in the special case in which Φi0=0, the equations governing the perturbed fields Ψi can be decoupled from the metric perturbation hμν and vice versa. The reason why this happen is that when Φi0=0, the stress-energy tensor Tμν can be set to zero at first order in the perturbation, since Tμν 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 Tμν=0 is equivalent to studying the test fields Ψi in the background gμν0.

In what follows, let us consider a specific matter field Ψ propagating in a generalized version of the Nariai spacetime described in Ref. [31]. Here, Ψ 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 dS2 with d1 copies of the unit spheres S2 possessing different radii Rj. Thus, the natural line element of the higher-dimensional version of the Nariai spacetime is given by


where fr is a function of the coordinate r and dΩj2 is the line element of the jth unit sphere S2 as follows


The radii R1 and Rj are given by


where Q1 is an electric charge and Qj are magnetic charges, while Q is defined by


This spacetime is a locally static solution of Einstein’s equation with a cosmological constant Λ and electromagnetic field =dA whose gauge field A is given by


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 t whose norm vanishes at r=±R1. Indeed, r=±R1 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 Ψ on the coordinates along the Killing vector t is assumed to be of the form eiωt. Usually, the articles consider that the coordinate r in de Sitter space assume values in the interval r0R1 [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 rR1R1. In such domain, it is useful to introduce the tortoise coordinate 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, rR1R1, into the interval x.

The QNMs accounting for an important class of fields are associated to Ψ 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=±R1, while the cones are the local light cones. Mathematically, the wavy arrow pointing to the right represents etx, while the wavy arrow pointing to the left represents et+x. For more details, see Ref. [39].

From the mathematical of view, since we are assuming that the time dependence of Ψ is eiωt, this boundary condition means that near the horizons r=±R1, that is, as x±, the radial component of the field Ψ should behave as et+x at x, while it should go as etx at x. 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:


where Aα stands for the components of the background gauge field. In D=2d dimensions, the Dirac matrices Γα represent faithfully the Clifford algebra by 2d×2d matrices obeying the relation


with Id standing for the 2d×2d identity matrix. The index α,β,γ run from 1 to 2d and label the vector fields of an orthonormal frame eα. 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


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


where a and a˜ are indices that range from 1 to d. The index a labels the first d vector fields of the orthonormal frame ea, while the index a˜ labels the remaining d vectors of the frame ea. The derivatives of the frame vector fields determine the spin connection according to the following relation:


Since the metric g is a covariantly constant tensor, it follows that the coefficients of the spin connection with all low indices ωαβγ=ωαβεδεγ are antisymmetric in their two last indices, ωαβγ=ωαγβ. Note that the indices of the spin connection are raised and lowered with δαβ and δαβ, respectively, so that frame indices can be raised and lowered unpunished. In particular, ωαβγ=ωαβγ, where indices inside the square brackets are antisymmetrized. The covariant derivative of a spinorial field Ψ is, then, given by


with α denoting the partial derivative along the vector field eα.

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 I denote the 2×2 identity matrix. Using this notation, a convenient representation of the Dirac matrices is the following:


where I stands for the 2×2 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 ξs given by


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


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


where each of the indices sa can take the values “+1” and “−1.” Since every sa can take just two values, it follows that the sum over ss1s2sd comprises 2d terms, which is exactly the number of components of a spinorial field in D=2d dimensions.

In the representation (Eq. (21)), the operator Γαα, called Dirac operator, is then represented by




is the Dirac operator on 2 with coordinates xaya. 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


where from the first to the second line we have changed the index sa to sa, which does not change the final result, since we are summing over all values of sa, which comprise the same list of the values of sa. Moreover, we have used sa2=1. Analogously, we have:


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


where each index sa can take the values sa=±1, the fields Ψ1s1tx satisfy the following differential equation (the reader is invited to demonstrate the equation below or consult more details in [39]):


The separation constant L in the above equation depends on the angular modes. In particular, in the special case of vanishing magnetic charges Qj, it is determined by the eigenvalues λj of the Dirac operator on unit sphere S2 according to the following relation


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


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 t is a Killing vector for this metric. So, it is useful to assume the following time dependence for the field Ψ1s1tx


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 ψs1 and ψs1. Eliminating ψs1 we obtain a second-order equation for ψs1. Indeed, we can prove that the fields ψs1 satisfy the following second-order ordinary differential equation


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 [37, 38]. It is straightforward to see that this potential satisfies the following properties:


In many cases, the potential function V is regular at r=0x=0, in particular 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 [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±, which define the QNFs in a unique way.

