Unstable circular null geodesics of static spherically symmetric black holes,
Regge poles and quasinormal frequencies
Abstract
We consider a wide class of static spherically symmetric black holes of arbitrary dimension with a photon sphere (a hypersurface on which a massless particle can orbit the black hole on unstable circular null geodesics). This class includes various spacetimes of physical interest such as Schwarzschild, SchwarzschildTangherlini and ReissnerNordström black holes, the canonical acoustic black hole or the Schwarzschildde Sitter black hole. For this class of black holes, we provide general analytical expressions for the Regge poles of the matrix associated with a massless scalar field theory. This is achieved by using thirdorder WKB approximations to solve the associated radial wave equation. These results permit us to obtain analytically the nonlinear dispersion relation and the damping of the “surface waves” lying close to the photon sphere as well as, from BohrSommerfeld–type resonance conditions, formulas beyond the leadingorder terms for the complex frequencies corresponding to the weakly damped quasinormal modes.
pacs:
04.70.s, 04.50.GhI Introduction
Quasinormal modes (QNMs) of black holes (BHs) have been studied for nearly 40 years due to their importance in the context of gravitational wave astronomy. In the last decade, there has been moreover an increase of activity in BH QNM studies motivated by potential applications in analog models of gravity, quantum gravity, string theory and related topics (TeVscale gravity, AdS/CFT correspondence, alternative theories of gravity, BH area quantization, phase transitions in BH systems,…). For excellent reviews on the status of QNMs prior to 1999 and on their relevance to gravitational wave astronomy, we refer to the articles by Kokkotas and Schmidt Kokkotas and Schmidt (1999) and by Nollert Nollert (1999). For a more recent review on BH QNMs, we refer to the article by Berti, Cardoso and Starinets Berti et al. (2009): it updates the two previously cited articles and it also presents the aspects of QNM physics linked to gaugegravity duality; it includes, furthermore, an interesting historical introduction on the subject as well as a useful impressive bibliography on all the aspects of BH physics linked to QNMs.
Immediately after the publication of one of the first papers on QNMs by Press Press (1971) where he identified the gravitational ringing of the Schwarzschild BH as due to its “free oscillations”, Goebel suggested a physically intuitive interpretation of the associated QNMs Goebel (1972): they could be interpreted in terms of gravitational waves in spiral orbits close to the unstable circular photon/graviton orbit at which decay by radiating away energy (here denotes the mass of the BH). This appealing interpretation has been developed by other authors for various field theories defined on BH backgrounds using the eikonal approximation, i.e., in a framework based on geodesics and bundle of geometrical rays (see Refs. Ferrari and Mashhoon (1984); Mashhoon (1985); Stewart (1989); Andersson and Onozawa (1996); Zerbini and Vanzo (2004); Cardoso et al. (2009); Dolan and Ottewill (2009); Hod (2009) as well as Ref. Sá Barreto and Zworski (1997) for a more mathematical approach). It has permitted them to obtain analytical approximations for the leadingorder terms of the characteristic complex frequencies of various BH spectra from an interpretation in terms of massless particles “trapped” near unstable circular null geodesics (see, more particularly, Ref. Cardoso et al. (2009) where the relation with the Lyapunov exponent corresponding to geodesic motion is clearly emphasized).
A potentially much richer implementation of the Goebel interpretation of BH QNMs which is not limited to purely geometrical considerations but based on wave/field theory and which goes beyond the leading oder terms has also been formulated Décanini et al. (2003); Décanini and Folacci (2009, 2010) (see also Ref. Dolan and Ottewill (2009)). It uses complex angular momentum (CAM) techniques (or, in other words, the Regge pole machinery) which play a central role in scattering theory. Since, as noted by Chandrasekhar and coworkers Chandrasekhar (1983) (see also Ref. Ferrari (1992)), BH perturbation theory can be formulated as a resonant scattering problem, CAM techniques arise naturally in BH physics. For reviews of the CAM method, we refer to the monographs of Newton Newton (1982), Nussenzveig Nussenzveig (1992) and Collins Collins (1977) as well as to references therein for various applications in quantum mechanics, nuclear physics, high energy physics, electromagnetism and seismology.
