# On the Classification of Asymptotic Quasinormal Frequencies for –Dimensional Black Holes and Quantum Gravity

###### Abstract:

We provide a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in dimensions. This includes all possible types of gravitational perturbations (tensor, vector and scalar type) as described by the Ishibashi–Kodama master equations. The frequencies for Schwarzschild are dimension independent, while for Reissner–Nordström are dimension dependent (the extremal Reissner–Nordström case must be considered separately from the non–extremal case). For Schwarzschild de Sitter, there is a dimension independent formula for the frequencies, except in dimension where the formula is different. For Reissner–Nordström de Sitter there is a dimension dependent formula for the frequencies, except in dimension where the formula is different. Schwarzschild and Reissner–Nordström Anti–de Sitter black hole spacetimes are simpler: the formulae for the frequencies will depend upon a parameter related to the tortoise coordinate at spatial infinity, and scalar type perturbations in dimension lead to a continuous spectrum for the quasinormal frequencies. We also address non–black hole spacetimes, such as pure de Sitter spacetime—where there are quasinormal modes only in odd dimensions—and pure Anti–de Sitter spacetime—where again scalar type perturbations in dimension lead to a continuous spectrum for the normal frequencies. Our results match previous numerical calculations with great accuracy. Asymptotic quasinormal frequencies have also been applied in the framework of quantum gravity for black holes. Our results show that it is only in the simple Schwarzschild case which is possible to obtain sensible results concerning area quantization or loop quantum gravity. In an effort to keep this paper self–contained we also review earlier results in the literature.

^{†}

^{†}preprint: hep-th/0411267

## 1 Introduction

Black holes are an undeniable landmark in the road that connects classical to quantum gravity. Having been first discovered as static solutions of classical general relativity, they were later shown to actually radiate and evaporate once quantum effects were properly taken into account [1, 2]. Thus, it seems that in a true quantum theory of gravity, non–extremal black holes will actually be unstable. In this setting, an important question that immediately comes to mind is whether the black hole solution under consideration was really a stable solution of the classical theory, to start off with.

Researchers first focused on analyzing the linear stability of four dimensional black hole solutions of general relativity in [3, 4, 5]. The linear perturbation theory for the Schwarzschild black hole was set up in [3], where the classical stability of the solution was also proven. The equation derived in that paper to describe the linear perturbations, and their frequencies, is a Schrödinger–like equation and is now known as the Regge–Wheeler (RW) equation. The procedure to derive the RW equation is based on a study of the linearized Einstein equations in the given background and proceeds with a decomposition of the perturbation in tensor spherical harmonics (for spherically symmetric backgrounds) in order to obtain a radial equation describing the propagation of linear perturbations. This procedure, as applied to the analysis of perturbations to the Schwarzschild metric, was brought to firmer grounds in [4] with an explicit construction of four dimensional tensor spherical harmonics. This work was further completed in [5] with an extension to the Reissner–Nordström (RN) black hole solution. There, it was shown that the complete Einstein–Maxwell system for each type of multipole (electric or magnetic) could be reduced to two second order Schrödinger–like equations, generalizing the RW equation of the Schwarzschild case.

Part of the physical picture that emerged from this study of linear perturbations to black holes is the following. After the onset of a perturbation, the return to equilibrium of a black hole spacetime is dominated by damped, single frequency oscillations, which are known as the quasinormal modes (we refer the reader to [6, 7] for recent reviews on quasinormal modes, and a more complete list of references). These modes are quite special: they depend only on the parameters of the given black hole spacetime, being independent of the details concerning the initial perturbation we started off with. Moreover, modes which damp infinitely fast do not radiate at all, and can thus be interpreted as some sort of fundamental oscillations for the black hole spacetime. We shall return to this point in a moment.

It was not until recently that the black hole stability problem was addressed within a –dimensional setting [8, 9, 10]. These papers tried to be as exhaustive as possible, studying in detail the perturbation theory of static, spherically symmetric black holes in any spacetime dimension and allowing for the possibilities of both electromagnetic charge and a background cosmological constant. The set of equations describing linear perturbations in –dimensions was derived in [8, 10]. These equations generalize the RW equation and will be denoted as the Ishibashi–Kodama (IK) master equations. The perturbations come in three types: tensor type perturbations, vector type perturbations and scalar type perturbations. It should be noted that this nomenclature refers to the tensorial behavior on the sphere, , of each Einstein–Maxwell gauge invariant type of perturbation, and is not related to perturbations associated to external particles. For instance, one should not confuse vector type perturbations with perturbations associated to the propagation of a spin–1 vector particle in the background spacetime, or scalar type perturbations with perturbations associated to the propagation of a spin–0 scalar particle. The IK master equations were used in [9, 10] to study the stability of –dimensional black holes, and although many known solutions were shown to be stable, stability of some other solutions is still an open problem. In this work we shall make use of the IK master equations to analytically compute asymptotic quasinormal frequencies, and thus focus on the –dimensional Einstein–Maxwell classical theory of gravity.