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


In particular, notice that y is defined on the domain y01 with the boundaries x± being given by y=0 and y=1. In addition to this change of independent variable, if we now set the Ansatz


with the parameters α and β being constants conveniently chosen as


the functions Hs1 must be solutions of the following differential equation


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 Hs1 satisfy a hypergeometric equation. Indeed, comparing with the standard hypergeometric differential equation


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 2F1 is the hypergeometric function and D and E are arbitrary integration constants. Given the hypergeometric solution for Hs1 is known, one can immediately find the general solution for ψs1. 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


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±, the relation between the coordinates x and y assumes the simpler form


Thus, taking into account the latter relation and using the fact that at y=0x the hypergeometric function 2F1abc0=1, one eventually obtains that near the boundary x the field ψs1 behaves as


On the other hand, in order to apply the boundary conditions at y=1x, it is useful to write the hypergeometric functions as functions of 1y, 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]:


where Γ stands for the gamma function. Doing so, and using Eq. (49), we eventually arrive at the following behavior of the solution at x+:


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 s1 is already chosen and fixed at s1=+ or s1= since the QNFs should not depend on the choice of s1=±. Let us impose, for instance, the boundary conditions for the component s1=+ 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:


Now, Figure 1 tells us that the field is assumed to move toward higher values of x at the boundary x, while at the boundary x it should move toward lower values of x. Then, since the time dependence of the field Ψ1+ is of the type eiωt, this means that Ψ1+ should behave as etx at x, while it should go as et+x at x+. Thus, from Eq. (55), we conclude that we must set D=0. In such a case, from Eq. (52), the field Ψ1+ becomes


Finally, to satisfy the QNM boundary condition near the boundary at x, we must eliminate the term etx 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, Γ1a= or Γ1b=. Since the gamma functions diverge only at nonpositive integers, we are led to the following constraint:


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 s1= of the spinorial field, we find that we must set E=0 at Eq. (50) and then ca=n or cb=n, with n being a nonnegative integer. This, in its turn, leads to the same spectrum obtained for the component s1=+ 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,


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].


where j and mj are integers, mjj, and 0. Due to the negative factor 1/4R12 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].


