a numerically accurate investigation of black-hole normal modes

A Numerically Accurate Investigation of Black-Hole Normal ModesAuthor(s): Nils AnderssonSource: Proceedings: Mathematical and Physical Sciences, Vol. 439, No. 1905 (Oct. 8, 1992), pp.47-58Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2646877 .

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A numerically accurate investigation of black-hole normal modes

BY NILS ANDERSSONt

Department of Theoretical Physics, Uppsala University, Thunbergsvdgen 3, S-752 38 Uppsala, Sweden

The oscillations of a Schwarzschild black hole, describing the late time ringing expected after, for example, a gravitational collapse, are discussed in terms of the characteristic normal-mode frequencies. A condition determining these frequencies is derived within the phase-amplitude method. The numerical results obtained using this condition are of very high accuracy, and the phase-amplitude analysis seems to provide a powerful alternative to the previous investigations of the normal-mode problem.

1. Introduction

One of the interesting questions in modern astrophysics concerns the existence of black holes. It is important to investigate the possible modes in which the presence of a black hole could be manifested, i.e. to determine whether processes such as particles falling through the event horizon, or a gravitational collapse, might produce observable events.

It is well known from pioneering work on the stability of Schwarzschild black holes (Regge & Wheeler 1957; Thorne & Campollataro 1967; Zerilli 1970a, b; Vishveshwara 1970 a, b; Edelstein & Vishveshwara 1970) that, when perturbed away from spherical symmetry the black hole will radiate. An interesting property of the induced radiation occurs at late times, after an initial wave burst, when it is dominated by a set of characteristic frequencies. These frequencies correspond to the normal-mode oscillations of the black hole. Since the oscillations must be damped as radiation is emitted, each normal-mode frequency is complex valued, such that the real part determines the physical oscillation frequency and the imaginary part corresponds to the damping rate of the mode. The normal-mode spectrum is an intrinsic characteristic of the geometry of space-time depending only on the mass and the angular momentum of the black hole (Price 1972a, b; Chandrasekhar & Detweiler 1975; Cunningham et al. 1978, 1979; Gaiser & Wagoner 1980). Consequently, the same set of frequencies will be of relevance in many different processes involving dynamical perturbations of a black hole, and the normal-mode frequencies could theoretically be used to identify a perturbed black hole.

From direct numerical integration of the equations governing the perturbations of a black hole it is known that the radiation process is energetically dominated by the lowest radiating multipoles (Davis et al. 1971, 1972; Press 1971; Ruffini 1973; Ferrari & Ruffini 1984). Hence, the slowest damped normal modes of the lowest multinoles are the nhvqieallv most imnortant- In nraptiet it should also be noted

t Present address: Department of Physics and Astronomy, University of Wales, College of Cardiff, Cardiff CF2 3YB, U.K.

Proc. R. Soc. Lond. A (1992) 439, 47-58 Printed in Great Britain 47

(? 1992 The Royal Society



48 N. Andersson

that, the frequencies of the electromagnetic radiation emitted are too low to propagate in an astrophysical environment. This is in contrast to the low frequency gravitational waves that can propagate, and therefore the slowest damped normal modes corresponding to the gravitational quadrupole are expected to be the most relevant from an observational point of view.

In recent papers by Leaver (1986, 1988) and Sun & Price (1988, 1990) the excitation of the normal-mode oscillations was investigated in detail. The possible inversion of theoretically determined normal-mode frequencies into the mass and angular momentum of the black hole was discussed by Echeverria (1989).

2. The black-hole normal-mode problem

Due to the symmetry of the background metric, first-order perturbations of a Schwarzschild black hole can be expressed in terms of spherical harmonics. If the perturbation is assumed to have a time dependence exp (- iwt), with w a complex frequency, the radial part of the perturbation is governed by the differential equation

d2f/dz2 +R(z) f = 0. (1)

The analytical function R(z) is explicitly given by (Fr6man et al. 1992)

R(z) = ( 1- 2/z)-2 [0-2 - V(z) + 2/z3 - 3/z4], (2) where V(z) is the effective black-hole potential. For convenience, I have introduced the dimensionless quantities o = Mwo and z = r/m, where r is the spatial radius. I also assume that the mass of the black hole, M, is expressed in geometrized units, such that c = G = 1.