Having thus acquired a list of stable black hole solutions, the next question to address within this general problem are the quasinormal modes—the damped oscillations which describe the return to the initial configuration. As we have said, modes which damp infinitely fast do not radiate, and they are known as asymptotic quasinormal modes. Besides their natural role in the perturbation theory of general relativity, asymptotic quasinormal modes have recently been focus of much attention following suggestions that they could have a role to play in the quest for a theory of quantum gravity [11, 12]. It was suggested in [11] that an application of Bohr’s correspondence principle to the asymptotic quasinormal frequencies could yield new information about quantum gravity, in particular on the quantization of area at a black hole event horizon. It was further suggested in [12] that asymptotic quasinormal frequencies could help fix certain parameters in loop quantum gravity. Both these suggestions lie deeply on the fact that the real part of the asymptotic quasinormal frequencies is given by the logarithm of an integer number, a fact that was analytically shown to be true, for Schwarzschild black holes in –dimensional spacetime, in [13, 14]. A question of particular relevance that immediately follows is whether the suggestions in [11, 12] are universal or are only applicable to the Schwarzschild solution. Given the mentioned analysis of [8, 10], one has at hand all the required information to address this problem and compute asymptotic quasinormal frequencies for --dimensional black holes^{1}^{1}1The least damped quasinormal modes associated to the IK master equations were addressed numerically in [15].. A preliminary clue is already present in [14], where the analysis of the four dimensional RN solution yielded a negative answer: the asymptotic quasinormal frequencies obeyed a complicated relation which did not seem to have the required form. Another clue was presented in [16], where the analysis of the four dimensional Schwarzschild de Sitter (dS) and Schwarzschild Anti–de Sitter (AdS) black holes also yielded a negative answer; again the asymptotic quasinormal frequencies did not seem to have the required form. It is the goal of this paper to carry out an extension of the techniques in [14, 16] to static, spherically symmetric black hole spacetimes in any dimension , including both electromagnetic charge and a background cosmological constant. Besides the intrinsic general relativistic interest of classifying these asymptotic quasinormal frequencies, it is also hoped that our results can yield conclusive implications for the proposals of [11, 12], dealing with the application of quasinormal modes to quantum gravity. We shall later see that it is only in the simple Schwarzschild case which is possible to obtain sensible results concerning area quantization or loop quantum gravity.

It is important to stress that even if the ideas in [11, 12] turn out not to be universal, it is still the case that quasinormal frequencies will most likely have some role to play in the quest for a theory of quantum gravity. Indeed, quasinormal frequencies can also be regarded as the poles in the black hole greybody factors which play a pivotal role in the study of Hawking radiation. Furthermore, the monodromy technique introduced in [14] to analytically compute asymptotic quasinormal frequencies was later extended, in [17], so that it can also be used in the computation of asymptotic greybody factors. It was first suggested in [17] that the results obtained for these asymptotic greybody factors could be of help in identifying the dual conformal field theory (CFT) which microscopically describes the black hole, and these ideas have been taken one step forward with the work of [18]. It remains to be seen how much asymptotic quasinormal modes and greybody factors can help in understanding quantum gravity.

The organization of this paper is as follows. In section 2 we provide a summary of our results for the reader who wishes to skip the technical details on a first reading. Section 3 represents the main body of the paper, where we use both the analytic continuation of the IK master equations to the complex plane and a method of monodromy matching at the several singularities in the plane, in order to analytically compute asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in dimension . This includes a brief review of quasinormal modes, as well as the Schwarzschild case of [14], for both completeness and pedagogical purposes. In section 4 we address some exact solutions for quasinormal frequencies, dealing with non--black hole spacetimes. We address Rindler, dS and AdS spacetimes, providing full solutions in all cases^{2}^{2}2Other exact calculations of quasinormal frequencies can be found in, e.g., [19, 20].. It should be noted that there has been some confusion in the literature concerning the computation of quasinormal modes using the monodromy technique, as well as on the computation of quasinormal modes in pure dS spacetime. We hope that this paper will serve to lay these confusions to rest. Section 5 reviews [11, 12] and the applications of asymptotic quasinormal frequencies to area quantization and loop quantum gravity. We show that these applications only seem to work in the simple Schwarzschild setting. In section 6 we conclude, listing some future directions of research. We also include three appendices. In appendix A we present a thorough list of conventions for black holes with mass , charge and background cosmological constant , in –dimensional spacetime. Appendix B includes all required formulae for the IK master equations, and appendix C makes a complete study of the tortoise coordinate in the spacetimes in consideration, alongside with analysis of the IK master equation potentials at several singularities in the complex plane. Throughout we will show that our results match earlier numerical calculations with great accuracy and will review some of the earlier literature on each case considered, as an effort to make this paper self–contained.

