Scattering of scalar waves by rotating black holes

Transcription

1 Scattering of scaar waves by rotating back hoes Kostas Gampedakis 1 and Nis Andersson 2 1 Department of Physics and Astronomy, Cardiff University, Cardiff CF2 3YB, United Kingdom 2 Department of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom We study the scattering of massess scaar waves by a Kerr back hoe by etting pane monochromatic waves impinge on the back hoe. We cacuate the reevant scattering phase-shifts using the Prüfer phase-function method, which is computationay efficient and reiabe aso for high frequencies and/or arge vaues of the anguar mutipoe indices (,m). We use the obtained phase-shifts and the partia-wave approach to determine differentia cross sections and defection functions. Resuts for off-axis scattering (waves incident aong directions misaigned with the back hoe s rotation axis) are obtained for the first time. Inspection of the off-axis defection functions reveas the same scattering phenomena as in Schwarzschid scattering. In particuar, the cross sections are dominated by the gory effect and the forward (Couomb) divergence due to the ong-range nature of the gravitationa fied. In the rotating case the overa diffraction pattern is frame-dragged and as a resut the gory maximum is not observed in the exact backward direction. We discuss the physica reason for this behaviour, and expain it in terms of the distinction between prograde and retrograde motion in the Kerr gravitationa fied. Finay, we aso discuss the possibe infuence of the so-caed superradiance effect on the scattered waves. I. INTRODUCTION Diffraction of scattered waves provides the expanation for many of Nature s most beautifu phenomena, such as rainbows and gories. It has ong been recognized that these optica phenomena have anaogies in many other branches of physics. They are of particuar reevance to quantum physics, where pane wave beams are routiney used to probe the detais of atoms, nucei or moecues. Such experiments provide a deep understanding of the scatterer s physics and can be used as a powerfu test of various theoretica modes. The anaogy can be extended aso to gravitationa physics and extreme astrophysica objects ike back hoes. In fact, back hoe scattering has been the subject of a considerabe amount of work carried out over the ast 3 years (see [1] for an extensive review). In the case of astrophysica back hoes it is unikey that the various diffraction effects wi ever be observed (athough it is not entirey impausibe that advances of current technoogy wi eventuay enabe us to study interference effects in gravitationay ensed waves). However, it is nevertheess usefu to have a detaied theoretica understanding of the scattering of waves from back hoes. After a, a study of these probems provides a deeper insight into the physics of back hoes as we as wave-propagation in curved spacetimes. The benchmark probem for back-hoe scattering is massess scaar waves impinging on a Schwarzshid back hoe. This probem is we understood [2 8], and it is known that it provides a beautifu exampe of the gory effect. Hander and Matzner [9] have shown that the situation remains amost unchanged if, instead of scaar waves, one decides to shoot pane eectromagnetic or gravitationa waves towards the back hoe. These authors have aso considered on-axis scattering of gravitationa waves in the case when the back hoe is rotating [9]. Their resuts suggest that the scattering cross sections consist of essentiay the same features as in the non-rotating case (there is a forward divergence due to the ong-range nature of the gravitationa fied and a backward gory). In addition, they find some pecuiar features that are, at the present time, not we understood. An expanation of these effects is compicated by the fact that they coud be caused by severa effects, the most important being the couping between the back hoe s spin and the spin/poarisation of the incident wave. Given that the avaiabe investigations have not been abe to distinguish between these various effects, we fee that our current understanding is somewhat unsatisfactory. This feeing is enhanced by the fact that no resuts for the most reaistic case, corresponding to off-axis incidence, have yet been obtained. This paper provides an attempt to further our understanding of the scattering from rotating back hoes. Our aim is to isoate those scattering effects that are due to the spin of the back hoe. In order to do this, we focus our attention on the scattering of massess scaar waves. For this case, the infaing waves have neither spin nor poarisation and therefore one woud expect the scattered wave to have a simper character than in the physicay more reevant case of gravitationa waves. However, one can be quite certain that the features discussed in this paper wi be present aso in the case of gravitationa waves. It is, after a, we known that the propagation of various fieds in a given back hoe geometry is described by very simiar wave equations. 1

