Wave Propagation in Nontrivial Backgrounds

Transcription

1 Wave Propagation in Nontrivia Backgrounds Shahar Hod The Racah Institute of Physics, The Hebrew University, Jerusaem 91904, Israe (August 3, 2000) It is we known that waves propagating in a nontrivia medium deveop tais. However, the exact form of the ate-time tai has so far been determined ony for a narrow cass of modes. We present a systematic anaysis of the tai phenomenon for waves propagating under the infuence of a genera scattering potentia V (x). It is shown that, genericay, the ate-time tai is determined by spatia derivatives of the potentia. The centra roe payed by derivatives of the scattering potentia appears not to be widey recognized. The anaytica resuts are confirmed by numerica cacuations. One of the most remarkabe features of wave dynamics in curved spacetimes is the deveopment of tais. Gravitationa waves (or other fieds) propagate not ony aong ight cones, but aso spread inside them. This impies that at ate times waves do not cut off sharpy but rather die off in tais. In particuar, it is we estabished that the ate-time evoution of massess fieds propagating in back-hoe spacetimes is dominated by an inverse power-aw behaviour. Price [1] was the first to provide a detaied anaysis of the mechanism by which the spacetime outside of a (neary spherica) coapsing star divests itsef of a radiative mutipoe moments, and eaves behind a Schwarzschid back hoe. The evoution of such waves (gravitationa, eectromagnetic, and scaar) in a curved spacetime is governed by the Kein-Gordon (KG) equation [2] [ 2 t 2 2 ( +1) + x2 x x 2 V (x/x s ) s ] Ψ =0, (1) where the term ( +1)/x 2 is the we-known centrifuga barrier ( is the mutipoe order of the fied), and V (x) is an effective curvature potentia (we henceforth take x s = 1 without oss of generaity). It was demonstrated that a radiative perturbations decay asymptoticay as an inverse power of time, the power indices equa Physicay, these inverse power-aw tais are associated with the backscattering of waves off the effective curvature potentia, V (x), at asymptoticay far regions [3,1]. The anaysis of Price has been extended by many authors. Bičák [4] generaized the resuts to the case of an eectricay neutra scaar fied propagating in a charged Reissner-Nordström spacetime. Leaver [5] demonstrated that the ate-time tai can be associated with the existence of a branch cut in the Green s function for the wave propagation probem. Gundach, Price, and Puin [6] showed that poweraw tais are a genuine feature of gravitationa coapse the existence of these tais was demonstrated in fu non-inear numerica simuations of the sphericay symmetric coapse of a sef-gravitating scaar fied (this was ater reproduced in [7]). Moreover, since the ate-time tai is a direct consequence of the scattering of the waves at asymptoticay far regions, it has been suggested that power-aw tais woud deveop independenty of the existence of an horizon [8]. This impies that tais shoud appear in perturbations of stars as we. Moreover, the KG wave equation has a much wider range of physica appicabiity. For instance, eectromagnetic waves propagating in an optica cavity can be described by this equation as we [9,10] [A nontrivia dieectric constant distribution, n 2 (x), provides a nontrivia medium, and is anaogous to the presence of a scattering potentia V (x).] For other reated works, see e.g., [10 20], and references therein. The dynamics of waves in rotating back-hoe spacetimes has received much attention recenty [21 31]. It has been demonstrated that power-aw tais present in the Kerr spacetime as we (However, the damping indices are generay different from those found in the Schwarzschid case.) Yet, a thorough understanding of the fascinating phenomenon of wave tais is far from being compete. In particuar, the exact form of the ate-time tai has so far been determined ony for a narrow cass of modes. In fact, most of previous anayses are restricted to ogarithmic potentias of the form V (x) n β x/x α (where α>2andβ =0, 1 are parameters). In a briiant work, Ching et. a. [10] have shown that, genericay, the atetime behaviour of waves propagating under the infuence of these specific potentias has the form Ψ n β t/t 2+α (with the we-known exception of a pure power-aw decay in the Schwarzschid spacetime). The purpose of this paper is to present a systematic anaysis of the tai phenomenon with a genera scattering potentia V (x). Ching et. a. [10] provided a heuristic picture of the scattering probem which is an important first step in this direction: Consider a wave from a source point y. The ate-time tai observed at a fixed spatia ocation, x, is a consequence of the wave first propagating to a distant point x >> y, x, being scattered by V (x ), and then returning to x at a time t (x y)+(x x) 2x. Hence, according to this scenario, the scattering ampitude (and thus the ate-time tai itsef) are expected to be proportiona to V (x ) V (t/2). However, it is we-known, at east in the specific case of 1

2 ogarithmic potentias, that this simpe picture requires two modifications. First, there is an extra suppression of the ate-time tai by a factor of t 2. Second, if α is an odd integer ess than 2 + 3, the eading term in the ate-time tai vanishes [10], and one shoud consider subeading terms (The we-known Schwarzschid spacetime beongs to this case.) There are severa interesting and important open questions regarding the tai phenomenon in a genera anaysis. What determines the ate-time tai is it simpy the asymptotic form of the scattering potentia itsef, as suggested by the heuristic picture? How generic is the suppression of waves with 1? Is it aways by a factor of t 2? What is the (most) genera cass of scattering potentias for which the eading term in the ate-time tai vanishes? (The Schwarzschid spacetime is ony one specific exampe to these potentias.) What is the genera form of the ate-time behaviour in these cases? These questions, and severa others ca for a study of the genera properties of wave tais. In this paper we present our main resuts for this fascinating phenomena. We consider the evoution of a wave fied whose dynamics is governed by a KG-type equation Φ ;ν ;ν + V (x)φ = 0. Resoving the fied into spherica harmonics Φ = m (t, r)y (θ, ϕ)/r (r being the circumferentia ra- Ψ m,m dius), one obtains a wave equation of the form Eq. (1) for each mutipoe moment [32] (For brevity we henceforth suppress the indices, m on Ψ.) It proofs usefu to introduce the doube-nu coordinates u t x and v t + x, which are a retarded time coordinate and an advanced time coordinate, respectivey. The initia data is in the form of some compact outgoing puse in the range u 0 u u 1, specified on an ingoing nu surface v = v 0. The genera soution to the wave-equation (1) can be written as a series depending on two arbitrary functions F and G [1] Ψ= + [ ] A kx k G ( k) (u)+( 1) k F ( k) (v) ] Bk [G (x) ( k 1) (u)+( 1) k F ( k 1) (v), (2) where A k =(+k)!/2k k!( k)!. For any function H, H (k) is its kth derivative; negative-order derivatives are to be interpreted as integras. The functions Bk (x) satisfythe recursion reation B k = 1 2 [ B k 1 ( +1)x 2 B k 1 V (A k x k + B k 1 ) ], (3) for k 1, where B db/dx, andb 0 = V (x)/2. For the first Born approximation to be vaid the scattering potentia V (x) shoud approach zero faster than 1/x 2 as x, see e.g., [10,16]. Otherwise, the scattering potentia cannot be negected at asymptoticay far regions [see Eq. (5) beow]. It is usefu to cassify the scattering potentias into three groups, according to their asymptotic behaviour: Group I: V approaches zero sower than V /x, and faster than V as x. Group II: V approaches zero at the same rate as V as x. Group III: V approaches zero at the same rate as V /x as x. Group I. For this case the recursion reation, Eq. (3), yieds B k (x) = 2 (k+1) V (k 1) (x). The first stage of the evoution is the scattering of the fied in the region u 0 u u 1. The first sum in Eq. (2) represents the primary waves in the wave front (i.e., the zeroth-order soution, with V 0), whie the second sum represents backscattered waves. The interpretation of these integra terms as backscatter comes from the fact that they depend on data spread out over a section of the past ight cone, whie outgoing waves depend ony on data at a fixed u [1]. After the passage of the primary waves there is no outgoing radiation for u>u 1, aside from backscattered waves. This means that G(u 1 ) = 