## 2 Summary of Results

For the reader who wishes to skip the main calculation on a first reading, we present in the following a summary of our results for the asymptotic quasinormal frequencies, in any spacetime dimension . In the appendices we review the black hole spacetime solutions we wish to consider, as well as the perturbation theory for these spacetimes which leads to the quasinormal mode analysis. The set of equations describing quasinormal modes in –dimensions was derived in [8, 10], and the perturbations come in three types: tensor type perturbations, vector type perturbations and scalar type perturbations. For each spacetime in consideration we compute asymptotic quasinormal frequencies, given each type of –dimensional perturbation. In the case of black hole spacetimes the computation involves a detailed monodromy analysis, alongside with some simple differential equations. In fact, the IK master equation describing quasinormal modes is of Schrödinger type, where the potential is associated to the perturbation under study. When using the monodromy technique one needs to solve the Schrödinger–like master equation at points where the potential is usually either zero (yielding simple plane wave solutions) or of Bessel type

(2.1) |

where is determined for each different case (for each type of perturbation and for each background considered). This equation can be solved in terms of Bessel functions, , with the result

(2.2) |

where and are constants. Let us list the values of which one has to deal with, when considering the black hole singularity region, at the origin of the coordinate frame. For all uncharged black hole solutions one finds for tensor type perturbations, for vector type perturbations and for scalar type perturbations. For all charged black hole solutions one finds for tensor type perturbations, for vector type perturbations and for scalar type perturbations. It is furthermore simple to observe that for all black hole spacetimes , , , and . This will ultimately imply that the asymptotic quasinormal frequencies will depend only on the background, being the same for the three types of perturbations. Let us also recall that one can prove (for spacetimes which are not asymptotically AdS), without explicit computation of the quasinormal frequencies, that and perturbations must have identical quasinormal spectra [21].

For spacetimes without a black hole (but with cosmological constant) the calculation of quasinormal frequencies follows in an analytic way, without any asymptotic restrictions. Here one need not use the monodromy method (even though it can be simply generalized to those cases as well) and we have proceeded by solving the wave equation directly. This is accomplished by first finding an appropriate change of variables that brings the quasinormal master equation to a hypergeometric form. Let us finally list our results on quasinormal frequencies:

The Schwarzschild Solution: This case was first studied in [13, 14] and we address it in this paper for the sake of completeness. For all types of perturbations, tensor, vector and scalar type perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where is the Hawking temperature in the Schwarzschild spacetime. As is well known, this case is particularly simple and one can moreover solve for the asymptotic quasinormal frequency as

The result is independent of spacetime dimension^{3}^{3}3By “independent” we mean that there is no explicit dependence on the spacetime dimension, , in the above formula. Of course if one wishes to compute the Hawking temperature in this Schwarzschild background, then one finds that it relates to the mass via an expression which also involves the dimension ..

The RN Solution: This case was studied, in the particular case, in [14]. Here we extend those results to arbitrary dimension. For all types of perturbations, tensor, vector and scalar perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where are the Hawking temperatures at outer and inner horizons (notice that ), and where

There is no known algebraic solution in for the above equation.

The Extremal RN Solution: It is important to realize that, in general, quasinormal frequencies of extremal solutions cannot be obtained from the corresponding expression for the non–extremal solution. In fact, the monodromy technique deployed in this paper is very sensitive to both the location of the complex horizons and the structure of the tortoise at the origin. Thus, as one changes the background solution there will be a change of topology in the complex plane and the solution to the quasinormal mode problem will be different. We present the extremal RN solution as an example, but one should keep this in mind if also interested in extremal solutions with a cosmological constant (which we do not address in this paper, but list the possibilities in appendix A). For all types of perturbations, tensor, vector and scalar perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where is as in the previous RN case and is not a temperature—in fact in the extremal case there is no Hawking emission. Rather, it is given by

where is related to the black hole mass (see appendix A). It is simple to solve for the asymptotic quasinormal frequency as

Observe that in dimension there is no solution for . This is in fact the only dimension where there is no solution for the asymptotic quasinormal frequencies of the extremal RN geometry.