Mathematically, a normal mode is a solution of the differential equation (1) satisfying boundary conditions of purely outgoing waves at spatial infinity, z = + oo, and purely ingoing waves at the event horizon, z = 2. The latter boundary condition follows directly from the requirement that no point inside the event horizon is to be communicable to the space outside.

There exist two, seemingly independent, classes of first-order perturbations of the Schwarzschild metric. The effective potential describing axial perturbations, initially derived by Regge & Wheeler (1957), is given by

V(z) = ( 1- 2/z) [1(1 + 1)/Z2 + 2,/z3]. (3) The integer I specifies the angular dependence, and the quantity f8 is related to the spin-weight s of the perturbing field by

/= 1-s2 (4)

For scalar, electromagnetic and gravitational perturbations fi has the integer value 1, 0, -3, respectively. All integer-spin perturbation fields are thus described by remarkably similar equations. This may be considered somewhat surprising since the emission of gravitational radiation is physically more complicated than the emission of electromagnetic or scalar radiation.

The polar perturbations corresponding to a gravitational perturbing field are governed by the Zerilli potential (1970a, b)

I

____ 211A+

3A7 V(z) = 1-

_ (A_+ 2 +-3] (5)

where A = (l-1)(+2). (6) Proc. R. Soc. Lond. A (1992)



Black-hole normal modes 49

As was rigorously shown by Chandrasekhar (1975), and is also discussed in a recent paper by Anderson & Price (1991), the solutions of (1) corresponding to the two potentials (3) and (5) are related in a simple way. The physical contents of the differential equations are the same, and the normal-mode frequencies belonging to axial and polar perturbations are identical. Likewise, the physical contents of the two effective potentials above can be proved equivalent to the contents of the potential derived by Bardeen & Press (1973). For further details on the mathematical theory of black holes, and perturbations of the background metric, we refer to the extensive work of Chandrasekhar (1983).

(In many investigations of the normal-mode problem the solutions corresponding to the Regge-Wheeler and the Zerilli potentials are described as odd and even parity perturbations, respectively. This nomenclature is rather unfortunate, since it does not agree with the well-defined scientific meaning of parity, e.g. from atomic theory. In fact, with the common definition, a perturbation corresponding to the Regge-Wheeler potential is of odd (or even) parity for an even (or odd) value of 1, and conversely for a perturbation described by the Zerilli potential. Since it is non- committal in this matter, the nomenclature proposed by Chandrasekhar (1983) is preferable. Hence, the Regge-Wheeler potential corresponds to axial perturbations, while the Zerilli potential gives rise to polar perturbations. Furthermore, the axial perturbations induce a dragging of the inertial frame, i.e. a rotation of the black hole, which is not the case for the polar perturbations. The difference between the two classes of perturbations is clearly established in the description of Chandrasekhar.)

3. The phase-amplitude normal-mode condition Naturally, the effective potentials (3) and (5) exclude an exact solution of the

differential equation (1) in terms of known functions. However, as was recognized by Froman et al. (1992), the normal-mode problem is analogous to the quantum mechanical problem of complex scattering resonances. Consequently, explicit formulae for the calculation of normal-mode frequencies could straightforwardly be derived within the phase-integral method (for a recent review see Fr6man & Frdman 1991). It was concluded that, while the phase-integral method had some deficiencies, the phase-amplitude method used for comparison generated very accurate normal- mode frequencies. Hence, a closer description of the phase-amplitude method, and its applicability to the problem of calculating normal-mode frequencies for a black hole, is appropriate. (The phase-integral formula derived by Fr6man et al. fails to determine reliable frequencies for all but the lowest lying normal modes. In the analysis only two transition points were considered, and it was suggested that further transition points had to be included in an accurate analysis. Work along these lines is in progress.)

