1. Introduction

The quasinormal modes (QNM) are the characteristic oscillations of a test field or of the metric perturbations that satisfy the appropriate boundary conditions near the event horizon and at the asymptotic region of the black hole^1,2. Thus the QNM appear when the black hole is perturbed from its equilibrium state and they depend on the physical properties of the black hole and the field. Recently the QNM have found many applications in different studies about the classical stability of the black holes^1,2, the area spectrum of the event horizon^3,4, and motivated by the AdS-CFT correspondence⁵, the QNM spectra of asymptotically anti-de Sitter black holes are extensively studied since they are useful in the calculation of the decay rates in dual theories^5-24.

Furthermore the exactly solvable systems are relevant in physics, since in these systems we can explore in detail their physical properties. Thus we think that in the research line of black hole perturbations is useful the search and study of exactly solvable systems. In particular for asymptotically anti-de Sitter black holes it is possible to calculate exactly the quasinormal frequencies (QNF) of several fields^18-25. We know that this is the case for the BTZ black holes^18-20, the massless topological black holes^21-24, and the five-dimensional topological black holes of Refs. 26 to 28. See also^29-31 for related examples of exact determination of QNF.

The QNF spectrum of the electromagnetic field in asymptotically anti-de Sitter black holes has been explored previously^8-12, ^14-17. Among the motivations we find that the electromagnetic field behaves in a different way than other classical fields and its analysis is physically more relevant than the study of the Klein-Gordon field. Additionally in the AdS-CFT correspondence the QNF of the electromagnetic field in asymptotically anti-de Sitter spacetimes are related to the poles of the retarded Green functions of the R-symmetry currents. Thus for the electromagnetic field we believe that it is convenient to determine its spectrum of QNF in asymptotically anti-de Sitter black holes.

Based on the results by Kodama and Ishibashi on the simplification of the Maxwell equations in static spacetimes³², (see also ³³) here we generalize previous results on the QNM of the Klein-Gordon field²⁵ in the five-dimensional topological black holes of Refs. 26 to 28 and calculate exactly the spectrum of QNF for the electromagnetic field. Using these results we study the stability of the topological black holes of Refs. 26 to 28 under perturbations. At this point it is convenient to mention that we discuss the possible existence of unstable QNM for the electromagnetic field for some values of the parameters.

We organize this work as follows. Following Kodama and Ishibashi³² (see also ³³) in Sec. 2 we recall some relevant results about the simplification of the vacuum Maxwell equations in static spacetimes to two differential equations, one for the vector type electromagnetic field and another for the scalar type electromagnetic field. In Sec. 3 we enumerate the relevant features of the five-dimensional topological black holes of Refs. 26 to 28 that we study in this work. In Sec. 4 we calculate exactly the QNF of the vector type electromagnetic field propagating on these five-dimensional topological black holes and discuss the stability of the QNM. We make a similar calculation for the scalar type electromagnetic field in Sec. 5, but in this case we need to make a careful study of the boundary condition at the asymptotic region since the usual boundary condition leads to a continuum of QNF and we modify the boundary condition to get a discrete set of QNF that depends on the parameters of the black hole and the field. Finally we discuss the main results in Sec. 6.

2. Maxwell equations

As is well known, we can write the line element of a D-dimensional generalization of the spherically symmetric spacetime in the form³²

where a,b=1,2,dΩD-22=γijdÿj,i,j=1,2,…,D-2 , i,j=1,2,…,D-2, is the line element of the (D-2)-dimensional maximally symmetric base manifold with metric γ^ij and whose Ricci tensor fulfills R^ij=(D-3)Kγ^ij, that is, the base manifolds are of Einstein type. Here K is a constant determined by the scalar curvature of the base manifold and can be normalized to the values K=0,±1 . In what follows we assume that the bidimensional line element that appears in the metric (1) is given by

with F and G functions of the radial coordinate r.