2 Athough we wi re-examine the case of axiay incident waves, our main attention wi be on the more interesting off-axis scattering cross sections. These cross sections turn out to be quite different from the ones avaiabe in the iterature. Obviousy, they have two degrees of freedom (corresponding to the two anges θ and ϕ in Boyer-Lindquist coordinates). In addition we wi show that the cross sections are asymmetric with respect to the incidence direction. In particuar, the gory moves away from the backward direction as a resut of rotationa frame dragging that provides a distinction between prograde and retrograde motion in the Kerr geometry. We construct our differentia cross sections using the we-known partia wave decomposition (for an introduction see [1]) the standard approach in quantum scattering theory. That this method is equay usefu in back-hoe scattering is we estabished [1]. We shoud point out, however, that aternatives (such as the compex-anguar momentum approach [11,12] and path-integra methods [7,13,14]) have aso been succesfuy appied to the back-hoe case. In the partia wave picture a scattering information is contained in the radia wavefunction s phase shifts. The cacuation of these phase-shifts must, apart from in exceptiona cases ike Couomb scattering, be performed numericay. Various techniques have been deveoped for this task. Basicay, one must be abe to determine the phaseshifts accuratey up to sufficienty arge partia waves that no interference effects are ost. This bois down to a need for many more mutipoes to be studied as the frequency of the infaing wave is increased. In back-hoe scattering severa methods have been empoyed for the phase-shift cacuation: Matzner and Ryan [2] numericay integrated the reevant radia wave equation (Teukosky s equation). Since the desired soution is an osciating function, this cacuation becomes increasingy difficut (and time consuming) as the frequency is increased. Consequenty, Matzner and Ryan restricted their study of eectromagnetic and gravitationa wave scattering to ωm.75 and 1. In order to avoid this difficuty, Hander and Matzner [9] combined a numerica soution in the region where the gravitationa curvature potentia varies rapidy, with an approximate WKB soution for reativey arge vaues of the radia coordinate. This trick aowed them to perform cacuations for 2 and ωm 2.5. Some years ago one of us used the phase-integra method [15,16] to derive an approximate formua for the phase-shifts in the context of Schwarzschid scattering [8]. This formua was shown to be reiabe and efficient even for high frequencies and/or arge vaues (in [8] resuts for ωm =1and 2 were presented). This means that the differentia cross sections determined from the phase-integra phase-shifts were reiabe aso for rather high frequencies. Even though the phaseintegra formua coud be generaised to scattering by a Kerr back hoe and therefore used for the purposes of the present study, we have chosen a different approach here. Our phase-shift determination is based on the so-caed Prüfer method (we-known in quantum scattering theory [17,18] and, in genera, in numerica treatments of Sturm-Liouvie probems [19] ) which, in a nutshe, invoves transforming the origina radia wavefunction to specific phase-functions and numerica integration of the resuting equations. In essence, this method is a cose reative of the phase-ampitude method that was devised by one of us to study back-hoe resonances [2]. The remainder of the paper is organised as foows. In Sections IIA and IIB the probem of scattering by a Kerr back hoe is rigorousy formuated. In Section IIC the important notion of the defection function is discussed. Section III is devoted to our numerica resuts. First, in Section IIIA our numerica method for cacuating phase-shifts is presented. In Section IIIB famiiar Schwarzschid resuts are reproduced as a code vaidation. Sections IIIC and IIID contain entirey new information: Differentia cross sections and defection functions for on and off-axis scattering respectivey. These are the main resuts of the paper. Furthermore, in Section IIIE we present numerica resuts concerning forward gories. The roe of superradiance for scattering of monochromatic waves is discussed in Section IIIF. Our concusions are briefy summarised in Section IV. Three appendices are devoted to technica detais, which are incuded for competeness. In Appendix A we discuss the notion of pane waves in the presence of a gravitationa fied. In Appendix B the partia-wave decomposition of a pane wave in the Kerr background is determined, and finay in Appendix C we briefy describe the method we have used to cacuate the spin- spheroida harmonics and their eigenvaues. Throughout the paper we adopt geometrised units (c = G = 1). II. SCATTERING FROM BLACK HOLES A. Formuation of the probem We consider a massess scaar fied in the Kerr back-hoe geometry. Then, first-order back-hoe perturbation theory, basicay the Teukosky equation [21], appies. The scaar fied satisfies the curved spacetime wave equation Φ =. Adopting standard Boyer-Lindquist coordinates we can aways decompose the fied as (since the spacetime is axiay symmetric) Φ(r, θ, ϕ, t) = 1 r2 + a 2 + m= φ m (r, θ, t)e imϕ (1)

3 In scattering probems it is customary to consider monocromatic waves with given frequency ω. Therefore we can further write φ m (r, θ, t) = = m c m u m (r, ω)s aω m (θ)e iωt (2) where c m is some expansion coefficient and Sm aω (θ) are the usua spin- spheroida harmonics. These are normaised as π dθ sin θ Sm(θ) aω 2 = 1 (3) 2π Finay, the function u m (r, ω) is a soution of the radia Teukosky equation: d 2 [ u m K 2 +(2amω a 2 ω 2 E m ) dr 2 + (r 2 + a 2 ) 2 dg ] G 2 u m = (4) dr where K =(r 2 +a 2 )ω am and G = r /(r 2 +a 2 ) 2. Furthermore, E m denotes the anguar eigenvaue, cf. Appendix C. As usua, = r 2 2Mr + a 2 and the tortoise radia coordinate r is defined as (with r ±, the two soutions to =, denoting the event horizon and the inner Cauchy horizon of the back hoe) r = r + 2Mr ( ) + r n 1 2Mr ( ) r n 1 + c (5) r + r r + r + r r Usuay, the arbitrary integration constant c is disregarded in this reation. However, in scattering probems it turns out to be usefu to keep it, as we sha see ater. We are interested in a causa soution to (4) which describes waves that are purey ingoing at the back hoe s horizon. This soution can be written u in m { e ikr as r r +, A out m + A in eiωr m as r +. e iωr where k = ω ma/2mr + = ω mω +. In addition, we want to impose an asymptotic scattering boundary condition. We want the tota fied at spatia infinity to be the sum of a pane wave pus an outgoing scattered wave. In other words, we shoud have Φ(r, θ, ϕ) Φ pane + 1 r f(θ, ϕ)eiωr as r + (7) where we have omitted the trivia time-dependence. A information regarding scattering is contained in the (compexvaued) scattering ampitude f(θ, ϕ). Note that, unike in axiay symmetric scattering the scattering ampitude wi depend on both anges: θ and ϕ. Up to this point, we have used the term pane wave quite oosey. In the presence of a ong-range fied such as the Kerr gravitationa fied (which fas off as 1/r at infinity) we cannot write a pane wave in the famiiar fat space form. This probem has been discussed in severa papers, see [23,24]. Remarkaby, it turns out that in a back hoe background the ong-range character of the fied is accounted for by a ogarithmic phase-modification of the fat space pane-wave expression. In practice, the substitution r r is made in the various exponentias. In order to make this paper as sef-contained as possibe, we discuss this point in some detai in Appendix A. (6) γ FIG. 1. A schematic iustration of the genera scattering probem. A pane wave impinges on a rotating back hoe making an ange γ with the rotation axis.