In the phase-amplitude method (Andersson 1991) the general solution of the differential equation (1) is given by a linear combination of the two functions

*f = q-2(z) exp[ + i q(z) dz]. (7)

This implies that the original equation, (1), can be replaced by the nonlinear differential equation

1 d q2q 3 (ddq) +q2_-R(Z) = O, (8)

Proc. R. Soc. Lond. A (1992)



50 N. Andersson

determining the function q(z). For practical purposes, it may be convenient to rewrite the differential equation (8) as a system of first-order equations. This can easily be achieved by introducing dq/dz = u, (8'a)

such that d2q/dz2 = du/dz = tu2/q-2q[q2 -R(z)]. (8'b)

Furthermore, an evaluation of the final phase-amplitude formula determining a normal-mode solution of the original differential equation (1) requires accurate knowledge of the integral of q(z) along the chosen contour of integration. Hence, we also introduce w = f qdz, i.e.

' ~~dw/dz = q, (8'c) to ensure that the desired integral is determined with the same accuracy as the function q(z) itself. This system of three coupled first-order differential equations may be integrated numerically by using any standard scheme. In this investigation, however, the real and the imaginary parts of each differential equation were integrated separately. In effect, the differential equation (8) was replaced by a system of six first-order differential equations. This system was integrated by using the Numerical Algorithms Group routine D02CBF.

The introduction of the normal-mode boundary conditions on the real coordinate axis inevitably gives rise to difficulties (cf. Fr6man et al. (1992) for a detailed discussion). This is due to an exponential increase in the desired normal-mode solutions along the real axis. To avoid these difficulties Fr6man et al. (1992) proposed that the conditions of outgoing waves as z - oo and waves travelling towards the event horizon, z = 2, should be imposed on appropriate anti-Stokes lines in the complex z-plane. The semiclassical anti-Stokes lines are defined as contours along which the quantity R2(z) dz is purely real. The pattern of anti-Stokes lines, corresponding to the function R(z), is a powerful aid in a semi-classical, or numerical, analysis of the normal-mode problem. For instance, as previously discussed (Andersson 1991), the numerically determined function q(z) is smooth as long as the integration of (8) is started from asymptotically accurate initial conditions and continued along anti-Stokes lines.

Furthermore, as is well-known from semi-classical theory, the same linear combination of the two functions (7) can not be used to represent the desired solution of (1) in the entire complex coordinate plane. If the function q(z) is numerically continued between two regions of the complex plane where the exact solution of (1) is expected to behave qualitatively different, q(z) will be oscillating (see the discussion by Andersson 1991). To avoid numerical instabilities in the integration of (8), and also obtain a continuous solution to (1), the numerically integrated solutions may be matched in the transition points, in this context zeros of the function R(z), that are considered as relevant. In the phase-amplitude analysis of the normal-mode problem we therefore introduce integration contours A1, A2 and A3 according to figure 1. The contour A1 corresponds to an anti-Stokes line that extends from the innermost of the two transition points considered, t1, towards the event horizon, z = 2. The contour A3 extends along an anti-Stokes line from the outer transition point t2 towards infinity, while the contour A2 connects the two transition points. Unfortunately, it may be difficult to realize exactly how A2 is to be chosen from, for example, figure 1. The only necessary requirement, however, is that it must not qualitatively differ too much from the two anti-Stokes lines that almost connect the transition points. In the analysis below, I introduce for clarity subscripts to Pr-oc. R. Soc. Lond. A (1992)




4-

0

-2 0 2 4 6 8

Re(r/M) Figure 1. The pattern of semiclassical anti-Stokes lines for the potential (3) and gravitational perturbations. The parameters correspond to 1 = 2 and n = 0 in ta,ble 3. The transition points considered in the phase-amplitude analysis are t1 and t2. Integration contours discussed are indicated by A1, A2 and A3. Since the contour A2 is not explicitly shown in the figure (it should not differ too much from the two anti-Stokes lines that almost connect the transition points) it has been put in a parenthesis. From each zero of the function R(z) emerge three semiclassical anti- Stokes lines.

indicate on which integration contour q(z) is to be numerically determined. Hence, the integration determining q1(z) is initiated well away from t1 on the contour A1. Likewise, q2(z) is to be determined by numerical integration from a point on A2 (presumably not too close to a transition point) towards t1 and t2, while the integrationof q3(Z) is initiated on A3 and continued towards t2.