0. Hence, for a arge x at u = u 1, the dominant term in Eq. (2) is Ψ(u = u 1,x)=B (x)g ( 1) (u 1 ). (4) This is the dominant backscatter of the primary waves. With this specification of characteristic data on u = u 1, we sha next consider the asymptotic evoution of the fied. We confine our attention to the region u>u 1, x x s. To a first Born approximation, the spacetime in this region is approximated as fat [1,8]. Thus, to first order in V (that is, in a first Born approximation) the soution for Ψ can be written as Ψ= [ ] A k x k g ( k) (u)+( 1) k f ( k) (v). (5) Comparing Eq. (5) with the initia data on u = u 1,Eq. (4), one finds f(v) = 2 1 V ( 1) (v/2)g ( 1) (u 1 ). For ate times t x one can expand g(u) = ( 1) n g (n) (t)x n /n! and simiary for f(v). With these expansions, Eq. (5) can be rewritten as [ ] Ψ= K n xn f (+n) (t)+( 1) n g (+n) (t), (6) n= where the coefficients K n are those given in [1]. They vanish for n if n is odd. 2

3 Using the boundary conditions for sma r (reguarity as x, at the horizon of a back hoe, or at x = 0, for a stear mode), one finds that at ate times g(t) =( 1) +1 f(t) to first order in the scattering potentia V (see e.g., [1,8] for additiona detais). That is, the incoming and outgoing parts of the tai are equa in magnitude at ate-times. This amost tota refection of the ingoing waves at sma r can easiy be understood on physica grounds it simpy manifests the impenetrabiity of the barrier to ow-frequency waves [1] (which are the ones to dominate the ate-time evoution [5]). We therefore find that the ate-time behaviour of the fied at a fixed radius (x t) is dominated by [see Eq. (6)] which impies Ψ 2K +1 f (2+1) (t)x +1, (7) Ψ Ψ 0 G ( 1) (u 1 )x +1 V (2) (t/2), (8) where Ψ 0 = 2 (2+1) K +1. Hence, the ate-time tai is determined by the 2th derivative of the scattering potentia. The anaysis for Groups II and III proceeds aong the same ines. That is, one shoud first sove Eq. (3) for B k (x) find f(v) using Eq. (5) which finay yieds Ψ(t ) through Eq. (6). In the foowing we present the main resuts for groups I and II. Group II. The dominant backscatter of the primary waves is Ψ(u = u 1,x)= B k (x)g ( k 1) (u 1 ), where k= the B k are the same as for group I. Using an anaysis aong the same ines as before, one finds Ψ n=+1,+3,... 2 n K n xn k= 2 k G ( k 1) (u 1 )V (n+k 1) (t/2), (9) at ate-times. Note that Eq. (9) is merey a generaization of Eq. (8), and reduces to it if V approaches zero faster than V (inwhichcasev (2) dominates at ate-times). Group III. Let V (x) W (x)/x α, where α > 2, and W (x)/x γ 0 as x for any γ > 0. The genera soution for B k (x) is now given by B k (x) b n (k, α)w (n) (x)/x α+k n 1, where the b n (k, α) are compicated numerica coefficients, which are determined by the recursion reation Eq. (3). The anaysis now proceeds aong the same ines as before; One finds that the ate-time behaviour of the wave is given by Ψ G ( 1) (u 1 )x +1 c n (, α)w (n) (t/2)t (α+2 n), (10) TABLE I. Late-time behaviour for various scattering potentias. Group V (x) Ψ (t ) I e xβ, 0 <β<1 e (t/2)β t 2(1 β) I sin(x β )/x α, 0 <β<1 sin[(t/2) β ]t α 2(1 β) II sin(x)/x α periodic fun. t α IIIa x 1/xβ /x α, β > 0 t (α+2) IIIb x 1/xβ /x α, α odd < 2 +3, β > 0 t (α+2+β) n t where the coefficients c n (, α) are constructed from the b n (, α). Group III is divided into three subgroups according to the asymptotic behaviour of the function W (x), and the vaue of α: Subgroup IIIa: W approaches zero faster than W /x as x, and α is not an odd integer ess than In this case, the dominant term in Eq. (10) is W (t/2)t (α+2). Reca now that V W/t α+1 as t, which impies Ψ G ( 1) (u 1 )x +1 V (2) (t/2), (11) at ate-times. Subgroup IIIb: W approaches zero faster than W /x as x, and α is an odd integer ess than (This subgroup of scattering potentias incudes the Schwarzschid spacetime as a specia case.) In this case one finds that the eading term of B (x) [proportiona to V ( 1) (x)] vanishes, and sub-eading terms shoud therefore be considered. Hence, the ate-time behaviour of the wave is dominated by is the (first) sub-eading term in the asymp- Namey, V (2) s W /t α+2 1. [Note Here V (2) s totic derivative. Ψ G ( 1) (u 1 )x +1 V (2) s (t/2). (12) that the resuts of [10] for