The Schwarzschild dS Solution: This case was studied, in the particular case, in [16]. Here we extend those results to arbitrary dimension. For all types of perturbations, tensor, vector and scalar perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where is the Hawking temperature at the black hole event horizon and is the [negative] Hawking temperature at the cosmological horizon. There is no known algebraic solution in for the above equation. The result is independent of spacetime dimension, except in dimension where the formula above must be replaced by:

Observe that for this solution one can actually take the Schwarzschild limit without provoking any topology change in the complex plane where the monodromy analysis is performed. Thus, the result obtained for Schwarzschild dS includes the pure Schwarzschild solution once one sets the cosmological constant to vanish (both expressions have the same, correct, Schwarzschild limit).

The RN dS Solution: For all types of perturbations, tensor, vector and scalar perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where is as in the previous RN case, are the Hawking temperatures at outer and inner black hole event horizons and is the Hawking temperature at the cosmological horizon. There is no known algebraic solution in for the above equation. While this result explicitly depends on the spacetime dimension (because depends on ), the formula is not valid in dimension , where it must be replaced by:

Similarly to the Schwarzschild dS case, one can take the pure RN limit without provoking any topology change in the complex plane where the monodromy analysis is performed. Thus, the result obtained for RN dS includes the pure RN solution once one sets the cosmological constant to vanish (both expressions have the same, correct, RN limit).

The Schwarzschild AdS Solution: This case was studied, in the particular case of and large black hole, in [16]. It was further studied, in the particular case of and large black hole, in [22, 23, 24, 25] using a variety of different analytical methods. Here we extend those results for arbitrary dimension and away from the large black hole approximation. For all types of perturbations, tensor, vector and scalar perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where is a parameter related to the tortoise coordinate at spatial infinity and is given by

where the are the complex horizons and the the surface gravities at each of these complex horizons. There is no general analytic solution for . However, in some cases, it can be computed exactly. For instance, for large black holes we compute it to be

where is the Hawking temperature in the Schwarzschild AdS spacetime. In spite of not having a general analytic solution for one can still solve for the asymptotic quasinormal frequency as

If one concentrates on large Schwarzschild AdS black holes, of particular relevance to describe thermal gauge theories within the AdS/CFT framework, then the formulae above lead to the following analytical result for the leading term, as , in the asymptotic quasinormal frequencies:

For the most popular AdS/CFT dimensions, , and , one obtains

Notice that the above formulae are not valid for scalar type perturbations in dimensions four and five. Instead, one finds for these perturbations in dimension

and more surprisingly, a continuous spectrum in five dimensions,

The RN AdS Solution: For all types of perturbations, tensor, vector and scalar perturbations, the algebraic equation for the asymptotic quasinormal frequencies is the same and is

where is as before (only one should recall that this time around there are complex horizons as we are in a charged situation) and is as in the previous RN cases. Again, there is no general analytic solution for . In spite of this, one can still solve for the asymptotic quasinormal frequency as

Notice that the above formula is not valid for scalar type perturbations in dimensions four and five. Instead, one finds for these perturbations in dimension

and more surprisingly, a continuous spectrum in five dimensions,

Besides the previous results, concerning black hole spacetimes, we have also addressed the case of spacetimes without a black hole but with a cosmological constant, i.e., the cases of AdS (where there are only normal modes) and of dS (where one finds quasinormal modes only in odd spacetime dimensions). Let us list those results as well:

The AdS Solution: This case was studied before in [26] for a massless scalar field, and in [27] for tensor and vector type perturbations. Here we extend those results to arbitrary dimension and perturbation. The first thing to notice is that AdS spacetime acts as an infinite potential well and the Schrödinger–like equation yields real frequencies only. In other words, there are no quasinormal modes in pure AdS, only normal modes. The normal frequencies one finds are (no asymptotic restrictions here, this is an exact result)

where is the angular momentum quantum number (eigenvalue of the spherical laplacian) and is for tensor type perturbations, for vector type perturbations and for scalar type perturbations. Thus, for fixed , tensor and scalar type perturbations yield the same normal frequencies, although different from the vector type frequencies. Notice that the above formula is not valid for scalar type perturbations in dimensions four and five. Instead, one finds for these perturbations in dimension