An obvious advantage of the phase-amplitude method is that long-range integration, and therefore also unwanted numerical errors, may be avoided. However, this demands that the integration of (8) is started from accurate initial conditions. This demand can be satisfied if the integration is started from analytical expressions for q(z) given by the phase-integral approximation (see Fr6man & Fr6man (1991) for references). By using this method, the approximate expressions replacing q(z) can be obtained with almost any desired accuracy well away from all transition points. In the calculations reported in this paper the asymptotic initial conditions for q1(z) and q3(Z) (i.e. on the contours A1 and A3), were obtained by using the ninth-order phase-integral expression for q(z), while the first order of approximation was found sufficient for initiating the integration determining q2(z) (on A2). The phase of the approximate expressions was chosen in the same way as in the phase-integral analysis of the normal-mode problem (Fr6man et al. 1992). Consequently, with the proposed time dependence, the function #f corresponds to the desired outgoing-wave solution on A3, and also to ingoing waves towards the horizon (along A1). Note that the two linearly independent solutions (7) are of the same order of magnitude on the anti-Stokes lines Al and 3 In efeet, the desired solution can, on each of these lines, easily be singled out from the undesired one. Furthermore, since each of these solutions will be exponentially increasing towards the real coordinate axis it is straightforward to show that they correspond to the expected exponential growth on the real axis.

As already mentioned above, it is not safe to continue the integration of (8) through a region of the coordinate plane than contains transition points. To avoid




52 N. Andersson

numerical instabilities, the determination of q1(z) is never continued through the transition point t1. Instead the condition that the solution to (1) and its derivative should be continuous at the transition point is used. This enables us to express the solution that corresponds to the normal-mode boundary condition on A1 in terms of the function q2(z), which is nicely behaving on the contour A2. From the general phase-amplitude formulae (Andersson 1991) it follows that

Cq-2(z) exp iJ q,(z) dzg zon A1, (9a)

qq(t)cos (z) Cos q2(z)dz-Oi1 zonA2, (9b)

where C is an undetermined constant. The condition of continuity implies that the connection phase 01 is given by

q2tan01 = iq1- (q-1dq1/dz-q-1dq2/dz), z = t1. (10)

In general, it is necessary to introduce branch cuts to ensure that the complex phase 01 is single valued. I assume that cuts extend from tan 01 = + i upwards along the imaginary axis, and from tan 01 = - i downwards.

Likewise, it follows directly from the desired boundary condition of purely outgoing waves on A3 that

Dq-)exp i q3(z) dz] z on A] (la)

=D qq(t2) q2(z) cos[ Jq2(z)dz-02] zon A2 (llb)

where D is a constant. This is, of course, provided that the initial conditions for the integration determining q3(z) are properly chosen. The connection phase 02 is given

by qq2 tan 02 = iq3 - 1(q1 dq3/dz - q-1 dq2/dz), z = t2. (12)

Since the equations (9 b) and (11 b) are to represent the same solution on the contour A2, except for a constant factor, we must have

t2

q2(z) dz = nn+ 0 - 02, n = 0, 1, 2, .., (13)

with the integration performed along the contour A2. This is the exact phase- amplitude condition determining the normal-mode frequencies of a Schwarzschild black hole, and n is a non-negative integer labelling the modes.

As already mentioned above, it is imperative for the numerical success of the condition (13) that the integration contours are chosen with care. For example, on the contour A2, i.e. in the region of the coordinate plane 'between' the two transition points t1 and t2, the exact solution to the differential equation (1) is expected to have a standing-wave behaviour. Meanwhile, on the contour A1 the exact solution must consist of travelling waves. In choosing the contour A2, it is important that it does not enter the region of the coordinate plane where the solution to (1) consists of travelling waves. If it does, the numerically determined function q(z) may exhibit





(a) (b)

t2 4

0 t1X

-2 0 2 0 2 4 Re(r/M)

Figure 2. The pattern of semiclassical anti-Stokes lines for the potential (3) and gravitational perturbations. The parameters correspond to 1 = 2 and (a) n = 2 (b) n = 6 in table 3. The transition points considered in the analysis of the normal-mode problem are t1 and t2.

poles (Andersson 1991). In effect, unwanted multiples of t may add to the integral of q(z). Therefore, it is crucial for the reliability of the condition (13), i.e. the uniqueness of n, that A2 never passes too close to A1. As Im oj increases, however, the anti-Stokes line defining A1 curls up in a tight spiral around the horizon, see figure 2 b. Consequently, the determination of the integration contours requires great rigour for the highly damped normal modes.