If we denote the Maxwell tensor by Fμν then the Maxwell equations in vacuum are

It is well known that if we make a harmonic sum on the scalar and vector eigenfunctions of the Laplacian on the base manifold dΩD-22, the Maxwell equations in a spacetime of the form (5) simplify to^32,33.

for the vector type electromagnetic field and

for the scalar type electromagnetic field. Here the symbol D _a denotes the covariant derivative for the bidimensional metric gab(x), the functions ΦV and ΦS depend on the coordinates xa of the two-dimensional space with metric gab and they contain the relevant information about the dynamics of the vector type and scalar type electromagnetic fields in spacetimes of the form (5). In the previous formulas k^V₂ (kS2) are the eigenvalues of the vector harmonics Vi (scalar harmonics S) on the maximally symmetric base manifold with line element dΩD-22, that is, they satisfy³²

where D^i is the covariant derivative on the maximally symmetric base manifold. For F=G=f we point out that in Eqs. (4) and (5) the operator DaDaes the form

3. Five dimensional topological black holes

The five-dimensional topological black holes that we study in this work have the line element

where dΩK2 is the line element of the three-dimensional maximally symmetric base manifold and the function f takes the form^26-28

where Λ is a negative constant, K=0,±1, and 𝑐 0 is a non negative constant. We notice that for the three-dimensional base manifold the scalar curvature is equal to 6K. The solution with positive sign of c0 is usually called the plus branch, whereas the solution with negative sign of c0 is usually known as the minus branch²⁸. The topological black holes (8) are solutions of several gravity theories as the five-dimensional Chern-Simmons theory²⁶, the five-dimensional Gauss-Bonnet gravity with special Gauss-Bonnet coefficient²⁷, and the five-dimensional z=4 Hořava-Lifshitz gravity²⁸.

The event horizon of the topological black holes (5) is determined by

Since we impose that the radius of the event horizon is positive, we need that the parameters c0 and K fulfill

Notice that depending on the value of K we have one or two positive values of the event horizon according to the following list.

For K=0 and K=1 the black hole exists in the minus branch.

For K=-1 to have a black hole in the plus branch we require that c0<1.

For K=-1 in the minus branch we always have a black hole.

Thus for K=-1 we have black holes in the two branches of the solution ([8]) .

In the following sections it is useful to define the quantity

and if the black hole exists, then p<0. We also notice that for K=-1 the parameter p satisfies |p|>1 in the minus branch and |p|<1 in the plus branch.

4. Quasinormal frequencies of the vector type electromagnetic field

To extend the previous results of ²⁵⁾ on the QNM spectrum of the five-dimensional topological black holes (4) in what follows we calculate exactly the QNF of the electromagnetic field moving on these black holes. Following Refs. 18 to 25 for the topological black holes (7) we define their QNM as the oscillations that satisfy the boundary conditions

i)

The electromagnetic field is purely ingoing near the black hole horizon.

ii)

The electromagnetic field goes to zero as r→+∞.

We notice that the line elements of the five-dimensional topological black holes (8) are of the form (4) with F=G=f, therefore we can use Eqs. (4) and (5) to study the propagation of the electromagnetic fields on these backgrounds. Here we begin with the vector type electromagnetic field.

Since the topological black holes (8) are static and taking into account the expression (7) for the operator DaDa, we propose that the function ΦV takes the form

and therefore from Eq. (4) we obtain that the radial function RV must be a solution of the differential equation

To solve exactly the previous differential equation, as in Ref. 25, it is convenient to define the variable i

with p already given in the formula (12). Using the variable 𝑣 we find that Eq. (14) transforms into

where we define the constants F and G by

Taking the function RV in the form

with the parameters A_v and B_v being solutions to the algebraic equations

we find that the function R_2v is a solution of the differential equation

This equation is of hypergeometric type^34-36

with parameters αV, βV, γV equal to

In what follows we choose

Where

Notice that q>0. For these values of A_v, B_v we get that γV-αV-βV=-1.

From these results we find that the function R_v is equal to^34-36

where K₁, K₂ are constants and 2F1(α,β;γ;v) denotes the hypergeometric function^34-36. Taking into account that v≈0 near the horizon of the black hole, from the expression (25) for R_v, we observe that near the horizon this function behaves as

where r* denotes the tortoise coordinate of the five-dimensional topological black hole (8)

Notice that r*→-∞ near the horizon and r*→0 as r→+∞.