Some years ago, the CAM method was used in gravitational wave physics by Chandrasekar and Ferrari Chandrasekhar and Ferrari (1992) to express the flow of energy due to nonradial oscillations of relativistic stars and by Andersson and Thylwe to describe scattering from the Schwarzschild BH Andersson and Thylwe (1994) as well as to interpret the Schwarzschild BH glory Andersson (1994). In this context, Andersson established, for the Schwarzschild BH of mass , the existence of a family of “surface waves” (each one associated with a Regge pole of the matrix) orbiting close to the unstable photon orbit at (see also Ref. Décanini et al. (2003) for a more rigorous approach). Recently, from these “surface waves”, we have been able to theoretically and numerically construct the spectrum of the weakly damped complex frequencies of the Schwarzschild BH QNMs Décanini et al. (2003); Décanini and Folacci (2010) and to interpret them as BreitWigner resonances. This has been achieved by obtaining analytically the nonlinear dispersion relation as well as the damping of the “surface waves” propagating close to the photon sphere. Let us also note two related papers concerning analytical or numerical determinations of the Regge poles of the Schwarzschild BH Glampedakis and Andersson (2003); Dolan and Ottewill (2009) and that, this last year, Regge poles have also been used to understand the resonant aspects of the BTZ BH Décanini and Folacci (2009) as well as to analyze some aspects of selfforce calculations Casals et al. (2009).
In the present paper, we extend the analysis developed for the Schwarzschild BH to more general BHs and we establish, from Regge pole considerations, a precise connection between the existence of a photon sphere and the properties of the “surface waves” propagating close to it. More precisely, we consider a wide class of static spherically symmetric BHs of arbitrary dimension with a photon sphere, i.e., a hypersurface on which a massless particle can orbit the BH on unstable circular null geodesics. For more rigorous definitions of the photon sphere concept in static spherically symmetric spacetime, we refer to the article by Claudel, Virbhadra and Ellis Claudel et al. (2001). This class of BHs includes various spacetimes of physical interest such as Schwarzschild, SchwarzschildTangherlini and ReissnerNordström BHs, the canonical acoustic BH or the Schwarzschildde Sitter BH. For this class of BHs, we provide general analytical expressions beyond the leadingorder terms for the Regge poles of the matrix associated with a massless scalar field theory. These results permit us to obtain analytically the nonlinear dispersion relation and the damping of the “surface waves” lying close to the photon sphere as well as, from BohrSommerfeld–type resonance conditions, the complex frequencies corresponding to the weakly damped QNMs.
Our paper is organized as follows. In Sec. II, we display our general working assumptions and we justify them physically. We then explain how to construct the matrix permitting us to analyze the resonant aspects of a scalar field theory defined on an asymptotically flat static spherically symmetric BH of arbitrary dimension with a photon sphere and we finally define its Regge poles as well as its complex quasinormal frequencies. In Sec. III, we provide a general analytical expression for the Regge poles. This is achieved by using and extending the WKB approach developed in the context of the determination of the QNMs by Schutz and Will Schutz and Will (1985) and by Will and Iyer Iyer and Will (1987); Iyer (1987) (see also Ref. Bender and Orszag (1978) for general aspects of WKB theory and for particular aspects connected with eigenvalue problems). Our result permits us to describe the Regge trajectories of a general asymptotically flat static spherically symmetric BH of arbitrary dimension with a photon sphere and to obtain, from semiclassical formulas, analytical expressions for the QNM complex frequencies. Our WKB analysis permits us moreover to show that (i) the dispersion relation of the th “surface wave” is nonlinear and depends on the index and that (ii) the damping of the th “surface wave” is frequency dependent. In Sec. IV, we apply the general theory developed in Sec. III to particular BHs (Schwarzschild, SchwarzschildTangherlini, ReissnerNordström and canonical acoustic BHs). In a brief conclusion, we consider some consequences of our work as well as possible extensions. In Appendix A, we establish the semiclassical connection between the Regge poles of a static spherically symmetric BH of arbitrary dimension with a photon sphere and the complex frequencies of its weakly damped QNMs. In Appendix B, we consider the particular case of the Schwarzschildde Sitter BH. Indeed, even if such a gravitational background is not asymptotically flat, the formalism developed in Secs. II and III naturally applies to it.