Finally, it should be mentioned that the normal-mode frequencies lie symmetrically distributed, with respect to the imaginary axis, in the complex plane. Normal-mode frequencies with Re o- < 0 and Re o- > 0 are in one-to-one correspondence. In the analysis above it is assumed that Re o- > 0 and that the initial conditions for the numerical integration of (8) are chosen such that the solution (II a) corresponds to an outgoing wave on the contour A3 of figure 1, i.e. a wave travelling away from the transition point t2. Although the identification of incoming and outgoing waves is reversed for frequencies with Re o- < 0, the general analysis of the normal-mode problem remains unchanged.

4. Numerical results

I have calculated normal-mode frequencies for the first 11 modes and the two lowest multipoles using the condition (13). The lowest radiating multipole correspond to a value of I equal to the spin-weight of the perturbing field. Calculations have been made for scalar, electromagnetic and gravitational perturbations. The numerical results are listed in tables 1-3. To ensure the reliability of the results, calculations for a gravitational perturbating field were made by using the Regge-Wheeler potential (3) and the Zerilli potential (5). Since the normal-mode frequencies determined from the two different potentials should theoretically be equivalent (Chandrasekhar 1975), a comparison of such calculations effectively confirms the accuracy of the results. In fact, the numerical results obtained from calculations by using the Regge-Wheeler and the Zerilli potentials are in perfect agreement, at least to the accuracy quoted in table 3. Proc. R. Soc. Lond. A (1992)



54 N. Andersson

Table 1. Normal-mode frequencies for scalar perturbations

(The normal-mode frequency o is given in units of (M)-')

n 1=0 1= 1

0 0.1104543 -0.1048943i 0.2929361 -0.0976600i 1 0.0861169 -0.3480524i 0.26444865 -0.306257 39i 2 0.075741 936-0.601 078590i 0.229539335-0.540 133425i 3 0.070410138-0.853677318i 0.203258386-0.788297823i 4 0.067074304- 1.105631 880i 0.185 109020- 1.040762 113i 5 0.064741 722- 1.357 139465i 0.172076811 - 1.294 119698i 6 0.062993830- 1.608341 366i 0.162225829- 1.547439894i 7 0.061621 213- 1.859326830i 0.154457052- 1.800490298i 8 0.060506253-2.110 153350i 0.148 126049-2.053236464i 9 0.059 577 195 - 2.360 859 381 i 0. 142 833 832 - 2.305 699 787i

10 0.058787470-2.611471606i 0.138320334-2.557914625i

Table 2. Normal-mode frequencies for electromagnetic perturbations

(The normal-mode frequency o is given in units of (M)-')

n 1=1 1=2

0 0.248263272-0.092487709i 0.457595512-0.095004426i 1 0.214515420-0.293667646i 0.436542386-0.290710 143i 2 0.174 773 568 - 0.525 187 599i 0.401 186 734 - 0.501 587 346i 3 0.146 176699-0.771 908924i 0.362595032-0.730 198514i 4 0.126554 146- 1.022550284i 0.328736671 -0.971 609379i 5 0.112252791-1.273925619i 0.301492996-1.219715250i 6 0.101 214705- 1.525266334i 0.279844918- 1.470793202i 7 0.092323 748 - 1.776399233i 0.262379088- 1.723 071 572i 8 0.084 934971 -2.027 306071 i 0.247 978 764 - 1.975 763447 i 9 0.078 649 642 - 2.278 008 820 i 0.235 855 298 - 2.228 523 247 i

10 0.068426557-2.778919402i 0.225460819-2.481201326i

It is worth mentioning that, in the phase-amplitude analysis no other transition points than t1 and t2 are of any consequence. This is, of course, provided that the integration contours are appropriately chosen. Hence, the deficiencies of the phase- integral method, cf. the analysis by Fr6man et al. (1992), can be avoided. Furthermore, the numerical results of the present paper allow us to explain the difficulties inherent in the phase-integral treatment.