and more surprisingly, a continuous spectrum in five dimensions,

The dS Solution: This case was studied in several different works, discussed in the main text, all with contradictory results. Here, we solve this apparent confusion in the literature with the following results. There are no dS quasinormal modes when the spacetime dimension is even. However, when the spacetime dimension is odd, there are quasinormal modes, with the quasinormal frequencies (no asymptotic restrictions here, this is an exact result)

where is the angular momentum quantum number (eigenvalue of the spherical laplacian) and is for tensor type perturbations, for vector type perturbations and for scalar type perturbations. Thus, for fixed , tensor and scalar type perturbations yield the same quasinormal frequencies, although different from the vector type frequencies. The above formula is independent of spacetime (odd) dimension.

Finally, we have studied the Rindler solution where we found—without great surprise—that there are no quasinormal modes. A detailed discussion concerning applications of these results to quantum gravity is included in the text, and we shall make no attempt of summarizing it in here.

## 3 Asymptotic Quasinormal Frequencies

In this section we first wish to review and set notation on both the perturbation theory for spherically symmetric, static --dimensional black holes^{4}^{4}4Throughout this work we consider only dimension ., with mass , charge and background cosmological constant , and the computation of quasinormal modes and quasinormal frequencies. We refer the reader to appendix A for a full list of conventions on the black hole spacetimes we shall consider. In the general case, there are two different types of fields which can be excited: these are the electromagnetic vector field and the gravitational metric tensor field , and we shall study perturbations to both these fields. Because there is no scalar field present in the Einstein–Maxwell system, there are no scalar field perturbations to consider. Nevertheless, we shall start by studying the scalar wave equation in our black hole backgrounds, in order to set notation on quasinormal modes.

Consider a massless, uncharged, scalar field, , in a background spacetime described by a metric . Its wave equation is well known

(3.3) |

where is the determinant of . The question we wish to address is what is the form of the wave equation for a background spacetime metric of the type

i.e., in a –dimensional spherically symmetric background. This issue was addressed in [28], with the following result. First perform a harmonic decomposition of the scalar field as , where the are the angles and the are the –dimensional spherical harmonics. Then, if

is the [time] Fourier decomposition of the scalar field, the wave equation can be recast in a Schrödinger–like form as

(3.4) |

where is the so–called tortoise coordinate and is the potential, both determined from the function in the background metric. The tortoise coordinate is defined so that [6, 7]

and is thus given by

(3.5) |

This new coordinate keeps infinity (or the cosmological horizon, , in the dS case) at and sends the black hole event horizon, , to (in the charged cases this refers to the outer horizon). The region of positivity of thus becomes the real line when in tortoise coordinates: for . The potential in the Schrödinger–like equation can be determined as one moves from the general form of the wave equation, (3.3), to its Schrödinger–like form, (3.4), and is given by [28]

Here, (with ) are the eigenvalues of the Laplacian on the sphere, and the potential still needs to be re–written in terms of the tortoise coordinate in order to be used in the Schrödinger–like equation. Once all this is done, the question still remains on what are the allowed values for , i.e., what is the spectrum of the Schrödinger–like operator above. It turns out that the spectrum will contain both a continuous and a discrete part, this last one being found when imposing “out going” boundary conditions: nothing arrives neither from infinity nor from within the black hole horizon. The [spherical] waves are out going at both extrema in :

Solutions to the Schrödinger–like equation with the previous “out going” boundary conditions lead to a discrete set of allowed frequencies, the quasinormal frequencies, with corresponding solutions, the quasinormal modes [6, 7]. These quasinormal frequencies, , are complex numbers, the real part representing the actual frequency of oscillation, the imaginary part representing the damping. It can be further shown for many cases that frequencies of quasinormal modes with negative imaginary part do not exist, meaning that these solutions are actually stable. Our interest here is to study the asymptotic behavior of quasinormal frequencies, i.e., the case where the imaginary part of grows to infinity. It turns out that in some cases the real part of the frequency approaches a finite limit, while the imaginary parts grow linearly without bound. These generics on quasinormal modes will obviously hold true for the vector field perturbations and metric tensor field perturbations, the main difference being the change in the potential of the Schrödinger–like equation. A complete description of the required potentials to describe the most general situation, as derived in [8, 10], is presented in appendix B, to which we refer the reader for further details.

