Open access peer-reviewed chapter - ONLINE FIRST

Quasinormal Modes of Dirac Field in Generalized Nariai Spacetimes

By Joás Venâncio and Carlos Batista

Submitted: May 23rd 2019Reviewed: August 14th 2019Published: November 11th 2019

DOI: 10.5772/intechopen.89179

Downloaded: 72

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, 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 [29, 30, 31] is used to show that the Dirac equation is separable [32] and can be reduced to a Schrödinger-like equation [33] whose potential is contained in the Rosen-Morse class of integrable potentials, which has the so-called Pöschl-Teller potential as a particular case [34, 35]. Finally, the boundary conditions leading to QNMs are analyzed, and the quasinormal frequencies (QNFs) are analytically obtained [5, 36].

2. Presenting the problem

In D dimensions, the dynamics of general relativity in spacetimes with a cosmological constant Λis described by the Einstein-Hilbert action1

S=116πdDxgRD2Λ+Sm,E1

where Ris the Ricci scalar and Smstands for the action of all matter fields Φicoupled 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 gμνand Φiwhich are given, respectively, by

Rμν12Rgμν+D22Λgμν=8πTμν,δSmδΦi=0,E2

where Tμνis the symmetric stress-energy tensor associated to Φidefined by the equation

Tμν=2gδSmδgμν.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

ds2=gμνxdxμdxν,Φi=Φix.E4

Now, let the pair gμν0and Φi0be 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 gμν0and Φi0and the small perturbations hμνand Ψi

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

In what follows, let us consider a specific matter field Ψpropagating in a generalized version of the Nariai spacetime described in Ref. [28]. Here, Ψis an electrically charged spinorial field of mass mthat 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 dS2with d1copies of the unit spheres S2possessing different radii Rj. Thus, the natural line element of the higher-dimensional version of the Nariai spacetime is given by

ds2=gμν0dxμdxν=frdt2+1frdr2+j=2dRj2dΩj2,E6

where fris a function of the coordinate rand dΩj2is the line element of the jth unit sphere S2as follows

fr=1r2R12,dΩj2=dθj2+sin2θjdϕj2.E7

The radii R1and Rjare given by

R1=Λ12Q12+Q2D21/2,Rj=Λ+12Qj2+Q2D21/2,E8

where Q1is an electric charge and Qjare magnetic charges, while Qis defined by

Q=Q12j=2dQj2.E9

This spacetime is a locally static solution of Einstein’s equation with a cosmological constant Λand electromagnetic field =dAwhose gauge field Ais given by

A=Q1rdt+j=2dQjRj2cosθjdϕj.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 twhose norm vanishes at r=±R1. Indeed, r=±R1define 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 [36]. For this reason, the dependence of all the components of the field Ψon the coordinates along the Killing vector tis 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[37, 38, 39]. However, this is just justified for de Sitter with D>2, but not for D=2; see [36] 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

dx=1frdrx=R1arctanhrR1,E11

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

ds2=1cosh2x/R1dt2+dx2+j=2dRj2dΩj2,E12

and the gauge field can be rewritten as

A=Q1R1tanhx/R1dt+j=2dQjRj2cosθjdϕj.E13

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 Ψwhich 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, 40, 41] 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 = ± R 1 , while the cones are the local light cones. Mathematically, the wavy arrow pointing to the right represents e − iω t − x , while the wavy arrow pointing to the left represents e − iω t + x . For more details, see Ref. [36].

From the mathematical of view, since we are assuming that the time dependence of Ψis eiωt, this boundary condition means that near the horizons r=±R1, that is, as x±, the radial component of the field Ψshould behave as et+xat x, while it should go as etxat 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 [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 1/2with electric charge q and mass m propagating in such spacetime is a spinorial field obeying the following version of the Dirac equation:

ΓααiqAαΨ=mΨ,E14

where Aαstands for the components of the background gauge field. In D=2ddimensions, the Dirac matrices Γαrepresent faithfully the Clifford algebra by 2d×2dmatrices obeying the relation