In this paper, we shall use units with .
Ii Quasinormal frequencies and Regge poles of static spherically symmetric black holes: General theory
We consider a static spherically symmetric spacetime of arbitrary dimension with metric
(1) 
Here denotes the line element on the unit sphere . On , we introduce the usual angular coordinates with and . We have
(2) 
Of course, a metric such as (1) does not describe the most general static spherically symmetric spacetime but it will permit us to consider a wide class of BHs of physical interest.
In Eq. (1), we shall furthermore assume that is a function of the usual radial coordinate with the following properties:

(i) There exists an interval with such as for .

(ii) is a simple root of , i.e.,
(3) and moreover satisfies
(4) 
(iii) There exists a value for which
(5) and
(6)
We shall now briefly discuss assumptions (i)(iii) previously introduced. Assumptions (i) and (ii) indicate that the spacetime considered is an asymptotically flat BH with an event horizon at , its exterior corresponding to . Assumption (iii) implies the existence of a photon sphere which is the support of unstable circular null geodesics (see below for more details). It should be noted that, as a consequence of (i) and (ii), the tortoise coordinate defined for by the relation and the condition provides a bijection from to .
Let us consider a freefalling massless particle orbiting the BH. Without loss of generality, we can consider that its motion lies on the equatorial hyperplane defined by for . Because it moves along a null geodesic, we have [cf. Eqs. (1) and (2)]
(7) 
where is an affine parameter and, of course, there exist two integrals of motion respectively associated with the Killing vectors and and given by
(8a)  
(8b) 
Here and denote respectively the energy and the angular momentum of the massless particle. Inserting Eqs. (8a) and (8b) into (7), we obtain
(9) 
where
(10) 
From these last two equations and from assumption (iii), one can easily remark that the massless particle can orbit the BH on an unstable circular geodesic defined by . Indeed, we have in particular
(11a)  
and  
(11b) 
On this orbit, the massless particle takes the time
(12) 
to circle the BH. This result can be obtained by integrating Eq. (7).
The wave equation for a massless scalar field propagating on a general gravitational background is given by
(13) 
If the spacetime metric is given by (1), after separation of variables and the introduction of the radial partial wave functions with , this wave equation reduces to the ReggeWheeler equation
(14) 
[Here we have assumed a harmonic time dependence for the massless scalar field.] In Eq. (14), is the ReggeWheeler potential given by
(15)  
It should be noted that

and and therefore the solutions of the radial equation (14) have a behavior in at the horizon and at infinity.

For , has a local maximum at because, in this limit, and are similar.

For any finite value of , the local maximum of is close to .
For a given angular momentum index , the matrix element is defined by seeking the solution of the ReggeWheeler equation (14) which has a purely ingoing behavior at the event horizon , i.e., which satisfies
(16) 
and which, at spatial infinity , presents an asymptotic behavior of the form
(17) 
We recall that the matrix permits us to analyze the resonant aspects of the considered BH as well as to construct the form factor describing the scattering of a monochromatic scalar wave (see Appendix A).