the specific famiy of ogarithmic potentias (with W n β x), coincide with the genera expressions, Eqs. (11) and (12).] Subgroup IIIc: W approaches zero at the same rate as W /x as x. In this case, the ate-time dynamics of the fied is given by Eq. (10). Numerica cacuations. It is straightforward to integrate Eq. (1) using the methods described in [8,19]. The ate-time evoution of the fied is independent of the form of the initia data used. The resuts presented here are for a Gaussian puse. Tabe I gives a seected ist of scattering potentias, chosen as representative of the various different groups (We have studied other potentias as we, which are not shown here.) The tempora evoutions of the waves (under the infuence of the various scattering potentias) are shown in Figs. 1 and 2. We find an exceent agreement between the anaytica resuts and the numerica cacuations. 3

4 FIG. 1. Tempora evoution of the fied for scattering potentias which beong to Group I. Top pane: V (x) =e (x/x 0) β (the resuts presented here are for the parameters β =1/2 andx 0 =1/2). We dispay the quantity n Ψt 2(1 β) vs t β. There is a definite inear dependence at ate-times, with sopes of 0.99 and 1.02 for =0(ower graph) and = 1, respectivey. These are in exceent agreement with the anayticay predicted vaue of 1. Bottom pane: V (x) = sin[(x/x 0) β ]/x α (the resuts presented here are for the parameters α =3,β =1/2, and x 0 =1/2). An osciatory power-aw fa off is manifest at ate times. The power-aw indices (determined from the maxima of the osciations) are 4.05, and 5.07, for = 1 (upper graph), and = 2, respectivey. These are in exceent agreement with the anayticay predicted vaues of 4, and 5. Summary and physica impications. We have given a systematic anaysis of the tai phenomena for waves propagating under the infuence of a genera scattering potentia. It was shown that the ate-time tai is governed by spatia derivatives of the scattering potentia (genericay, by the 2th derivative). In particuar, the potentia function itsef does not enter into the expression of the ate-time tai (with the exception of the monopoe case). The centra roe payed by derivatives of the scattering potentia appears not to be widey recognized. The anaytica resuts are in exceent agreement with numerica cacuation. In addition, we have demonstrated that the (extra) suppression of waves by a factor of t 2 (which adds to the basic ate-time decay), a phenomena we-known in back-hoe spacetimes, is actuay not a generic feature of the scattering probem. In particuar, for scattering potentias that beong to group I the suppression of the FIG. 2. Top pane: Tempora evoution of the fied for a scattering potentia of the form V (x) = sin(x/x 0)/x α,which beongs to Group II (the resuts presented here are for the parameters α =3andx 0 = 100/π). The power-aw indices are 3.02 and 3.04 for = 1 (upper graph) and =2,respectivey. These vaues are in exceent agreement with the anayticay predicted vaue of 3forboth =1and =2. The osciations period agrees with the predicted vaue to within 0.4%. Bottom pane: Tempora evoution of the fied for scattering potentias of the form V (x) = x 1/xβ /x α, which beong to Group III (the resuts presented here are for the case β = 1). We present the quantity n Ψt ɛ n δ t vs t, where:(i) ɛ =3,δ =0forα =3with = 0, (ii) ɛ =6,δ = 1 forα =3 with = 1, and (iii) ɛ =6,δ =0forα =4with = 1 (from bottom to top). These quantities are anayticay predicted to approach a constant vaue at asymptoticay ate-times. waves is smaer, whie for scattering potentias that beong to group II there is no (extra) suppression at a. Moreover, it was shown that the famiiar case of the Schwarzschid spacetime beongs to a wider group of scattering potentias, in which the eading term in the tai [proportiona to V (2) (t/2)] vanishes (and thus, subeading terms dominate the ate-time dynamics). We are at present extending the anaysis to incude: (i) time-dependent scattering potentias, and (ii) scattering potentias that ack spherica symmetry (in which case the scattering probem is of dimensions). ACKNOWLEDGMENTS I thank Tsvi Piran for discussions. This research was supported by a grant from the Israe Science Foundation. 4

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