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Quasi-Normal Mode Expansion for Linearized Waves in Gravitational Systems

The quasinormal modes (QNM's) of gravitational systems modeled by the Klein-Gordon equation with effective potentials are studied in analogy to the QNM's of optical cavities. Conditions are given for the QNM's to form a complete set, i.e., for the Green's function to be expressible as a sum over QNM's, answering a conjecture by Price and Husain [Phys. Rev. Lett. {\bf 68}, 1973 (1992)]. In the cases where the QNM sum is divergent, procedures for regularization are given. The crucial condition for completeness is the existence of spatial discontinuities in the system, e.g., the discontinuity at the stellar surface in the model of Price and Husain.

be in general negative, as the system giving rise to QNM's is nonhermitian. Yet Price and Husain [7] have given a model of relativistic stellar oscillations that does have a complete set of QNM's.
In this Letter, we consider this question in a general context, and give conditions for completeness to hold. The crucial condition is the existence of spatial discontinuities. (In the model of [7], this discontinuity corresponds to the stellar surface [8].) Such a condition is not surprising: a discrete sum of QNM's can at best be complete over a finite interval; the discontinuities provide a natural demarcation of this interval, analogous to the boundaries of an optical cavity.
The propagation of waves in curved space is often modeled by the Klein-Gordon (KG) equation [9] Dφ(x, t) ≡ ∂ 2 with the outgoing wave boundary condition at infinity. The potential V (x), assumed to be positive, bounded and vanishing at infinity, describes the scattering of wave by the background geometry. We show that the QNM's of (1) form a complete set, in the sense that the Green's function can be expressed as a sum over QNM's (cf. (10) below), provided the following three conditions hold: (i) V (x) is everywhere finite, and vanishes sufficiently rapidly, as x → ∞, in a sense to be defined below; (ii) there are spatial discontinuities demarcating a finite interval; (iii) consideration is limited to certain domains of spacetime.
In particular, for propagation from a source point y to an observation point x in a time t, the QNM's give a complete description for (a) y inside the interval, x outside the interval, and t > t p (x, y), for a certain t p ; or (b) both y and x inside the interval, and t > 0 (with the retarded Green's function being zero for t ≤ 0 ). Case (b) is analogous to the completeness of normal modes in hermitian systems defined on a finite interval.
The above conditions allow dispersion, backscatter, as well as differences in the damping times, the absence of which has been conjectured to be important [7]. The results are valid to all orders in the rate of leakage. (Lowest order results would be trivial, since the system becomes hermitian in that limit.) These results extend our previous work on the wave equation [10].
The spatial coordinate x in (1) often represents a radial variable, so first consider a half-line problem (x ≥ 0) with φ(x = 0, t) = 0. The QNM's are eigen-solutions to the time-independent KG equation with the boundary conditionsφ(0) = 0 and the outgoing wave condition at infinity. The eigenfunctions and eigenvalues are denoted asφ(x) = f j (x) and ω = ω j .
The retarded Green's function for the system is defined by DG(x, y; t) = δ(x − y)δ(t) with G = 0 for t ≤ 0, and the boundary conditions are (i) G = 0 for either x = 0 or y = 0, and (ii) the outgoing wave condition as either x → ∞ or y → ∞. In the frequency domain, and henceforth choosing y ≤ x,G(x, y; ω) = f (ω, y)g(ω, x)/W (ω), where f and g are solutions to (2) with the boundary conditions f (ω, x = 0) = 0; f ′ (ω, x = 0) = 1 [11], and The Wronskian W (ω) = gf ′ − f g ′ is independent of x, and its zeros are the QNM . For simple zeros [12], the residues of Multiple zeros can be handled readily. To analyse the denominator ∂W/∂ω, start with the defining equation for f (ω j , x) and g(ω, x). The usual manipulations lead to where the integral is taken along any contour from x = 0 to x = X. Use the outgoing wave condition to evaluate f and g at the upper limit, and take ω → ω j by l'Hospital's rule. This then gives for X → ∞. Since f and g are proportional at these poles, a generalized norm of the QNM's can be defined as with which the residue ofG at ω j can be expressed as The generalized norm, introduced in other contexts [1,13,14], involves f 2 rather than |f | 2 , and is therefore complex. Although each term on the right of (5) does not have a limit as X → ∞, the limit exists for the sum.
Next write G in terms ofG by a Fourier integral, and distort the contour to a large semicircle in the lower half ω plane. One then sees that The sum comes from the zeros of W (ω), I c is the integral along a semicircle at infinity, and The first step is straightforward, by appealing to well-known results in the quantum theory of scattering [15]. Since f and g satisfy the Schroedinger equation with energy ω 2 , they are analytic functions of ω if the potential is bounded and "has no tail" [15], in the Our work on potentials that have a tail, e.g., inverse power laws, will be reported elsewhere.