Since we take a time dependence of the form exp(-iωt) (see the expression (13) we find that the first term of the expression (35) is an outgoing wave near the horizon, whereas the second term is an ingoing wave near the horizon. Thus to satisfy the boundary condition i) of the QNM we must take K1=0 and therefore the function 𝑅 𝑉 fulfilling the boundary condition near the horizon takes the form

With

To study the behavior of the field as r→+∞ (v→1) the usual procedure for many exactly solvable problems^18-25 is to use the Kummer property of the hypergeometric function^34-36, but for the vector type field, since the parameters of the hypergeometric function that appears in the solution (26) fulfill γ^V-β^V-α^V=-1 and the Kummer property is not valid when the values of the parameters satisfy this condition^34-36, we cannot employ the usual procedure. Nevertheless for γ-α-β=-m, m=0,1,2,…, the hypergeometric function satisfies

when the parameters α and β are different from negative integers^34-36. In the previous formula ψ is the logarithmic derivative of the gamma function and (m)s is the Pochhammer symbol. For m=0 we must delete the finite sum. Notice that for the vector type electromagnetic field we have m=1.

To analyze the behavior of the field as r→+∞ we use the property (30) to write the radial function (28) as

From this expression we obtain that the second factor goes to zero as r→+∞, but in this limit the first factor behaves in the form

and therefore diverges as v→1. In a similar way to previous works^18-25,33, to cancel this term we like to impose the conditions

with n=0,1,2,… But if we impose the conditions (33), then we contradict the assumptions under which the property (30) is true. Thus if we use the property (30) then we cannot impose the conditions (33) because we contradict the assumptions under which this property is valid.

Another path is to impose the conditions (33) on the parameters of the solution (28) and see whether the radial function 𝑅 𝑉 satisfies the boundary condition ii) of the QNM³³. Thus assuming that α^V=-n, we find that

where we use that the hypergeometric function fulfills³⁶

In a straightforward way we obtain that near v=1 the function RV of the expression (34) behaves as

and therefore it satisfies the boundary condition ii) of the QNM as v→1. A similar thing happens for β^V=-n. Thus imposing the conditions (33) we obtain that the radial function RV satisfies the boundary condition ii) of the QNM.

From the conditions (33) and taking into account the values of the parameters (22) and (23) we get that the QNF of the vector type electromagnetic field are equal to

It is convenient to notice that the QNF (37) of the vector type electromagnetic field depend on the value of the parameter K related to the scalar curvature of the base manifold. We recall that p<0 to have a positive radius of the event horizon. Furthermore we remember that the eigenvalues kV2 are non negative and they are discrete for K=1, whereas they are continuous for K≤0 ³². From the values (36) for the QNF we get that for K=0 and 𝐾=1 the real part of the QNF for the vector type field is different from zero and the imaginary part is negative, that is, the QNM decay in time since we have a time dependence exp(-iωt), and therefore they are stable for K=0 and K=1. We remark that for K=0 and K=1 the QNF (37) are complex.

For K=-1 we obtain that the QNF (37) simplify to

Thus for kV2>2 the QNF (38) have real and imaginary parts different from zero and they are stable since Im(ωV)<0. Nevertheless if the base manifold allows eigenvalues of the vector harmonics satisfying 0<kV2<2, we obtain that the QNF (38) transform into

that is, they are purely imaginary. From the expressions (38) and (39) we remark that for K=-1 we get complex or purely imaginary QNF depending on the value of the eigenvalues kV2. The QNF (40) with the minus sign of the square root are stable since Im(ωV)<0, but the QNF with the plus sign of the square root are unstable if

We notice that this inequality is not satisfied for |p|>1, but for kV2 sufficiently small, we can fulfill the inequality (40) for sufficiently small |p|<1, that is, for the plus branch of the topological black holes (8) with K=-1. Thus, for these topological black holes the fundamental mode (n=0) is unstable for

For the overtones we find that the condition (40) becomes

Thus for the topological black holes (8) with K =−1 in the plus branch with sufficiently small values of the parameter |𝑝| we find unstable QNM for the vector type electromagnetic field if the base manifold has eigenvalues of the vector harmonics in the range 0<kV2<2.