For a four dimensional Schwarzschild black hole, one has the asymptotic quasinormal frequencies

where the real part of the offset is the frequency of the emitted radiation, and the gap are the quantized increments in the inverse relaxation time. Here, the gap is given by the surface gravity. One can try to extend this analysis to more general situations and also include spacetimes with two horizons, but then generic results become much harder to obtain [29, 30, 31, 32] (curiously, for spacetimes with multiple horizons, there is a unique definition of temperature only when the ratio of surface gravities is in [32, 33]). Another property of quasinormal modes is a reflection symmetry which changes the sign of . Indeed, is a quasinormal mode corresponding to .

We have chosen the time dependence for the perturbation to be , so that for stable solutions (implying that the perturbation vanishes exponentially in time). This implies that as and thus, while solutions can oscillate, they must exponentially increase with . In other words, is not a normalizable wave function and this is, ultimately, the reason why we speak of quasinormal modes rather than normal modes (which would form a complete set of stationary eigenfunctions for the IK master differential operator).

Before embarking in the actual calculation, let us address the general strategy of the method introduced in [14], which begins with the question of how to properly define and impose quasinormal boundary conditions. This is known to be somewhat complicated, at least at an operational level, because as long as we restrict the quasinormal boundary conditions above amount to distinguishing between an exponentially vanishing term and an exponentially growing term. The idea of [14] is to do an analytic continuation to the complex plane, taking both and . We will see that in the complex plane one can impose quasinormal boundary conditions in a completely new way. Indeed, if one picks the complex contour in , then the asymptotic behavior of is always oscillatory on this line and there will never be any problems with exponentially growing versus exponentially vanishing terms. One should thus restrict to studying the boundary conditions on the so–called Stokes line, .

For Schwarzschild, RN, Schwarzschild dS and RN dS black hole spacetimes, numerical tests generically indicate that the asymptotic quasinormal frequencies are such that . In other words, one should have , with , and , in the asymptotic case. Thus, in the limit, the contour can be approximated by the curve which is immediate to plot in . This replacement of contours is selecting asymptotic conditions for the quasinormal modes. For Schwarzschild AdS and RN AdS things are different, as numerical tests generically indicate that asymptotic quasinormal frequencies behave as , and thus one should have , with , and , in the asymptotic case. We shall later see with greater detail how to plot the Stokes lines for each black hole spacetime under consideration.

To fully understand the advantage of the analytic continuation in the exact computation of asymptotic quasinormal frequencies, let us first address regions of the complex –plane around an event horizon (be it a black hole horizon or a cosmological horizon). The horizons themselves are defined by the zeroes of , i.e., , and besides the physical real horizons there can be other, non–physical complex horizons . We shall denote horizons which are not real as fictitious horizons. Power series expansion near any of these horizons simply yields , and it follows for the tortoise coordinate

(3.6) |

locally near the chosen horizon (provided that the horizon is nondegenerate, i.e., that is a simple zero of ). Here is the surface gravity and is the Hawking temperature.

One learns that around any nondegenerate horizon the tortoise coordinate will thus be multivalued and it makes sense to ask for the monodromy of tortoise plane waves in clockwise contours around any given horizon . Let us consider an horizon , and a clockwise contour centered at and not including any other horizon. As we take the tortoise coordinate around it increases by . This immediately implies the monodromy of the plane waves along the selected contour:

This result is quite interesting as it now allows one to recast quasinormal boundary conditions as monodromy conditions [14]; if one wants, as quasinormal mode monodromy conditions. At the black hole event horizon the quasinormal boundary condition is

where as , with the black hole horizon. One immediately re–writes this boundary condition as a monodromy condition for the solution of the master equation at the black hole event horizon:

with a clockwise contour. The other quasinormal boundary condition,

lives at , as when (for all spacetimes except asymptotically dS spacetimes). Thus, this boundary condition cannot be recast as a quasinormal monodromy condition. Instead, if one considers asymptotically dS spacetimes, the above boundary condition is located at the cosmological event horizon, as when . Then, it is immediate to recast this boundary condition as a monodromy condition for the solution of the master equation at the cosmological event horizon:

with a clockwise contour. We shall see in the following how these simple ideas about boundary conditions will allow for analytic calculations of asymptotic quasinormal frequencies in all static, spherically symmetric black hole spacetimes.

### 3.1 Asymptotically Flat Spacetimes

[10] discusses the stability of black holes in asymptotically flat spacetimes to tensor, vector and scalar perturbations. For black holes without charge, all types of perturbations are stable in any dimension. For charged black holes, tensor and vector perturbations are stable in any dimension. Scalar perturbations are stable in four and five dimensions but there is no proof of stability in dimension . As we work in generic dimension we are thus not guaranteed to always have a stable solution. Our results will apply if and only if the spacetime in consideration is stable.