4 The asymptotic expression for a pane wave traveing aong a direction making an ange γ with the back hoe s spin axis, see Figure 1, is Φ pane = e iωr (sin γ sin θ sin ϕ+cos γ cos θ) (8) where without any oss of generaity we have assumed an ampitude of unity. We can decompose this pane wave in a way simiar to (1) and (2); Φ pane 1 r c () m u() m m= = m where u () m are asymptotic soutions of (4). For r we have (see Appendix B), c () m u() m (r, ω)saω m(θ)e imϕ (9) (r, ω) 2πSaω m(γ) { ( i) m+1 e iωr + i m+1 ( 1) +m e iωr } (1) Simiary, the fu fied at infinity can be approximated as : Φ 1 r m= = m c m (A in m e iωr + A out m eiωr )S aω m (θ)eimϕ (11) By imposing the scattering condition (7), we can fix c m by demanding that the ingoing wave piece of Φ Φ pane vanishes. After some straightforward manipuations we get for the scattering ampitude f(θ, ϕ) = 2π [ ] ( i) m S aω iω m(θ)sm(γ)e aω imϕ +1 Aout m ( 1) 1 (12) m= = m By defining the scattering matrix eement S m =( 1) +1 A out m /Ain m we can equivaenty write S m = e 2iδ m (13) where we have introduced the phase-shift δ m. Thus we see that the phase-shifts δ m contain a reevant information regarding the scattered wave. It is worth emphasising that for non-axisymmetric scattering the phase-shifts wi depend on both and m. Aso,theδ m are in genera compex vaued in order to account for absorption by the back hoe. For ater convenience, we aso point out that the fu fied at infinity can be written A in m Φ sin(ωr + δ m π 2 ) as r (14) In the case of on-axis incidence (γ = ) the scattering ampitude simpifies consideraby, and we get f(θ) = 2π [ S aω iω (θ)saω () ( 1) = +1 Aout A in Here we see that the outcome is no onger dependent on m, which is natura given the axia symmetry of the probem. Furthermore, it is easy to see that we recover the famiiar Schwarzschid expression [8] by setting a =. The differentia cross section (often simpy caed the cross section in this paper) is the most important observabe in a scattering probem. It provides a measure of the extent to which the scattering target is visibe from a certain viewing ange. As demonstrated in standard textbooks [1], the differentia cross section foows immediatey from the scattering ampitude dσ dω = f(θ, ϕ) 2. (16) This cross section corresponds to eastic scattering ony, that is, it describes the anguar distribution of the waves escaping to infinity. We can simiary define an absorption cross section but we sha not be concerned with this issue here. Nevertheess, as we have aready pointed out, back hoe absorption has an effect on the phase-shifts that are used to compute the cross section (16). The strategy then for a cross section cacuation (for given back hoe parameters and wave frequency) invoves three steps: i) cacuation of the phase-shifts δ m, (or, equivaenty, of the asymptotic ampitudes A out/in m ), ii) cacuation of the spheroida harmonics Sm aω (θ), and finay iii) evauation of the sums in (12) and/or (15) incuding a sufficienty arge number of terms. ] 1 (15)

5 B. Approximating the scattering ampitude In practice, the partia-wave sum cacuation is probematic as it converges sowy. In fact, the sum is divergent for some anges. This is just an artifact due to the ong-range nature of the gravitationa fied. No matter how far from the back hoe a partia wave may trave, it wi aways fee the presence of the gravitationa potentia (that fas off as 1/r). A simiar behaviour is known to exist in Couomb scattering. The divergence aways occurs at the ange that specifies the incident wave s propagation direction. That this wi be the case is easiy seen from the identity Sm aω (θ)eimϕ Sm aω (γ)e imπ/2 = δ(cos θ cos γ)δ(ϕ π/2), (17),m which foows directy from the fact that the functions Sm aω(θ)eimϕ form an orthonorma set. The corresponding identity for on axis incidence is S aω (θ)saω () = 1 δ(cos θ 1). (18) 2π From (17) we can deduce a pecuiar feature: Athough the scattering probem is physicay insensitive to the actua ϕ of the incidence direction, the specific vaue ϕ = π/2 is imposed by the above reations. Of course, this has no physica reevance since the probem at hand is axiay symmetric and we can, without any oss of generaity, assume an incoming wave traveing aong the direction (θ, ϕ) =(γ,π/2). The fact that the Kerr gravitationa fied behaves asymptoticay as a Newtonian one consideraby simpifies the scattering ampitude cacuation. We woud expect that arge partia waves (stricty speaking when /ωm 1) to essentiay fee ony the far-zone Newtonian fied. In terms of the phase-shifts, we expect them to approach their Newtonian counterparts δ m δ N asymptoticay. In order to secure this matching we add to our phase-shifts an integration constant 2ωM n(4ωm) +ωm. In this way, we aso get r r c,wherer c = r +2Mn(2ωr) is the respective tortoise coordinate of the Couomb/Newtonian probem. Such a manipuation is admissibe given the arbitrariness in the choice of the constant c in (5). In cacuating the partia-wave sum for the scattering ampitude, it is convenient to spit it into two terms: f(θ, ϕ) =f D (θ, ϕ)+f N (θ, ϕ) (19) Here, f D (θ, ϕ) represents the part of the scattering ampitude that carries the information of the main diffraction effects, whie f N (θ, ϕ) denotes the Newtonian (Couomb) ampitude. Expicity we have f N (θ, ϕ) = 2π iω,m [ ] Y m (θ)e imϕ Y m (γ)( i) m e 2iδN 1 where we have deiberatey forgotten the spherica symmetry of the Newtonian potentia (which woud had aowed us to write f N as a function of θ ony, and thus in terms of a sum over ). However, the Newtonian phase-shifts δ N are sti given by the we-known expression [1], (2) e 2iδN = Γ( +1 2iωM) Γ( +1+2iωM) (21) After simpe manipuations we get f N (ξ) = 1 (2 +1)P (cos ξ)(e 2iδN 1) (22) 2iω = where cos ξ =cosθ cos γ +sinθ sin γ sin ϕ. The sum in (22) is known in cosed form [1]; [ Γ(1 2iωM) f N (ξ) =M sin ξ ] 2+4iωM (23) Γ(1 + 2iωM) 2 From this we can see that f N (θ, ϕ) diverges in the ξ = direction. Let us now focus on the diffraction ampitude f D (θ, ϕ). It has the form