Nevertheless, for three-dimensional closed base manifolds and three-dimensional open base manifolds such that the quantity VjD^iVj fall off sufficiently rapidly at infinity, it is known that the eigenvalues of the vector harmonics satisfy kV2≥2|K| . Thus for these three-dimensional base manifolds there is no instability of the QNM of the vector type electromagnetic field. We do not know an example of a three-dimensional base Einstein manifold with eigenvalues in the range 0<kV2<2.

5. Quasinormal frequencies of the scalar type electromagnetic field

Here we extend the results of the previous section and calculate exactly the QNF of the scalar type electromagnetic field propagating on the topological black holes (8). In a similar way to the previous section we take

and from Eq. (5) we obtain that the function RS must be a solution of the differential equation

To solve exactly this equation we use the variable v defined in the formula (15) to find that it transforms into

with F defined in the expression (17) and

To solve Eq. (45) we propose that the radial function RS takes the form

with the quantities AS and BS being solutions of the algebraic equations

to find that the function RS is a solution of the differential equation

This equation is of hypergeometric type (see Eq. (21)) with parameters

In what follows we choose AS=1/4, BS=iωq/2, and we notice that the parameters αS,βS,γS of the hypergeometric function fulfill γS-βS-αS=0. From these results we obtain that the radial function RS is equal to^34-36

where K3 and K4 are constants. Near the event horizon (v=0) of the topological black holes (8) this function behaves in a similar way to Eq. (26). To fulfill the boundary condition i) of the QNM we must take K3=0 and the function RS that satisfies this boundary condition is

With

It is convenient to notice that the new parameters α^S,β^S,γ^S also fulfill γ^S-α^S-β^S=0. As for the vector type electromagnetic field, due to this fact we cannot use the Kummer property of the hypergeometric function^34-36 to analyze the behavior of the scalar type electromagnetic field as r→+∞ (v→1). Taking into account the property (30) of the hypergeometric function we find that the function RS satisfying the boundary condition near the horizon takes the form

in the variable 1-v. As r→+∞ (v→1) this function behaves as

where L1 and L2 are constants. We notice that in the previous formula both terms go to zero as r→+∞. Thus the function RS of the expression (52) that fulfills the boundary condition near the horizon, for all the frequencies, it also satisfies the boundary condition ii) of the QNM, thus, we shall obtain a continuous spectrum of QNF for the scalar type electromagnetic field (see for example Refs. 37 and 38). Nevertheless we expect to obtain a discrete set of quasinormal frequencies determined by the physical parameters of the black hole and the field^1,2. Therefore we must make a careful analysis of the behavior of the radial function RS as r→∞ before we impose the boundary condition ii) of the QNM.

We find a similar example in the exact calculation of the QNF for the electromagnetic field propagating on an asymptotically Lifshitz black hole³³. In the previous reference, to calculate the QNF of the electromagnetic field the boundary condition in the asymptotic region was modified to get a discrete spectrum of QNF. We see that in Ref. 33 it is proposed that for calculating the QNF of the electromagnetic field in the cases where the function 𝑅 𝑆 is well behaved as r→+∞ we must impose as a boundary condition that the leading term of the asymptotic behavior must be canceled.

Following Ref. 33 to calculate the QNF of the scalar type electromagnetic field in the topological black holes (8) we propose that instead of the boundary condition ii) of the previous section, in the asymptotic region we must impose as boundary condition that the leading term in the asymptotic behavior (55) of the radial function 𝑅 𝑆 must be canceled. We notice that for the vector type electromagnetic field when we calculate its QNF in the previous section we also cancel the leading term in the asymptotic behavior of the radial function 𝑅 𝑉 , but for the vector type electromagnetic field the leading term is divergent as r→+∞. Thus for the vector type electromagnetic field the new boundary condition as r→+∞ also can be used to compute its QNF and we obtain the same results of Sec. 4. We think that the new boundary condition that we impose at the asymptotic region is a natural generalization of the boundary condition ii).