Let us therefore briefly discuss gravitational perturbations corresponding to the Regge-Wheeler potential (3) with 1 = 2. It was concluded by Fr6man et al. (1992) that the phase-integral treatment of the normal-mode problem was very accurate for the lowest lying modes, when lIm ol << IRe ol. However, phase-integral calculations using the potential (3) had the deficiency of breaking down already at moderately low-lying modes. A probable reason for this unsatisfying behaviour is that the two transition points t1 and t2, considered as relevant in the phase-integral analysis, are not the only transition points that must be included in a rigorous approximate analysis, cf. figure 2a. For highly damped modes, when JIm oi > Re oj, the phase- integral treatment completely broke down. It is evident that a rigorous analysis demands that further transition points are considered, cf. figure 2b. It was also concluded by Fr6man et al. (1992) that the accuracy of the phase-integral results for a given value of n improves with increasing 1. This is not really surprising, since the Proc. R. Soc. Lond. A (1992)




Table 3. Normal-mode frequencies for gravitational perturbations

(The normal-mode frequency o- is given in units of (M)-'). Calculations made for the Regge-Wheeler potential (3) and the Zerilli potential (5) agree to the accuracy quoted.)

n 1=2 1=3

0 0.373671 684-0.088962315i 0.599443288-0.092703048i 1 0.346 710 997 - 0.273 914 875 i 0.582 643 803 - 0.281 298 113 i 2 0.301053455-0.478276983i 0.551684901-0.479092751i 3 0.251 504962-0.705 148202i 0.511961911 -0.690337096i 4 0.207 514580 - 0.946 844 891 i 0.470 174006- 0.915 649393i 5 0.169299403- 1.195608054i 0.431 386479- 1.152 151 362i 6 0.133252340- 1.447910626i 0.397659524- 1.395912243i 7 0.092822336-1.703841 172i 0.368992276-1.643844528i 8 - 0.344618319- 1.894032804i 9 0.063 263 505 - 2.302 644 765 i 0.323 683 132 - 2.145 398 950 i

10 0.076553463-2.560826617i 0.305460902-2.397354550i

lowest-lying mode follows the approximate formula IRe ui (27)-2l (Press 1971; Davis et al. 1972; Ferrari & Mashoon 1984) while lIm ul 0.1 independently of 1. Consequently, the phase-integral treatment will break down for a higher value of n if I is increased.

In discussing table 3 it is also important to mention the algebraically special solutions investigated by Couch & Newman (1973), and in greater detail discussed by Chandrasekhar (1984). The algebraically special modes correspond to purely imaginary frequencies and are said to require boundary conditions of either purely ingoing or purely outgoing waves. Such perturbations give rise to a special eigenvalue for the Schwarzschild black hole

G=- 1iA(A+2), (14)

with A given by (6), i.e. o- 2i for 1 = 2. However, by using the analysis of the present paper, it is impossible to determine a normal mode corresponding to a purely imaginary frequency. This is due to the fact that, when Re o vanishes, the semiclassical anti-Stokes lines form closed contours in the region of the event horizon. Hence, the desired boundary condition of ingoing waves at the event horizon cannot be imposed on an anti-Stokes line. Consequently, the special normal mode, for I = 2 corresponding to n = 8, has been excluded from table 3.

5. Conclusions

Many previous investigations of the normal-mode problem have been published during the past two decades. Direct numerical integration of the differential equation governing the perturbations has been used to investigate the spectrum of radiation by several authors (see, for example, de la Cruz et al. 1970; Davis et al. 1971, 1972; Press 1971; Cunningham et al. 1978, 1979). To avoid the numerical difficulties inherent in such calculations a Riccati equation associated with (1) was integrated by Chandrasekhar & Detweiler (1975) to generate the normal-mode frequencies. However, due to numerical instabilities on the real coordinate axis, cf. the discussion of the normal-mode boundary conditions by Froman et al. (1992), this method completely fails to determine frequencies for which lIm 61 > 0.1.