6 f D (θ, ϕ) = 2π iω m= = m { } ( i) m e imϕ Sm aω (θ)saω m (γ)(e2iδ m 1) Y m (θ)y m (γ)(e 2iδN 1) (24) The corresponding on-axis expression is, f D (θ) = 1 2iω = { } 4πS aω (θ)saω ()(e2iδ 1) (2 +1)P (cos θ)(e 2iδN 1) (25) One woud expect the sums in (24) and (25) to converge. This foows from the fact that for /ωm we have δ m δ N and Sm aω(θ)eimϕ Y m (θ, ϕ) [9]. We introduce a negigibe error by truncating the sums at a arge vaue max (say). In practice, max need not be very arge. We find that a vaue 3 5 for ωm < 2 typicay suffices. Since each partia wave can be abeed by an impact parameter b() (see Section IID), the criterion for max to be a good choice, is that b( max ) b c,whereb c is the (argest) critica impact parameter associated with an unstabe photon orbit in the Kerr geometry. The truncation of the partia-wave sums wi introduce interference osciations in the fina cross sections (roughy with a waveength 2π/ max [9]). These unphysica osciations can be eiminated by foowing the approach of Hander and Matzner [9]. For a chosen max we add a constant β to a the phase-shifts in (24) and (25). This constant is chosen such that δ max,m + β = δ N. This means that the resuting cross section is effectivey smoothed. C. Defection functions It is we-know that the so-caed defection function is of prime importance in scattering probems. It arises in the semicassica description of scattering, as discussed in the pioneering work of Ford and Wheeer [25]. Athough these authors considered scattering in the context of quantum theory, their formaism is readiy extended to the back-hoe case. In the semicassica paradigm, the phase-shifts are approximated by a one-turning point WKB formua (typicay usefu for much arger than unity). In a probem which has ony one cassica turning point, the defection function is defined as Θ() =2 dδwkb (26) d where is assumed to take on continous rea vaues. As a convention, the defection function is negative for attractive potentias. The right-hand side of this equation resembes the expression for the defection ange of cassica motion in the given potentia, provided that we define the foowing effective impact parameter b for the wave motion [1] b = +1/2 (27) ω The back-hoe effective potentia has two turning points, but for /ωm the scattering is mainy due to the outer turning point and one can derive a one turning point WKB approximation for the phase shifts. For a Schwarzschid back hoe this expression is [8] δ WKB = + t [ Q s ( 1 2M r ) 1 ω] dr ωt +(2 +1) π 4 (28) where Q 2 s = ( 1 2M r ) 2 [ ( ω 2 1 2M r ) ( +1) r 2 + M 2 ] r 4 (29) Here t denote the vaue of the tortoise coordinate corresponding to the (outer) turning point t. We now define the defection function as [ ] dδ Θ() =2Re (3) d where ony the rea part of the phase shift is considered, as the whoe discussion is reevant for eastic scattering ony. Using (28), we find that the rea scattering ange is

7 ( ) [ +1/2 + ( dr Θ() =π 2 ω r M r t )( ) ] 2 1/2 +1/2 + O(M 2 /r 2 ) (31) ωr This WKB resut shoud be compared to the defection ange for a nu geodesic in the Schwarzschid geometry, which is given by + [ ( dr Θ c (b) =π 2b t r M ) ] b 2 1/2 r r 2 (32) where b = L z /E is the orbit s impact parameter (L z and E denote, respectivey, the orbita anguar momentum component aong the back hoe s spin axis and the orbita energy) and t is the (cassica) turning point. In writing down these expressions we have chosen the signs in such a way that the defection ange is negative for attractive potentias. Ceary, it is possibe to match the defection function (3) with the cassica defection ange, abeit ony at arge distances. Since the effective impact parameter wi be given by (27), it is cear that in (31) the integra wi be over arge r ony. Owing to its cear geometrica meaning the defection function is an exceptionay usefu too in scattering theory. It can be used to define diffraction phenomena ike gories, rainbows etc. [25]. For exampe, in axisymmetric scattering backward gories are present if the defection function takes on any of the vaues Θ = nπ, wheren a positive odd integer. It seems natura to try and define defection functions for Kerr scattering as we. In genera, we anticipate the need for two defection functions Θ(, m) andφ(, m) (with ony the first being reevant for the specia case of on-axis scattering). The WKB phase-shift formua becomes in the Kerr case: where δ WKB m = + t [ Q k ( r 2 + a 2 ) ] ω dr ωt +(2 +1) π 4 (33) Q 2 k = 1 2 [ K 2 λ +M 2 a 2] (34) where λ = E m + a 2 ω 2 2amω. The next step is to derive the defection anges for nu geodesics approaching a Kerr back hoe from infinity. Such orbits are studied in detai in [26]. From the resuts in [26] it is cear that it is not easy to write down a genera expression for the defection ange in the Kerr case. But we can obtain usefu resuts in two particuar cases. We begin by considering a nu ray with L z = (which woud correspond to an axiay incident partia wave). For such an orbit we find that the defection ange Θ c obeys the foowing reation Θc(η) π ] 1/2 + ( dθ [1+ a2 dr η 2 cos2 θ = 2η [1 t r 2 η2 r 2 1 2M r ) ] 1/2 + a2 r 2 + a2 r 2 + 2a2 M r 3 (35) where η = C 1/2 /E, withc denoting the orbit s Carter constant. This expression is vaid provided the ray s θ- coordinate varies monotonicay during scattering. This shoud be true in the cases we are interested in, at east for arge impact parameters such that η M. The ray wi aso be defected in the ϕ-direction but this defection carries no information regarding pane-wave scattering due to the axisymmetry of the probem. We next consider a nu ray traveing in the back hoe s equatoria pane. This situation wi be particuary reevant for a pane wave incident aong γ = π/2. The net azimutha defection Φ c (b) for an impact parameter b = L z /E is + dr Φ c (b) =π 2b [1 a2 t r + 2aMr ][ ( r 2 + a 2 + 2a2 M b 2 1 2M b r r ) 4aMb ] 1/2 (36) r Working to the same accuracy in terms of M/r as in the Schwarzschid case, we can match (35) and (36) to δ m /. This matching becomes possibe if we use the foowing approximate expression for the eigenvaue E m [9] E m ( +1) 1 2 a2 ω 2 + O( a3 ω 3 ) (37) As in the Schwarzschid case we assume that the effective impact parameter is given by (27). Athough there is no occurrence of the mutipoe m in the above expressions, one can argue (from the symmetry of the various spheroida