  1. 1. Vishveshwara CV. Scattering of gravitational radiation by a Schwarzschild black-hole. Nature. 1970;227:936
  2. 2. Regge T, Wheeler JA. Stability of a Schwarzschild singularity. Physical Review D. 1957;108:1063
  3. 3. Abbott BP et al. (LIGO scientific and virgo collaborations), observation of gravitational waves from a binary black hole merger. Physical Review Letters. 2016;116:061102
  4. 4. Cardoso V. Quasinormal Modes and Gravitational Radiation in Black Hole Spacetimes [doctoral thesis]. Universidade Técnica de Lisboa; 2004. [arXiv:gr-qc/0404093]
  5. 5. Berti E, Cardoso V, Starinets AO. Quasinormal modes of black holes and black branes. Classical and Quantum Gravity. 2009;26:163001
  6. 6. Kokkotas KD, Schmidt BG. Quasinormal modes of stars and black holes. Living Reviews in Relativity. 1999;2:2
  7. 7. Hod S. Bohr’s correspondence principle and the area spectrum of quantum black holes. Physical Review Letters. 1998;81:4293
  8. 8. Dreyer O. Quasinormal modes, the area spectrum, and black hole entropy. Physical Review Letters. 2003;90:081301
  9. 9. Maggiore M. The physical interpretation of the spectrum of black hole quasinormal modes. Physical Review Letters. 2008;100:141301
  10. 10. Domagala M, Lewandowski J. Black hole entropy from quantum geometry. Classical and Quantum Gravity. 2004;21:5233
  11. 11. Konoplya RA, Zhidenko A. Quasinormal modes of black holes: From astrophysics to string theory. Reviews of Modern Physics. 2011;83:793
  12. 12. Frolov VP et al. Massive vector fields in Kerr-NUT-(A)dS spacetimes: Separability and quasinormal modes. arXiv: 1804.00030
  13. 13. Zhidenko A. Massive scalar field quasi-normal modes of higher dimensional black holes. Physical Review D. 2006;74:064017
  14. 14. Zhidenko A. Linear perturbations of black holes: Stability, quasi-normal modes and tails. [doctoral thesis]. Universidade de São Paulo; 2009. [arXiv:0903.3555]
  15. 15. Liu LH, Wang B. Stability of BTZ black strings. Physical Review D. 2008;78:064001
  16. 16. Mukhi S. String theory: A perspective over the last 25 years. Classical and Quantum Gravity. 2011;28:153001
  17. 17. Emparan R, Reall HS. Black holes in higher dimensions. Living Reviews in Relativity. 2008;11:6
  18. 18. Csáki C. TASI lectures on extra dimensions and branes. In: Shifman M, Vainshtein A, Wheater J, editors. From Fields to Strings: Circumnavigating Theoretical Physics. Vol. 2. Singapore: World Scientific; 2005. p. 967
  19. 19. Maldacena JM. The large-N limit of superconformal field theories and supergravity. International Journal of Theoretical Physics. 1999;38:1113
  20. 20. Horowitz GT, Polchinski J. Gauge/gravity duality. In: Oriti D, editor. Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge, England: Cambridge University Press; 2009. p. 169
  21. 21. Hubeny VE. The AdS/CFT correspondence. Classical and Quantum Gravity. 2015;32:124010
  22. 22. Horowitz GT, Hubeny VE. Quasinormal modes of AdS black holes and the approach to thermal equilibrium. Physical Review D. 2000;62:024027
  23. 23. Birmingham D, Sachs I, Solodukhin SN. Conformal field theory interpretation of black hole quasinormal modes. Physical Review Letters. 2002;88:151301
  24. 24. Nunez A, Starinets AO. AdS/CFT correspondence, quasinormal modes, and thermal correlators in N = 4 SYM. Physical Review D. 2003;67:124013
  25. 25. Keranen V, Kleinert P. Thermalization of Wightman functions in AdS/CFT and quasinormal modes. Physical Review D. 2016;94:026010
  26. 26. David JR, Khetrapal S. Thermalization of green functions and quasinormal modes. Journal of High Energy Physics. 2015;07:041
  27. 27. López-Ortega A. Dirac quasinormal modes of D-dimensional de sitter spacetime. General Relativity and Gravitation. 2007;39:1011
  28. 28. Brady PB, Chambers CM. Radiative falloff in Schwarzschild-de sitter spacetime. Physical Review D. 1999;60:064003
  29. 29. Abdalla E et al. Support of dS/CFT correspondence from perturbations of three dimensional spacetime. Physical Review D. 2002;66:104018. arXiv:hep-th/0204030
  30. 30. Nollert HP. Quasinormal modes: The characteristic’ sound’ of black holes and neutron stars. Classical and Quantum Gravity. 1999;16:159
  31. 31. Batista C. Generalized charged Nariai solutions in arbitrary even dimensions with multiple magnetic charges. General Relativity and Gravitation. 2016;48:160
  32. 32. Venâncio J. The spinorial formalism, with applications in physics [Master dissertation]. Federal University of Pernambuco; 2017. Available from:;
  33. 33. Benn I, Tucker R. An Introduction to Spinors and Geometry with Applications in Physics. Adam Hilger; 1987. Available from:;
  34. 34. Cartan E. The Theory of Spinors. Dover; 1966. Available from:;
  35. 35. Venâncio J, Batista C. Separability of the Dirac equation on backgrounds that are the direct product of bidimensional spaces. Physical Review D. 2017;95:084022
  36. 36. Güven R. Wave mechanics of electrons in Kerr geometry. Physical Review D. 1977;16:1706
  37. 37. Dutt R, Khare A, Sukhatme UP. Supersymmetry, shape invariance, and exactly solvable potentials. American Journal of Physics. 1988;56:163
  38. 38. Pöschl G, Teller E. Bemerkungen zur Quantenmechanik des anharmonischen Oszillators. Zeitschrift für Physik. 1933;83:143
  39. 39. Venâncio J, Batista C. Quasinormal modes in generalized Nariai spacetimes. Physical Review D. 2018;97:105025
  40. 40. Hartman T. Lecture notes on classical de Sitter space. 2017. arXiv:1205.3855 [hep-th]
  41. 41. Anninos D. de Sitter Musings. 2013. arXiv:1205.3855 [hep-th]
  42. 42. Bengtsson I, Sandin P. Anti de sitter space, squashed and stretched. Classical and Quantum Gravity. 2006;23:971
  43. 43. López Ortega A. The Dirac equation in D-dimensional spherically symmetric spacetimes. arXiv:0906.2754
  44. 44. Zhidenko A. Linear perturbations of black holes: Stability, quasi-normal modes and tails [doctoral thesis]. Universidade de São Paulo; 2009. ArXiv:0903.3555
  45. 45. Du ED, Wang B, Su R. Quasinormal modes in pure de sitter spacetimes. Physical Review D. 2004;70:064024. arXiv:hep-th/0404047
  46. 46. Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover; 1972
  47. 47. Rosa JG. Superradiance in the sky. Physical Review D. 2017;95:064017


  • 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 gtt=1−Λ/3r2, as occurs in the case D=4, this coefficient should be D−1D−2.
  • In D=2d+1, besides the 2d Dirac matrices Γa and Γa˜, we need to add one further matrix, which will be denoted by Γd+1 given by Γd+1=σ3⊗σ3…⊗σ3⏟dtimes.

Written By

Joás Venâncio and Carlos Batista

Reviewed: August 14th, 2019 Published: November 11th, 2019