In several recent papers (Schutz & Will 1985; Iyer & Will 1987; Iyer 1987; Will




56 N. Andersson

& Guinn 1988), a higher-order WKB method was developed to investigate the normal modes of a Schwarzschild black hole. In this method a local Taylor expansion of the effective potential is used to bridge the region between the relevant transition points. If these transition points are not close-lying, i.e. for but the lowest-lying mode, this is a poor approximation. Consequently, the proposed WKB method inevitably breaks down as n is increased. In an attempt to remedy this failure Guinn et al. (1990) proposed a different WKB condition determining the normal-mode frequencies. The proposed formula is analogous to the well-known Bohr-Sommerfeld quantization condition in quantum mechanics. Guinn et al. never attempted to derive the formula in the complex frequency case. Instead, they assumed that it would follow from an analytical extension of the real frequency situation. However, since the formula proposed is, to the lowest orders of approximation, identical to the formula derived by Froman et al. (1992) the analysis is justified. There has been several other semi-analytical approaches to the problem (Fackerell 1971; Chung 1973; Gaiser & Wagoner 1980). The most interesting of these exploit the relation between the normal modes and the bound states of the inverted black-hole potential (cf. Blome & Mashoon 1984; Ferrari & Mashoon 1984; Zaslavskii 1991). Un- fortunately, the numerical results of all these investigations are poor.

The so far most reliable numerical results were obtained by Leaver (1985), who successfully generated normal-mode frequencies from an exact infinite-series representation of the solutions of a differential equation equivalent to (1), together with a numerical solution involving continued fractions. Leaver concluded that for each value of 1, there exists an infinite number of normal modes, such that for large n the normal-mode frequencies lie evenly spaced along an asymptote parallel to the imaginary axis. Although a method combining the series solution of Leaver with the Hill-determinant method was recently suggested by Majumdar & Panchapakesan (1989), the analysis of Leaver remains numerically superior.

It is obvious that determining the normal-mode frequencies of a Schwarzschild black hole is far from a trivial problem. Although the phase-integral treatment of Froman et al. (1992) generated more reliable results than any previous semi- analytical approach, it completely failed to determine modes for which lIml? > IRe o1. Of course, it must still be remembered that the physically most important, since they are expected to energetically dominate the radiation process, are the least damped modes for the electromagnetic dipole and the gravitational quadrupole. The breakdown of the phase-integral treatment indicates the demand for a more sophisticated phase-integral approach to the highly damped normal-mode problem. It is probable that the phase-integral treatment can be improved in accuracy by using the results of a uniform treatment of clusters of transition points. Work along these lines is in progress.

Finally, the high numerical accuracy of the phase-amplitude treatment discussed in the present paper must be stressed. The phase-amplitude calculations confirm and improve the published results of Leaver (1985). It should be noted, though, that Leavers calculations were limited to an accuracy of seven decimal places due to the computer used. Recent double-precision calculations using the continued fraction method give even more accurate results (Leaver, personal communication). The new continued fraction results, for gravitational perturbations and I = 2, are in perfect agreement with the phase-amplitude results of the present paper, down to possible round-off errors.

Conclusively, the phase-amplitude method provides a reliable and powerful tool Proc. R. Soc. Lond. A (1992)




that may be of great importance also in other problems where a high numerical accuracy is desired. In a future paper the method will be used to discuss the normal- mode frequencies of Reissner-Nordstr6m black holes.

I am grateful to Dr Edward Leaver for sending me his Ph.D. thesis and, most timely as this paper was almost finished, discussing his recent results with me. I also thank Professor Subrahmanyan Chandrasekhar, especially for sharing his thoughts on the description of the different kinds of I-nv+ -II l-- ;rvn ti n

References Anderson, A. & Price, R. H. 1991 Phys. Rev. D 43, 3147-3154. Andersson, N. 1991 Accurate investigation of scattering phenomena. Doctoral thesis, Uppsala

University, Sweden. Bardeen, J. M. & Press, W. H. 1973 J. math. Phys. 14, 7-19. Blome, H. J. & Mashoon, B. 1984 Phys. Lett. A 100, 231-234. Chandrasekhar, S. 1975 Proc. R. Soc. Lond. A 343, 289-298. Chandrasekhar, S. 1983 The mathematical theory of black holes. New York: Oxford University