Secondly, on the large semicircleG can be estimated by the WKB approximation.
As |ω| → ∞ WKB approximation is valid except at points where the potential has discontinuities, which can be handled by connecting the WKB solutions across them. Let G can then be obtained in terms of φ(x) and k(x).
Consider a potential with a step discontinuity at x = a. The reflection coefficient is For simplicity we give the argument only for Case b: 0 < y ≤ x < a. It is straightforward to show that where . Now on the semicircle ω = ω R + iω I = Ce iθ , π < θ < 2π, as ω I = C sin θ → −∞, both the numerator and the denominator ofG are dominated by the term proportional to R, and G(x, y; ω)e −iωt ≃ e −iω(t+x−y) /ω .
As C → ∞, this vanishes for t > 0. This conclusion remains valid if V (x) has a discontinuity only in its p-th derivative (p = 0, 1, 2, ...), since the reflection coefficient R would vary as ω −(p+2) [16]. Thus we have proved (apart from a technical hitch mentioned below) that for a discontinuous potential, the QNM's are complete for t > 0. For Case a, i.e. 0 < y < a < x, a similar estimate gives, in place of (9) However, there is one technical hitch. The QNM sum is in fact a divergent series, but converges to the correct answer when regularised in a standard way [17]. The simplest regularization is to first invoke symmetry and only keep modes with Re ω j > 0 (hereafter denoted as j > 0) in the sum in (7); secondly replace all times t by t − iτ , with τ → 0 + . This implies the use of a regulating factor I j (τ ) = exp(−ω j τ ) multiplying the contribution of the j-th QNM. The need for regularization can be seen as follows. The proof that the integrand vanishes on the large semicircle as C → ∞ is not valid near the real axis, in particular, where Im ω = O(log C). In this part, a factor such as exp(Im ωσ) (σ real and positive, e.g.
behaves like a power of C rather than exponentially, and it is necessary to examine the integrand more carefully [18]. If R ∼ ω −q , the powers of ω in the integrand for G are inadequate to control the contribution along the semicircle if q ≥ 1 [19]. For large p, the divergence is more severe, the regulated sum converges more slowly to the correct result as τ → 0, and the use of a small finite τ becomes increasingly inaccurate.
Thus, in practice, the QNM sum is only useful for relatively strong discontinuities, e.g.
To summarize, we have proved, under the conditions stated, that the Green's function G is expressible, in a certain domain of x, y, t, as the regulated QNM sum ConsideringĠ as t → 0 + , valid for x, y ∈ [0, a]. This is perhaps the more familiar notion of completeness [19].
We now focus on (11), noting that such a resolution of the identity will be useful for a variety of problems similar to those of hermitian systems. The QNM representation of G for t > 0 will be further discussed elsewhere in the context of initial value problems. As an example, consider a potential defined on the full line, with V (x) = V 1 > 0 for 0 ≤ x ≤ a, and zero otherwise. This potential has discontinuities and no tail, thus satisfying the conditions necessary for our results. The QNM's of this system are easily found. We consider (11), but as an approximate equality for some small nonzero regularization parameter τ , and in a distribution sense, e.g.
for x, y 1 , y 2 ∈ [0, a]. Figure 1 shows that for small τ , the partial sum indeed converges close to the right hand side. Without the regulator, the sum does not converge.
This example further illustrates why the series needs to be regulated. Suppose that for a full-line problem, the reflection coefficients on both sides behave as R(ω) ∼ ω −q at high frequencies (q = 2 for the present situation). By setting the denominator in (8) to zero [20], it is straightforward to show that Im ω j a ∼ −q log |j|. In the region 0 < x, y < a, a typical term in the product f j (x)f j (y) goes as where Λ = max(|x + y − a|, |x − y|). Since the maximum value of Λ is a, without the regulator the worst behavior of the summand in (10) is |j| q−1 . The sum would not converge [19] without a regulating factor if q ≥ 1. The culprit is Im ω j a ∼ −q log |j|, from which one can say unequivocally that the need for regulating the sum is an intrinsic property of KG open systems.
We have shown that for potentials that have no tail but which contain spatial discontinuities, the QNM sum is complete, if suitably regulated. This result settles a question in the literature [7]. The generic need for regularization, in particular, distinguishes the KG equation from the wave equation. When there is a tail, there will be scattering of the wave from asymptotic regions of space, leading to power-law type long time behavior, which will be discussed elsewhere.