8 harmonics, which is simiar to that of the spherica harmonic of the same (, m)) that the cassica anges (35) and (36) are reated to partia waves with m = and incidence γ = and partia waves with m = ± and incidence γ = π/2, respectivey. Hence, we define the atitudina defection function [ ] δm Θ() =2Re (m =) (38) and the azimutha ( equatoria ) defection function Φ() =2Re [ δm ] (m = ±) (39) III. NUMERICAL RESULTS A. Phase-shifts cacuation via the Prüfer transformation In order to determine the required scattering phase-shifts we have used a sighty modified version of the simpe Prüfer transformation, we-known from the numerica anaysis of Sturm-Liouvie probems [19]. The method is best iustrated by a standard second order ordinary differentia equation: d 2 ψ + U(x)ψ = (4) dx2 where we can think of x as being a radia coordinate, spanning the entire rea axis, and U an effective potentia (in our probem corresponding to a singe potentia barrier) with asymptotic behaviour U(x) { k 2 as x, ω 2 as x +. (41) with ω and k rea constants. (The back hoe probem we are interested in does, of course, have exacty this nature.) The soution of (4) wi take the form of osciating exponentias for x ±. Let us assume that we are ooking for a soution to (4) with purey ingoing behaviour at the eft boundary (x ) and mixed ingoing/outgoing behaviour as x + : ψ { e ikx as x, B sin[ωx + ζ] as x +. where ζ and B are compex constants. We can then write the exact soution of (4) in the form P (x)dx ψ(x) =e (42) (43) The function P (x) is the ogarithmic derivative of ψ(x) (a prime denotes derivative with respect to x) ψ ψ = P (44) which obeys the boundary condition P (x) ik for x. Simiary, we can express the function ψ and its derivative via a Prüfer transformation; ψ(x) =B sin[ωx + P (x)] (45) ψ (x) =Bω cos[ωx + P (x)] (46) with P (x) aprüfer phase function which has ζ as its imiting vaue for x +. Direct substitution in (4) yieds the equations dp dx + P 2 + U(x) = (47)

9 d P [ dx + ω U(x) ] sin 2 ( ω P + ωx) = (48) The idea is to numericay integrate (47) and (48) instead of the origina equation (4). The motivation for this is that, whie the origina soution may be rapidy osciating, the phase-functions P and P are expected to be sowy varying functions of x. We expect this integration scheme to be consideraby more stabe, especiay for high frequencies, than any direct approach to (4). Moreover, eqs. (47) and (48) are we behaved aso at the cassica turning points and are we suited for barrier penetration probems. However, if we want to ensure that the phase-functions are smooth and non-osciatory we must account for the so-caed Stokes phenomenon the switching on of sma exponentias in the soution to an equation of form (4). To do this we simpy shift from studying P (x) (which is cacuated from x = up to the reevant matching point x m )to P (x) (which is cacuated outwards to x =+ ). In practice, the cacuation is stabe and reiabe if the switch is done in the vicinity of the maximum of the back-hoe potentia barrier (the essentia key is to not use one singe representation of the soution through the entire potentia barrier). The two phase functions are easiy connected by P (x) = ωx + 1 [ ] ip ω 2i n (49) ip + ω Finay, the desired phase-shift can be easiy extracted as δ m = ζ + π/2. A major advantage of the adopted method is that it permits direct cacuation of the partia derivatives δ m / and δ m / m, which are required for the evauation of the defection functions from Section IID. The equations for the, m-derivatives of P and P are simpy found by differentiation of (47) and (48). For the case of scattering by a Kerr back hoe, cacuation of δ m and its derivatives with respect to, m requires knowedge of the anguar eigenvaue E m (see Appendix C) and its, m-derivatives. We have used an approximate formua which is a poynomia expansion in aω (formua of [28]). This expression (and its derivatives) is we behaved for a integer vaues of and m. However, it is divergent for the haf-integer vaues =1/2, 3/2, 5/2. Hence, the numerica cacuation of the defection function wi fai at these points, and wi be generay i-behaved in their neighbourhood. For the fu cross section cacuation, we additionay need to cacuate the spin- spheroida harmonics Sm aω (θ). This cacuation is discussed in detai in Appendix C. B. Schwarzschid resuts In this section we reproduce phase-shifts and cross sections for Schwarzschid scattering. The purpose of this exercise is to vaidate, and demonstrate the reiabiity of, our numerica methods. We compare our numerica integration resuts to ones obtained using the phase-integra method [8]. As a first crucia test we compare, in Fig. 2, the first 1 phase-shifts for ωm = 1. Because of the muti-vaued nature of the phase-shifts we aways pot the quantity S = e 2iδ. As is evident from Fig. 2, the agreement between our numerica phase-shifts and the phase-integra ones is exceent. This is equay true for a a frequencies examined (up to ωm = 1). It shoud be noted that S is essentiay zero (δ has a positive imaginary part) for those vaues of for which absorption by the back hoe is important. That this is the case for the owest mutipoes is cear from Fig. 2. As increases δ becomes amost rea and as a consequence S is amost purey osciating

10 FIG. 2. Comparison of numerica phase-shifts for a Schwarzschid back hoe (cross), against phase-integra data (pus). We show the rea (upper frame) and imaginary (ower frame) parts of the scattering matrix eement S = e 2iδ as functions of. The agreement between the two methods is ceary exceent. In Fig. 3 we present an ωm = 1 cross section generated from our numerica phase-shifts. For this particuar cacuation we have used max = 2. The resuting cross section matches the one constructed using approximate phase-integra phase shifts perfecty. This demonstrates the efficiency of our approach in the high frequency regime, and it is cear that our study of Kerr scattering wi not be imited by the ack of reiabe phase shifts. However, the Kerr study is nevertheess imited in the sense that ω cannot be taken to be arbitrariy arge. This restriction is imposed by the cacuation of the spheroida harmonics (Appendix C). However, it is important to emphasize that the most interesting frequency range, as far as diffraction phenomena is concerned, is ωm 2 [8]. Thus, we expect that our investigation shoud be abe to reiabe unvei a reevant rotationa effects in the scattering probem og 1 [M 2 dσ/dω] θ(π) FIG. 3. Differentia cross section for scattering of a wave with reativey high frequency, ωm = 1, from a Schwarzschid back hoe, based on the the first 2 partia wave phase-shifts. The backward gory osciations are prominent. We aso find that the numerica vaues for dδ /d are in good agreement with the respective phase-integra resuts. As a fina remark we shoud emphasize that the numerica approach adopted in this work is very efficient from a computationa point of view. C. On-axis Kerr scattering Having confirmed the reiabiity of our numerica resuts we now turn to the study of scattering of axiay incident scaar waves by a Kerr back hoe, cf. Fig. 4. In principe, we woud expect the corresponding cross sections to be quaitativey simiar to the Schwarzschid ones. The main reason for this is the inabiity of axiay impinging partia waves to distinguish between prograde and retrograde orbits. However, examination of the orbita equations [26] reveas that the critica impact parameter (associated with the unstabe photon orbit) decrease sigthy from the vaue 3 3M as the back hoe spins up. For exampe, for a=.99m we have b c =4.74M.