ΓαΓβ+ΓβΓα=geαeβId,E15

with Idstanding for the 2d×2didentity matrix. The index α,β,γrun 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

e1=icoshx/R1t,ej=1Rjsinθjϕj,e1˜=coshx/R1x,ej˜=1Rjθj,E16

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

geαeβ=δαβgeaeb=δab,geaeb˜=0,gea˜eb˜=δa˜b˜,E17

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:

αeβ=ωαβγeγ.E18

Since the metric g is a covariantly constant tensor, it follows that the coefficients of the spin connection with all low indices ωαβγ=ωαβεδεγare antisymmetric in their two last indices, ωαβγ=ωαγβ. Note that the indices of the spin connection are raised and lowered with δαβand 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 Ψis, then, given by

αΨ=αΨ14ωαβγΓβΓγΨ,E19

with αdenoting 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

σ1=0110,σ2=0ii0,σ3=1001,E20

are the Hermitian Pauli matrices and Idenote the 2×2identity matrix. Using this notation, a convenient representation of the Dirac matrices is the following:

Γa=σ3a1timesσ1IIdatimes,Γa˜=σ3a1timesσ2IIdatimes,E21

where Istands for the 2×2identity 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 ξsgiven by

ξ+=10,ξ=01,E22

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

σ1ξs=ξs,σ2ξs=isξs,σ3ξs=sξs.E23

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

Ψ=sΨs1s2sdξs1ξs2ξsd,E24

where each of the indices sacan take the values “+1” and “−1.” Since every sacan take just two values, it follows that the sum over ss1s2sdcomprises 2d terms, which is exactly the number of components of a spinorial field in D=2ddimensions.

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

Γαα=a=1dΓaa+Γa˜a˜=a=1dσ3a1timesDaIIdatimes,E25

where

Da=σ1a+σ2a˜,E26

is the Dirac operator on 2with 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

ΓaΨ=ss1s2sa1Ψs1s2sdξs1ξs2ξsa1ξsaξsa+1ξsd=ss1s2saΨs1s2sa1sasa+1sdξs1ξs2ξsa1ξsaξsa+1ξsd,E27

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

Γa˜Ψ=ss1s2sa1isaΨs1s2sdξs1ξs2ξsa1ξsaξsa+1ξsd=iss1s2sasaΨs1s2sa1sasa+1sdξs1ξs2ξsa1ξsaξsa+1ξsd.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. [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

Ψs1s2sd=Ψ1s1txΨ2s2Φ2θ2ΨdsdΦdθd,E29

where each index sacan take the values sa=±1, the fields Ψ1s1txsatisfy the following differential equation (the reader is invited to demonstrate the equation below or consult more details in [36]):

1˜+ω11˜12iqA1˜is11+ω1˜11˜2iqA1Ψ1s1=Lis1mΨ1s1.E30

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 λjof the Dirac operator on unit sphere S2according to the following relation

L=λ22+λ32++λd2,λj=±1,±2,±3,,E31

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

ω11˜1=ω111˜=1R1sinhx/R1,ωjj˜j=ωjjj˜=1Rjcotθj,E32

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

A1=iQ1R1sinhx/R1,Aj=QjRjcotθj.E33

Now, since the components of the metric are independents of the coordinate t, the vector tis a Killing vector for this metric. So, it is useful to assume the following time dependence for the field Ψ1s1tx

Ψ1s1tx=eiωtψs1x.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:

ddx+is1ω+isqQ1R112R1tanhx/R1ψs1=Lis1mcoshx/R1ψs1.E35

In order to solve these equations, we should first decouple the fields ψs1and ψs1. Eliminating ψs1we obtain a second-order equation for ψs1. Indeed, we can prove that the fields ψs1satisfy the following second-order ordinary differential equation

d2dx2+ω2Vxψs1=0,E36

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

Vx=A+Btanhx/R1+Ccosh2x/R1,E37

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

A=14R12qQ1is1+qQ1R12,B=ωR1is1+2qQ1R12,C=m2+L2+14R12+q2Q12R12.E38

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:

VA+Batx+,ABatx.E39

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

VA+Catx0,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 [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, x±, which define the QNFs in a unique way.