#### 3.1.1 The Schwarzschild Solution

For completeness, we first present a computation of the asymptotic quasinormal frequencies for the Schwarzschild –dimensional black hole, using the monodromy method. This calculation was first done in [14]. The interesting result shown in [14] is that, if one is only interested in the asymptotic quasinormal modes, there is no need to solve the IK master equation exactly. Rather, there is a method which explores the analytic continuation of the master equation to the complex plane and demands only for approximate solutions near infinity, near the origin, and near the black hole event horizon. Knowledge of the solutions in these regions, together with monodromy matching along a specially chosen contour, then yields the quasinormal frequencies. Let us carefully explain this method in the simplest Schwarzschild example, as we shall employ it several times in the following.

We consider solutions of the Schrödinger–like master equation (3.4)

in the complex –plane. Let us begin at infinity. Since the potential vanishes for , we will have

in this region. The boundary condition for quasinormal modes at infinity is then

(3.7) |

Next we study the behavior of near the singularity . In this region, the tortoise coordinate is

and the potential for tensor and scalar type perturbations is

with (see appendix C). The Schrödinger–like master equation approximates to

whose solution is (this is the solution for —more on this in a moment)

where represents a Bessel function of the first kind and are (complex) integration constants. One would next like to link this solution at the origin with the solution at infinity.

For the Schwarzschild asymptotic quasinormal modes one has , with , and hence is very large and approximately purely imaginary. Consequently, one has for ; in a neighborhood of the origin, the above relation between and tells us that this happens for

with and . These are half–lines starting at the origin, equally spaced by an angle of . Notice that the sign of on these half–lines is ; in other words, starting with the half–line corresponding to , the sign of is alternately positive and negative as one goes anti–clockwise around the origin.

Precisely because we are interested in these asymptotic modes, we may consider the following asymptotic expansion of the Bessel functions

(3.8) |

from where we learn that

(3.9) | |||||

in any one of the lines corresponding to positive , and where we have defined

This asymptotic expression for near the origin is ideal to make the matching with its asymptotic expression at infinity. This matching must however be done along the so–called Stokes line, defined by (or ), so that neither of the exponentials dominates the other. In this Schwarzschild case the Stokes line definition corresponds to or .

To trace out the Stokes line let us first observe that we already know its behavior near the origin. Furthermore, this is the only singular point of this curve: indeed, since is a holomorphic function of , the critical points of the function are the zeros of

(i.e., only). We have an additional problem that is a multivalued function: each of the “horizons”

is a branch point. In fact, near such points one has (see (3.6))

Thus we see that although the function is well defined around with and (if is odd), as is real in these cases, it will be multivalued around all the other fictitious horizons.

For one has (see appendix C). Consequently, is holomorphic at infinity and we can choose the branch cuts to cancel out among themselves. Therefore is well defined in a neighborhood of infinity, and moreover will be approximately parallel to in this neighborhood. Two of the branches of the Stokes line starting out at the origin must therefore be unbounded. The remaining can either connect to another branch or end up in a branch cut. On the other hand, it is easy to see that the Stokes line must intersect the positive real axis exactly in one point, greater than . Using this information plus elementary considerations of symmetry and the sign of , one can deduce that the Stokes line must be of the form indicated in Figure 1. These results are moreover verified by the numerical computation of the same Stokes lines, as indicated in Figure 2.

Let us now consider the contour obtained by closing the unlimited portions of the Stokes line near , as shown in Figure 1. At point we have , and therefore the expansion (3.9) holds at this point. Imposing condition (3.7) one obtains

(3.10) |

For one has the expansion

(3.11) |

where is an even holomorphic function. Consequently, as one rotates from the branch containing point to the branch containing point , through an angle of , rotates through an angle of , and since

one has (notice that )

at point . As one closes the contour near , one has and hence ; since , it follows that is exponentially small in this part of the contour, and therefore only the coefficient of should be trusted. As one completes the contour, this coefficient gets multiplied by

On the other hand, the monodromy of going clockwise around this contour is , where

is the surface gravity at the horizon. Since

for , will increase by , and hence will get multiplied by as one goes clockwise around

Thus the clockwise monodromy of around the contour depicted in Figure 1 is

The important point to realize now is that one can deform this chosen contour—without crossing any singularities—so that it becomes a small clockwise circle around the black hole event horizon . Near the horizon , we have again , and hence again