11 θ FIG. 4. A schematic drawing iustrating the case of on-axis scattering from a rotating back hoe. Because of the axia symmetry of the probem, the scattering resuts are quaitativey simiar to those for a Schwarzschid back hoe. Two rays, which emerge having been scattered by the same ange θ are indicated. Indeed, Fig. 5 confirms our expectations. The data in the figure corresponds to a back hoe with spin a =.99M and a wave frequency of ωm = 2. For purposes of comparison, we aso show the corresponding Schwarzschid cross section. The two cross sections are very simiar. In particuar, they are both dominated by the backward gory. It is we-known [1,7] that for Schwarzschid scattering the gory effect can be described in terms of Besse functions. It has been shown that for θ π the gory cross section can be approximated by dσ dω (θ) gory J 2 [ωb c sin θ] (5) A simiar resut hods for the backward gory in the case of on-axis Kerr scattering. Since b c gets smaer, one woud expect the zeros of the Besse function (the diffraction minima) to move further away from θ = π as a increases. This effect is indicated by the data in Fig og 1 [M 2 dσ/dω] θ(π) FIG. 5. Differentia cross section for on-axis scattering from a Kerr back hoe (soid ine). The back hoe s spin is a =.99M and ωm = 2. The graph is based on data for max = 3. The dashed ine represents the corresponding Schwarzschid cross section. The above concusions are further supported by the resuts for the defection function (38), as shown in Fig. 6. Because of the inaccuracies inherent in our method of cacuating the defection function for the owest -mutipoes, see the discussion in Section IIIA, we do not show resuts for this regime. This is, however, irreevant as the corresponding partia waves are expected to be more or ess competey absorbed by the back hoe.

12 .5 Θ/π FIG. 6. Defection function (in units of π) for on-axis scattering from a Kerr back hoe. The back hoe spin is a =.99M and ωm = 1. The dashed curve is the corresponding Schwarzschid defection function. This figure confirms the behaviour expected from the geometric optics considerations, namey, the sight decrease of the critica impact parameter with increasing a. D. Off-axis Kerr scattering The concusions of our study of on-axis scattering are perhaps not very exciting. Once the Schwarzschid case is understood, the on-axis resuts for Kerr come as no surprise. This is, however, not the case for off-axis scattering, cf. Fig. 7, where severa new features appear. θ ϕ= π/2 ϕ π/2 ϕ= FIG. 7. A schematic drawing iustrating off-axis scattering from a rotating back hoe. We show (as thick dashed ines) two rays, one of which corresponds to motion in the back hoe s equatoria pane. Our study of the off-axis case provides the first resuts for non-axisymmetric wave scattering in back hoe physics. Since this probem has not been discussed in great detai previousy, it is worthwhie asking whether we can make any predictions before turning to the numerica cacuations. Two effects ought to be reevant: First of a, the partia waves now have orbita anguar momentum which coupes to the back hoe s spin. As a resut the partia waves can be divided into prograde (m >) and retrograde (m <) ones. We expect prograde waves to be abe to approach coser to the horizon than retrograde ones. In the geometric optics imit, prograde and retrograde rays tend to have increasingy different critica impact parameters as a M. As a second feature, we expect to find that arge partia waves wi effectivey fee ony the sphericay symmetric (Newtonian) gravitationa potentia. In other words, partia waves with the same (arge) and different vaues of m wi approximatey acquire the same phase-shift. Our numerica resuts essentiay confirm these expectations, as is cear from the phase-shifts (cacuated for a =.9M and ωm = 1) shown in Fig. 8. As above, we have graphed the singe-vaued quantity S m = e 2iδ m as a function of. For each vaue of we have incuded a the phase-shifts for m +. The soid (dashed) ine corresponds to m =+ (m = ) and the intermediate vaues of m ead to resuts in between these two extremes. For 1,

13 partia waves with different vaues of m have amost the same phase-shift. This is easy to deduce from the fact that the two curves approach each other as increases. On the other hand, for the first ten or so partia waves we get very different resuts for the various vaues of m. In particuar, we see that phase-shifts with m> become amost rea (that is, S m becomes non-zero) for a smaer -vaue as compared to the m< ones. As anticipated, this is due to the different critica impact parameters associated with prograde/retrograde motion, and the fact that a arger number of prograde partia waves are absorbed by the back hoe FIG. 8. Off-axis phase-shifts for ωm = 1anda =.9M. We iustrate the rea (upper pane) and imaginary (ower pane) parts of S m = e 2iδ m as functions of, incuding a permissibe vaues of m. The two curves correspond to m = (soid ine) and m = (dashed ine). We now turn to the cross section resuts for the off-axis case. We have considered a pane wave incident aong the direction γ = ϕ = π/2. Even though our formaism aows incidence from any direction we have focussed on this case, which is iustrated in Fig. 7. The motivation for this is that there wi then be partia waves (specificay the ones with m = ±) that are mainy traveing in the back hoe s equatoria pane. These partia waves are important because one woud expect them to experience the strongest rotationa effects. Besides, we can obtain an understanding of these waves by studying equatoria nu geodesics in the geometric optics imit. Equatoria nu rays are much easier to describe than nonequatoria ones. This proves vauabe in attempts to decipher the off-axis cross sections, and the obtained concusions provide an understanding aso of the genera case. In Fig. 9 we present a series of cross sections as functions of ϕ for the specific vaues θ = π/8,π/4, 3π/8,π/2. These resuts correspond to viewing the scattered wave on the circumference of cones (ike that shown in Fig. 7) with increasing opening anges. Two different frequencies ωm = 1 and ωm = 2 have been considered for a back hoe with spin a =.9M. A first genera remark concerns the asymmetry of the cross sections with respect to the incidence direction (note, however, that as a consequence of our particuar choice of incidence direction there is sti a refection symmetry with respect to the equator). We can aso easiy distinguish the Couomb forward divergence in the direction θ = ϕ = π/2. Another obvious feature in Fig. 9 is the markedy different appearance of the cross section for different vaues of θ. As we move away from the equatoria pane the cross sections becomes increasingy featureess. This behaviour is artificia in the sense that as θ decreases, we effectivey observe aong a smaer circumference. At θ = this circumference degenerates into a point, cf. Fig. 7.