To describe semiclassically resonance phenomena, the dual structure of the matrix plays a crucial role. Indeed, the matrix is a function of both the frequency and the angular momentum index . It can be analytically extended into the complex plane as well as into the complex plane (CAM plane) with . From now on, we shall denote by this double analytical extension. For , the simple poles lying in the fourth quadrant of the complex plane [let us recall that they are also simple poles of ] are the complex frequencies of the QNMs. These modes are therefore solutions of the radial wave equation (14) which are purely outgoing at infinity and purely ingoing at the horizon. We shall denote by where the quasinormal frequencies. We recall that and represent respectively the frequency of the oscillation and the damping corresponding to the associated QNM. We assume that, in the immediate neighborhood of , has the BreitWigner form, i.e.,
(18) 
For a given value of the frequency, the simple poles lying in the first quadrant of the complex plane are the socalled Regge poles. It should be noted that they are also poles of and therefore the associated modes (Regge modes) are purely outgoing at infinity and purely ingoing at the horizon. We shall denote the Regge poles by , the index permitting us to distinguish each pole.
The structure of the matrix in the complex plane allows us, by using integration contour deformations, Cauchy’s theorem and asymptotic analysis, to provide a semiclassical description of scattering (see Appendix A for more precisions). The curves traced out in the CAM plane by the Regge poles as a function of the frequency are the socalled Regge trajectories. They permit us to interpret Regge poles in terms of “surface waves” (see Appendix A): provides the dispersion relation for the th “surface wave” while corresponds to its damping. Furthermore, from the Regge trajectories, we can semiclassically construct the resonance spectrum [see, in Appendix A, formulas (88), (90) and (91)]. The semiclassical formula (a BohrSommerfeld–type quantization condition)
(19) 
provides the location of the excitation frequencies of the resonances generated by th “surface wave”, while a second semiclassical formula gives the widths of these resonances
(20) 
It should be moreover noted that this formula reduces, in the frequency range where the condition is satisfied, to
(21) 
Iii WKB approximations for the Regge poles and semiclassical expressions of the complex quasinormal frequencies
In general, it is not possible to solve exactly the ReggeWheeler equation (14) and therefore we can obtain only analytical approximations for the Regge poles and for the complex quasinormal frequencies. For example, the WKB approach developed in the general context of eigenvalue problems (for more details see Ref. Bender and Orszag (1978)) has been adapted for the determination of the Schwarzschild BH QNMs by Schutz and Will Schutz and Will (1985) and by Will and Iyer Iyer and Will (1987); Iyer (1987) and, in Ref. Décanini and Folacci (2010), for the determination of the Schwarzschild BH Regge poles. It can be extended to the more general case considered in this paper. By using thirdorder WKB approximations Iyer and Will (1987); Iyer (1987) for the Regge modes of Eq. (14), we find that the Regge poles are the complex solutions of the equation
(22) 
with and . Here
(23a)  
and  
(23b) 
In Eqs. (III) and (23), we have introduced the notations
(24) 
and, for ,
(25) 
with which denotes the maximum of the function .
In order to solve Eq. (III), we need to express the asymptotic expansions for of and and of the various ratios appearing in Eqs. (23) and (23). In order to simplify the results we have introduced the notations
(26) 
and
(27) 
It is worth noting that the parameter is directly linked to the second derivative (11b) of the effective potential (10) taken at . As a consequence, it represents a kind of measure of the instability of the circular orbits lying on the photon sphere. In fact, it can be expressed in terms of the Lyapunov exponent corresponding to these orbits introduced in Ref. Cardoso et al. (2009) and which is the inverse of the instability time scale associated with them: we have
(28) 
We will say no more about this connection because, as already mentioned in Sec. I, we intend to go beyond purely geometrical considerations in our analysis of the resonant behavior of BHs.
After a tedious calculation, we obtain
(29a)  
and  
(29b)  
as well as
(30a)  
(30b)  
(30c)  
(30d)  
(30e) 
We can now solve Eq. (III) by assuming as well as . We obtain for the solutions a family with given by the approximation
(31) 
where
and
(33) 
with