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

y=12+12tanhx/R1.E41

In particular, notice that y is defined on the domain y01with the boundaries x±being given by y=0and y=1. In addition to this change of independent variable, if we now set the Ansatz

ψs1x=yα1yβHs1y,E42

with the parameters α and β being constants conveniently chosen as

α=R12ABω2,β=R12A+Bω2,E43

the functions Hs1must be solutions of the following differential equation

y1yd2Hs1dy2+2α+12+2α+2βydHs1dyCR12+α+β1+α+βHs1.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 Hs1satisfy a hypergeometric equation. Indeed, comparing with the standard hypergeometric differential equation

y1yd2Hs1dy2+c1+a+bydHs1dyabHs1=0,E45

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

a=12+α+β+14CR12,b=12+α+β14CR12,c=2α+1.E46

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

Hs1y=D2F1abcy+Ey1c2F11+a+c1+b+c2cy,E47

where 2F1is the hypergeometric function and D and E are arbitrary integration constants. Given the hypergeometric solution for Hs1is 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

ψs1=1y12a+bc[Dy12c12F1abcy+Ey12c12F11+ac1+bc2cy].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±, the relation between the coordinates x and y assumes the simpler form

ye+2x/R1atx,1ye2x/R1atx+.E49

Thus, taking into account the latter relation and using the fact that at y=0xthe hypergeometric function 2F1abc0=1, one eventually obtains that near the boundary xthe field ψs1behaves as

ψs1xDec1x/R1+Eec1x/R1.E50

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]:

2Fabcy=ΓcΓcabΓcaΓcb2Faba+bc+11y+ΓcΓa+bcΓaΓb1ycab2Fcacbcab+11y,E51

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

ψs1x+DΓcabΓcΓcaΓcb+EΓcabΓ2cΓ1aΓ1bea+bcx/R1+DΓa+bcΓcΓaΓb+EΓa+bcΓ2cΓac+1Γbc+1ea+bcx/R1.E52

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

a=iR1μ2+q2Q12R12+L2+1+s114R12i1s1qQ1R122,b=iR1μ2+q2Q12R12+L2+1+s114R12i1s1qQ1R122,c=12+is1qQ1R12ωR1.E53

In particular, the following relations hold

c1/R1=is1ω+is1qQ1R112R1,E54
a+bc/R1=iqQ1R1+s112R1.E55

Now we are ready to impose the boundary conditions. Obviously, without loss of generality, we can consider that the spin s1is 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:

Ψ1+txx=Det+xeiqQ1R112R1x+EetxeiqQ1R112R1x.E56

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 xit 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 etxat x, while it should go as et+xat x+. Thus, from Eq. (55), we conclude that we must set D=0. In such a case, from Eq. (52), the field Ψ1+becomes

Ψ1+x+EΓcabΓ2cΓ1aΓ1betxeiqQ1R112R1x+EΓa+bcΓ2cΓac+1Γbc+1et+xeiqQ1R1+12R1x.E57

Finally, to satisfy the QNM boundary condition near the boundary at x, we must eliminate the term etxof 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:

1a=nor1b=n,wheren0,1,2.E58

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

ω=±m2+q2Q12R12+L2+iR1n+12,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 s1=of the spinorial field, we find that we must set E=0at Eq. (50) and then ca=nor 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,

ωD=±m2+q2Q12R12+L2+iR1n+12,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 [36].

ωKG=±m2+j=2djj+1Rj214R12iR1n+12,ωM=±j=2djj+1Rj214R12iR1n+12,E61

where jand mj are integers, mjj, and 0. Due to the negative factor 1/4R12inside 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 1/2field. 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 [33]. Thus, it would be interesting to investigate whether an analogous thing happens in the background considered here.

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 .

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Joás Venâncio and Carlos Batista (November 11th 2019). Quasinormal Modes of Dirac Field in Generalized Nariai Spacetimes [Online First], IntechOpen, DOI: 10.5772/intechopen.89179. Available from:

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