14 og 1 (dσ/dω)(m 2 ) θ/π ϕ/π θ/π ϕ/π FIG. 9. Off-axis cross sections (θ = π/8,π/4, 3π/8,π/2) for a back hoe with spin a =.9M and scattered waves with frequency ωm = 1 (upper pane) and ωm = 2 (ower pane). The incident wave is traveing in the (θ, ϕ) =(π/2,π/2) direction. og 1 (dσ/dω)(m 2 ) In order to understand the features seen in Fig. 9 further, we focus on the θ = π/2 cross section. In Fig. 1 we show these equatoria cross sections for a sequence of spin rates a/m =.2,.5,.7,.9. As before, we have considered two different wave frequencies, ωm =1andωM = 2. From the resuts shown in Fig. 1 it is cear that that the gory maximum is typicay not observed in the backward (ϕ = π/2) direction. In fact, it is cear that the maximum of the gory osciations move away from the backward direction as the spin of the back hoe is increased. A simiar shift is seen in a interference osciations. This behaviour is easy to expain in terms of the anticipated rotationa frame-dragging. In order to iustrate this argument, we consider the geometric optics imit where partia waves are represented by nu rays. Reca that in axisymmetric scattering the backward gory is associated with the divergence of the cassica cross section at Θ = π in such a way that ( ) dσ dω c = b sin Θ ( ) 1 dθ (51) db The divergence is a resut of the intersection of an infinite number of rays. For simpicity, et us consider rays traveing in a specified pane. Scattering near the backward direction by a Schwarzschid back hoe is iustrated in Fig. 11

15 (eft pane). Two rays with different impact parameters emerge at any given ange. These two waves make the main contribution to cross section at that particuar ange. It is cear that for scattering at θ = π the two rays in Fig. 11 wi foow symmetric trajectories. This means that when observed at infinity, after being scattered, the two waves wi have equa phases (provided their initia phases were equa). In effect, these two rays wi then constructivey interfere in the exact backward direction. As we move away from the backward direction, we shoud observe a series of interference maxima and minima the two rays wi now have an overa phase difference since they foow different orbits (see Fig. 11). A very crude estimate of the ocation of the successive maxima woud be θ n nπ/3ωm where n =, 1, 2,..., in reasonabe agreement with the exact resuts. og 1 (dσ/dω)(m 2 ) a/m ϕ/π og 1 (dσ/dω)(m 2 ) a/m ϕ/π FIG. 1. Off-axis cross sections for θ = π/2 and various back-hoe spins (a/m=.2,.5,.7,.9) and scattered wave frequencies ωm = 1 (upper pane) and ωm = 2 (ower pane). Simiar arguments appy in the case of a Kerr back hoe. We sha consider ony equatoria rays, cf. Fig 11 (right pane). As a resut of the discrimination between prograde and retrograde orbits, the two rays contributing to the cross section in the exact backward direction wi no onger foow symmetric paths. In fact, the ray symmetric to the prograde ray shown in Fig 11 wi foow a punging orbit. Therefore, we shoud not expect the interference maximum to be ocated in the exact backward direction. An estimate (based on a crude cacuation of the phase difference between the two nu rays) of the ocation of the main backward gory maximum yieds

16 ( ) rph+ r ph ϕ max π r ph+ + r ph (52) where r ph+ and r ph denote, respectivey, the ocation of the prograde and retrograde unstabe photon orbits (in Boyer-Lindquist coordinates). This ange is measured from the backward direction in the direction of the back hoe s rotation. This simpe prediction agrees reasonaby we with the resuts inferred from our numerica cross sections. In a simiar way, a other maxima and minima wi be frame-dragged in the back hoe s rotationa direction < < < < FIG. 11. Equatoria nu geodesics (viewed from above ) around a Schwarzschid (eft pane) and Kerr back hoe (right pane). The figures are scaed in units of M. The rays are assumed to arrive from infinity (they enter from the right side of each figure in the direction indicated by the arrows) in parae directions and exit at the same ange after being scattered. The dashed circes represent the unstabe photon circuar orbits. The Kerr back hoe, in the right pane, is taken to rotate counter-cockwise with a =.99M. To compete this discussion we consider the defection function Φ(, m) for equatoria partia waves (m = ±), an a =.9M back hoe and ωm = 1. The corresponding data is shown in Fig. 12. There are two distinct ogarithmic divergences which are associated with the existence of separate unstabe circuar photon orbits for prograde and retrograde motion. Note that for m> the defection function diverges steeper than it does for m<. The origin of this effect is the fact that prograde partia waves with b b c perform a greater number of revoutions (before escaping to infinity) than retrograde ones. Finay, for m 1 we recover, as expected, the Einstein defection ange Φ 4M/b (not expicity shown in the figure) Φ/π m FIG. 12. The defection function Φ for off-axis scattering is shown as a function of m for equatoria partia waves m = ±. The back hoe spin is a =.9M and the wave frequency is ωm =1. Thesma m region is not incuded because of the inaccuracies discussed in Section IIIA. E. Digression: forward gories The resuts presented in the preceeding sections ceary show that, in genera, back hoe cross sections are dominated by a Couomb divergence in the forward direction and (frame-dragged) gory osciations near the backward direction.