Press. Chandrasekhar, S. 1984 Proc. R. Soc. Lond. A 392, 1-13. Chandrasekhar, S. & Detweiler, S. 1975 Proc. R. Soc. Lond. A 344, 441-452. Chung, K. P. 1973 Nuovo Cim. B 14, 293-308. Couch, W. E. & Newman, E. T. 1973 J. math. Phys. 14, 285-286. Cunningham, C. T., Price, R. H. & Moncrief, V. 1978 Ap. J. 224, 643-667. Cunningham, C. T., Price, R. H. & Moncrief, V. 1979 Ap. J. 230, 870-892. Davis, M., Ruffini, R., Press, W. H. & Price, R. H. 1971 Phys. Rev. Lett. 27, 1466-1469. Davis, M., Ruffini, R. & Tiomno, J 1972 Phys. Rev. D 5, 2932-2935. de la Cruz, V., Chase, J. E. & Israel, W. 1970 Phys. Rev. Lett. 24, 423-426. Echeverria, F. 1989 Phys. Rev. D 40, 3194-3203. Edelstein, L. A. & Vishveshwara, C. V. 1970 Phys. Rev. D 1, 3514-3517. Fackerell, E. D. 1971 Ap. J. 166, 197-206. Ferrari, V. & Mashoon, B. 1984 Phys. Rev. D 30, 295-304. Ferrari, V. & Ruffini, R. 1984 Phys. Lett. B 98, 381-384. Frdman, N. & Frdman, P. 0. 1991 In Forty more years of ramifications. Spectral asymptotics and

its applications (ed. S. A. Fulling & F. J. Narcowich), pp. 121-159. Texas A&M University. Froman, N., Froman, P. O., Andersson, N. & Hokback, A. 1992 Phys. Rev. D 45, 2609-2616. Gaiser, B. D. & Wagoner, R. V. 1980 Ap. J. 240, 648-657. Guinn, J. W., Will, C. M., Kojima, Y. & Schutz, B. F. 1990 Class. quant. Grav. 7, L47-53. Iyer, S. & Will, C. M. 1987 Phys. Rev. D 35, 3621-3631. Iyer, S. 1987 Phys. Rev. D 35, 3632-3636. Leaver, E. W. 1985 Proc. R. Soc. Lond. A 402, 285-298. Leaver, E. W. 1986 Phys. Rev. D 34, 384-408. Leaver, E. W. 1988 Phys. Rev. D 38, 725. Majumdar, B. & Panchapakesan, N. 1989 Phys. Rev. D 40, 2568-2571. Press, W. H. 1971 Ap. J. 170, L105-108. Price, R. HI. 1972 a Phys. Rev. D 5, 2419-2438. Price, R. H. 1972b Phys. Rev. D 5, 2439-2454. Regge, T. & Wheeler, J. A. 1957 Phys. Rev. 108, 1063-1069. Ruffini, R. 1973 Phys. Rev. D 7, 972-976. Schutz, B. F. & Will, C. M. 1985 Ap. J. 291, L33-36. Sun, Y. & Price, R. H. 1988 Phys. Rev. D 38, 1040-1052. Sun, Y. & Price, R. H. 1990 Phys. Rev. D 41, 2492-2506.




58 N. Andersson

Thorne, K. S. & Campolattaro, A. 1967 Ap. J. 14, 591-611. Vishveshwara, C. V. 1970a Nature, Lond. 227, 936-938. Vishveshwara, C. V. 1970b Phys. Rev. D 1, 2870-2879. Will, C. M. & Guinn, J. W. 1988 Phys. Rev. A 37, 3674-3679. Zaslavskii, 0. B. 1991 Phys. Rev. D 43, 605-608. Zerilli, F. J. 1970a Phys. Rev. Lett. 24, 737-738. Zerilli, F. J. 1970b Phys. Rev. D 2, 2141-2160.

Received 4 February 1992; accepted 15 May 1992




a numerically accurate investigation of black-hole normal modes

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