The condition for quasinormal modes at the horizon is therefore . Again using the fact that

for , we can restate this boundary condition as the condition that the monodromy of going clockwise around the contour should be

Because the monodromy is invariant under this deformation of the contour, the condition for quasinormal modes at the horizon follows as

(3.12) |

The final condition for equations (3.10) and (3.12), which reflect quasinormal mode boundary conditions both at infinity and at the black hole horizon, to have nontrivial solutions is simply

(3.13) |

This equation is automatically satisfied for . This is to be expected, as for the Bessel functions coincide and do not form a basis for the space of solutions of the Schrödinger–like master equation near the origin. Following [14], we consider this equation for nonzero and take the limit as . This amounts to writing the equation as a power series in and equating to zero the first non–vanishing coefficient, which in this case is the coefficient of the linear part. Thus we just have to require that the derivative of the determinant above with respect to be zero for :

For vector type perturbations, the potential for the Schrödinger–like master equation near the origin is of the form

with (see appendix C). Repeating the same argument, one ends with the same equation as (3.13), except that now , . This equation is exactly the same as in the case, for which , , and consequently we end up with the precise same quasinormal frequencies.

In the calculation above we have obtained the asymptotic quasinormal frequencies with . However, one knows that if is a quasinormal frequency then so is . Consequently (for ) must also be a solution for the asymptotic quasinormal frequencies. To understand how these comes about, and why we did not obtain them above, we recall that we imposed condition (3.7) at point , where . Notice that we could instead have imposed it at point , where . It is in choosing one of these points that we automatically choose the sign of . Indeed, if then at point and at point . Consequently, only at point is it meaningful to impose condition (3.7). On the other hand, if then at point and at point , and only at point is it meaningful to impose condition (3.7). Had we done this, we would have obtained the second set of asymptotic quasinormal frequencies.

The results above have been thoroughly checked in the literature. Actually, an analytic calculation of the asymptotic quasinormal frequencies was first done in [13], in four dimensions, and then extended to –dimensions in [14], at least for tensor type perturbations (and already using the monodromy method). The result was shown to be dimension independent and equal to . It was already expected that these –dimensional frequencies would scale linearly with the Hawking temperature, from the earlier results in [34]. Later, in [35], it was shown that the same result of holds for both vector and scalar perturbations. All these results were later subject to a detailed numerical check in [36, 37], with fully positive results.

#### 3.1.2 The Reissner–Nordström Solution

Here we compute the asymptotic quasinormal modes of the RN –dimensional black hole using the monodromy method. This calculation was done for in [14]. We consider solutions of the Schrödinger–like equation (3.4) in the complex –plane. Since the potential vanishes for , we will again have

in this region, the boundary condition for quasinormal modes at infinity being

(3.14) |

Next we study the behavior of near the singularity . In this region, the tortoise coordinate is

and the potential for tensor type and scalar type perturbations is

with (see appendix C). The solution of the Schrödinger–like equation in this region is therefore well approximated by

where represents a Bessel function of the first kind and are (complex) integration constants.

For the asymptotic quasinormal modes one has , and hence is approximately purely imaginary. Consequently, one has for ; in a neighborhood of the origin, this happens for

with and . These are half–lines starting at the origin, equally spaced by an angle of . Notice that the sign of on these lines is ; in other words, starting with the line corresponding to , the sign of is alternately negative and positive as one goes anti–clockwise around the origin.

From the asymptotic expansion (3.8) we see that

(3.15) | |||||

in any one of the lines corresponding to positive , where again we define

This asymptotic expression for near the origin is to be matched with its asymptotic expression at infinity. This matching must again be done along the Stokes line (that is, ), so that neither of the exponentials dominates the other.

To trace out the Stokes line we observe that we already know its behavior near the origin. Furthermore, this is the only singular point of this curve: indeed, since is a holomorphic function of , the critical points of the function are the zeros of

(i.e., only). We have the additional problem that is a multivalued function: each of the “horizons”

(see appendix C) is a branch point. From (3.6), we see that although the function is still well defined around with and (if is odd), as is real in these cases, it will be multivalued around all the other fictitious horizons.

For one has and hence . Consequently is holomorphic at infinity, and we can choose the branch cuts to cancel out among themselves. Therefore is well defined in a neighborhood of infinity, and moreover will be approximately parallel to in this neighborhood. Two of the branches of the Stokes line starting out at the origin must therefore be unbounded. The remaining can either connect to another branch or end up in a branch cut. On the other hand, it is easy to see that the Stokes line must intersect the positive real axis exactly in two points, one in each of the intervals and <