In a similar way to Ref. 33 and motivated by the results of the previous section we assume that the parameters α^S and β^S take the values

and with these values of the parameters we verify whether the radial function RS of the formula (52) satisfies the new boundary condition of the QNM as r→+∞. Thus taking α^S=-n we get that this radial function becomes

Using the property (35) of the hypergeometric function we find that as r→+∞ the radial function (57) behaves as

that is, we cancel the leading term of the asymptotic expansion (55) for RS and as r→+∞ the radial function (57) fulfills the new boundary condition of the QNM. Something similar happens when we take β^S=-n. Therefore the QNF of the scalar type electromagnetic field are determined by the conditions (56) and they are equal to

In contrast to the QNF of the vector type electromagnetic field, the previous QNF do not depend on the parameter K. Recalling that for K=0,±1 the eigenvalues kS2 are non negative³², for the three values of K and for kS2>|p| we have complex QNF that are stable since Im(ωS)<0. For kS2<|p| the QNF (59) transform into

that are purely imaginary. Thus depending on the values of |p| and kS2 we get complex or purely imaginary QNF for the scalar type electromagnetic field. For the minus sign of the square root in the formula (60) we have stable QNM, but for the plus sign of the square root we obtain QNF with Im(ωS)>0 when we fulfill the following inequality

In a straightforward way we verify that this inequality cannot be satisfied for the allowed values of the physical parameters, that is, for kS2<|p| also we obtain stable QNM. Thus for the scalar type electromagnetic field we find only stable QNM in the five dimensional topological black holes (60).

6. Discussion

In the previous sections we calculate exactly the QNF of the vector type and the scalar type electromagnetic fields propagating on the topological black holes (8). It is convenient to notice that for the three values of the parameter K we state the radial problem in a common form and we solve simultaneously the differential equations. It is convenient to notice that the method previously used to calculate the QNF (37) and (59) is slightly different from the used in other references, since the special values of the parameters for the hypergeometric functions that we obtain in the topological black hole (8), force us to impose the conditions (33) and (56) and then verify that for these values of the parameters the radial functions satisfy the boundary condition at infinity. Usually it possible to employ the Kummer property of the hypergeometric function and to choose the appropriate behavior at the boundaries by imposing the analogue of the conditions (33) and (56)^18-25.

Depending on the physical parameters for the scalar type and vector type electromagnetic fields we obtain complex QNF or purely imaginary QNF. We find that the QNM of the electromagnetic field are stable, except for the topological black holes with K=-1 of the plus branch for which we find that for small values of the parameter |𝑝| and of the eigenvalues kV2 for the vector harmonics, the QNM of the vector type electromagnetic field would be unstable if the three-dimensional base manifold has eigenvalues kV2 satisfying 0<kV2<2. Therefore, as noted previously for other backgrounds in Refs. 32 and 39, the eigenvalues of the scalar and vector harmonics of the base manifold play a relevant role in the analysis of the classical stability of the black holes under perturbations.

Comparing the expressions (37) and (59) for the QNF of the vector type and scalar type electromagnetic fields, we see that they are not isospectral since the terms in square roots show a different dependence on the parameter 𝑝. Also, notice that the QNF of the vector type electromagnetic field depends on the scalar curvature K, whereas the QNF of the scalar type electromagnetic field are independent of K. Furthermore the QNF of the vector type field depend on the overtone number 𝑛 in the form (n+1), whereas the QNF of the scalar type field depend on 𝑛 in the form (n+1/2).

Finally considering that the Hawking temperature of the topological black holes (8) is ²⁸

for the vector type electromagnetic field we can write its QNF (37) as

and for the scalar type electromagnetic field its QNF (59) as

Thus for the five-dimensional topological black holes (8) the QNF are proportional to its Hawking temperature.