17 However, according to the predictions of geometrica optics [1], one woud expect to find gory osciations aso in the forward direction (see comments in [8]). For the case of Schwarzschid scattering, this effect woud be associated with partia waves scattered at anges Θ =, 2π, 4π,... Inspection of the reevant defection function (Fig. 6) indicates that a partia wave which has Θ = wi aso be strongy absorbed, since it has an impact parameter b<b c. Hence, we woud expect its contribution to the forward gory to be severey supressed. It thus foows that, as far as the possibe forward gory is concerned, the most important partia waves are those with Θ = 2π. These partia waves whir around the back hoe as they have b b c. Ford and Wheeer s semicassica approach [25] shows that the forward gory is we approximated (for θ ) by (5), athough with a sighty different proportionaity factor. However, we shoud obviousy not expect to see a pronounced forward gory in the cross section, as it wi drown in the forward Couomb divergence. Sti, as an experiment aimed at supporting our intuition, we can try to dig out the forward gory pattern. This has to be done in a somewhat artificia manner, but since the forward gory is due to scattering and interference of partia waves with b b c we can isoate their contribution by truncating the sum in f D at some max ωb c and at the same time negecting the Newtonian part f N entirey. It is, of course, important to reaize that this truncated cross section is not a physica (observabe) quantity. In Fig. 13 we show the resut of this truncated cross section cacuation for the case of a Schwarzschid back hoe and ωm = 2. We compare resuts for two eves of truncation, max = 1 and 15. In the first case, a cear Besse- function ike behaviour arises (it is straightforward to fit a J 2 (ωb c sin θ) function to the soid curve in Fig. 13). This confirms our expectation that there is, indeed, a forward gory present in the data. As more partia waves are incuded the cross section begins to deviate from the gory behaviour, and if we increase max further the forward gory is swamped by terms that contribute to the Couomb divergence og 1 [M 2 f D (θ) 2 ] θ(π) FIG. 13. Iustration of a forward gory in Schwarzschid scattering. The diffraction piece f D(θ) 2 of the cross section is shown in the vicinity of the forward direction, for wave frequency ωm = 2 and for max = 1 (soid curve) and max =15 (dashed curve). F. The roe of superradiance in the scattering of monochromatic waves. Superradiance is an interesting effect known to be reevant for rotating back hoes. It is easiy understood from the asymptotic behaviour (6) of the causa soution to the scaar-fied Teukosky equation (4). If we use this soution and its compex conjugate, and the fact that two ineary independent soutions to (4) must ead to a constant Wronskian, it is not difficut to show that (1 mω + /ω) T m 2 =1 S m 2. (53) wherewehavedefined T m 2 = 1 A in m 2. (54)

18 From the above resut it is evident that the scattered waves are ampified ( S m 2 > 1) if ω<mω +. This ampification is known as superradiance. In principe, one woud expect superradiance to pay an important roe in the scattering probem for rapidy spinning back hoes. For exampe, one coud imagine that some partia waves which woud otherwise be absorbed, coud escape back to infinity. These waves might then possiby make a noticeabe contribution to the diffraction cross section, provided that there were a sufficient number of them (as compared to the tota number max of partia waves contributing to the diffraction scattering ampitude). In order to investigate this possibiity, we have performed a number of off-axis cross section cacuations for a variety of wave frequencies ωm =.5 1 and for a M, i.e. back hoes spinning near the extreme Kerr imit. We have found no quaitive difference whatsoever between those cross-sections and the ones for a somewhat smaer spin vaue, a =.9M (say). In essence, we were unabe to find any effects in the cross section that coud be attributed to superradiance. Consequenty, we are ed to suspect that our intuition regarding the importance of superradiance for the scattering probem may be wrong. This suspicion is confirmed by the foowing simpe argument. In order for a partia wave to be superradiant we shoud have <ω<mω +. Considering an extreme Kerr back hoe (which provides the best case for superradiant scattering) and the fact that m, we have the condition < 2ωM < (55) As aready mentioned, the partia waves for which superradiance wi be important are the ones with impact parameters b < b c, i.e. those that woud be absorbed under different circumstances. This then requires that Combining (55) and (56) we arrive at the inequaity < ωb c 1/2 (56) < 2M < b c 1/2ω (57) The critica impact parameter (for prograde motion) for an a = M back hoe is b c 2M. Hence the condition (57) wi not be satisfied, and it is unikey that we woud get a significant number of (if any) superradiant partia waves that coud affect the cross section. This concusion may seem surprising given resuts present in the iterature [1,9]. In particuar, Hander and Matzner have briefy discussed the effect of superradiance on axiay incident gravitationa waves. They argue that (see figure 14 in [9]) superradiance has the effect of imposing a arge background over the pattern, fiing in the interference minima. Given our current eve of understanding (or ack thereof) we cannot at this point say whether superradiance can be the expanation for the effects observed by Hander and Matzner. After a, one shoud remember that superradiant scattering strongy depends on the spin of the fied that is being scattered. It is we known [22] that gravitationa perturbations can be ampified up to 138% compared to a tiny.4% ampification for scaar fieds (which is the case considered in this paper). This means that superradiance may significanty affect aso partia waves with b>b c in the gravitationa wave case, which coud ead to our simpe argument not being vaid. This issue shoud be addressed by a detaied study of the scattering of gravitationa waves from rotating back hoes. IV. CONCLUDING DISCUSSION We have presented an investigation of scattering of massess scaar waves by a Kerr back hoe. Our numerica work is based on phase-shifts obtained via integration of the reevant radia wavefunction with the hep of the Prüfer phase-function method. This method has been shown to be computationay efficient and to provide accurate resuts. Using the obtained phase-shifts we have constructed differentia cross sections for severa different cases. First we have discussed the case of waves incident aong the back hoe s rotation axis, for which we showed that the resuting cross sections are simiar to ones obtained in the (non-rotating) Schwarzschid case. We then turned to the case of off-axis incidence, where the situation was shown to change consideraby. In that case the cross sections are genericay asymmetric with respect to the incidence direction. The overa diffraction pattern is frame dragged, and as a resut the backward gory maximum is shifted aong with of the back hoe s rotation. Moreover, we have concuded that (at east for scaar waves) the so-caed superradiance effect is unimportant for monochromatic scaar wave scattering. To summarize, our study provides a compete understanding of the purey rotationa effects invoved in back-hoe scattering. Given this we are now we equipped to proceed to probems of greater astrophysica interest, particuary ones concerning gravitationa waves. In these probems one woud expect further features to arise as the spin and poarisation of the impinging waves interact with the spin of the back hoe. For incidence aong the hoe s spin axis,