normal-mode frequencies of reissner-nordstrom black holes

Normal-Mode Frequencies of Reissner-Nordstrom Black HolesAuthor(s): Nils AnderssonSource: Proceedings: Mathematical and Physical Sciences, Vol. 442, No. 1915 (Aug. 9, 1993), pp.427-436Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52291 .

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Normal-mode frequencies of Reissner-Nordstrom black holes

BY NILS ANDERSSONt Department of Theoretical Physics, Thunbergsvdgen 3, S-752 38 Uppsala, Sweden

The normal-mode frequencies of a Reissner-Nordstrom black hole are determined from a phase-amplitude formula, using numerical integration in the complex coordinate plane. The results obtained are numerically very accurate, extending previous higher-order WKB results of Kokkotas and Schutz as well as the continued fraction results of Leaver. The change in the characteristic frequency of each mode as the charge of the black hole increases is also discussed.

1. Introduction

The normal modes of a black hole, expected to be observable as a discrete spectrum after for example a supernova collapse, have attained much interest in modern

astrophysics. In a sense, the normal modes arise as source-free perturbations of the

background metric propagating outwards at spatial infinity and inwards across the event horizon. Due to radiation damping the characteristic normal-mode frequencies have complex values; the real part corresponds to the physical oscillation frequency and the imaginary part governs the damping rate of the mode.

From initial work (Zerilli 1974; Moncrief 1974a, b, 1975) it is known that the

equations governing small perturbations of Reissner-Nordstrom black holes

decouple into two differential equations of the second order. These equations can be studied separately. Assuming a time-dependence exp (- it), where wo is a complex valued frequency, the radial part of a perturbation is described by the equations (Chandrasekhar 1979, 1983, 1984)

d2 Z+ + [2 _ V] Z+ =0, v = 1,2. (1)

The tortoise coordinate, r*, is related to the spatial radius, r, by

d A d dr r2 dr'

where A is the horizon function (see, for example, Chandrasekhar 1983). Superscripts indicate the different kinds of perturbations; a positive sign corresponds to polar perturbations while a negative sign represents axial perturbations. (The two kinds of

perturbations are often said to be of even and odd 'parity', respectively. This

terminology is unfortunate, however, since it does not agree with the common

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

Proc. R. Soc. Lond. A (1993) 442, 427-436 ? 1993 The Royal Society Printed in Great Britain 427



definition of parity, in for example atomic physics. In this paper the terminology of Chandrasekhar (1983) will be used, primarily since it is non-committal in this matter.)

In the Schwarzschild limit, when the charge of the black hole vanishes, the eigenfunctions Z1 and Z+ correspond to purely electromagnetic and gravitational perturbations, respectively. A fundamental difference in a charged environment is that a variation in the electromagnetic field will induce gravitational radiation and vice versa. A normal mode of a Reissner-Nordstrom black hole therefore generally corresponds to emission of both gravitational and electromagnetic radiation. It should, of course, be remembered that the frequencies are so low (105 Hz or less) that the electromagnetic radiation is not expected to propagate in an astrophysical environment. The greatest prospect for detection of radiation originating in normal- mode oscillations of a black hole therefore relies on the future identification of gravitational waves.

Using geometrized units, c = G = 1, and choosing the mass, M, of the black hole equal to unity (this is basically equivalent to introducing dimensionless quantities o = wM and r = r/M) we have

A = r2-2r+ 2, (3)

where e is the charge of the black hole. The outer and inner horizons, r+ and r_, of the Reissner-Nordstrom black hole are solutions to A = 0, i.e. are explicitly given by

r+ =1 + V(1-e2). (4)

From this follows that lel must be smaller than, or equal to, unity. Although it is hard to imagine a physically realistic situation where e is

significant at all (a macroscopic body with a measurable net charge) an investigation of the situation where e is close to 1 does not lack interest. On the contrary, since the Reissner-Nordstr6m solution provides a more general framework than the Schwarzschild geometry, this case may contribute significantly to our understanding of space and time. The highly charged case may even give us some information about what to expect in an investigation of a perturbed Kerr black hole. That is; in the charged case we do not only have the event horizon, r+, but also an inner (Cauchy) horizon, r_. In this respect the effect of the charge is similar to that of the angular momentum in the Kerr case. It is conceivable that a method that successfully deals with this feature of the Reissner-Nordstrom problem may be used with confidence when the Kerr metric is considered. If, on the other hand, the method is not reliable in the Reissner-Nordstrom case it is hardly worth applying it to perturbations of the Kerr black hole at all.

After a substitution (Fr6man et al. 1992) =+ = (A/r2)iZ), v=1,2, (5)

the differential equation (1) can be written

dr ̂ +R+ + = 0, v = 1,2. (6)

The analytic function R+ is given by

R + = {)2- 2V+-22 rd2 dr +4 -dr = 1,2. (7) Proc. R. Sc. Lnd. Ar 4 (dr\r

Proc. R. Soc. Lond. A (1993)

N. Andersson 428



Reissner-Nordstrim black holes

One of the remarkable facts in black-hole perturbation theory is that the two potentials V7 and V+, describing axial and polar perturbations, are not really independent (Chandrasekhar 1979, 1980). The physical contents of the corresponding differential equations are, in fact, the same and the potentials are related in a simple way. By defining

Uw = A/r3(Ar + ^), v = 1,2, (8)

1,2 = 3T /(9+4Ae2), (9)

and A= (1-1)(1+2), (10)

where I is the integer index of the spherical harmonics that describe the angular dependence of the perturbation, the effective potentials can be written

A du 2 ?^ + dr?2u2 +(A + 2) Auv, = 1,2. (11)

It can readily be verified that the potentials V2 and V2 reduces to the Regge-Wheeler (Regge & Wheeler 1957) and the Zerilli (1970) potential, respectively, in the limit e-?0.

2. The phase-amplitude normal-mode condition

In two previous paper (Froman et al. 1992; Andersson 1992) the normal-mode oscillations of a Schwarzschild black hole have been investigated. Froman et al. (1992) derived a condition determining the normal-mode frequencies using the

phase-integral method (see Froman & Froman (1991) for references). The accuracy of their phase-integral formula was not satisfactory, however. That is; the numerical results obtained did not agree well with, what was believed to be, accurate results

published by Leaver (1985) for but the first few modes. The need for reliable

independent results, that could be used for comparison, was apparent. After

generalizing an idea of Newman & Thorson (1972) to use in the complex coordinate

plane, a numerical integration method was developed (Andersson 1991). An exact relation between the 'phase' and the 'amplitude' of a general solution to a

Schrodinger-like differential equation, such as (6), forms the basis of the method.

Consequently it was (somewhat ambiguously) referred to as the phase-amplitude method. This method was used to generate very accurate normal-mode frequencies for the Schwarzschild black hole (Andersson 1992). In fact, the results obtained agree perfectly with (and improve on) those of Leaver. It could be concluded that the

phase-integral treatment of Froman et al. (1992) was not, although more accurate than any previous semi-analytic approach, appropriate for investigating highly damped normal modes. In the phase-integral analysis only two of the so-called transition points (essentially the zeros of R+) were considered. It was suggested that the effect of further transition points (for axial perturbations of a Schwarzschild black hole there are four) must be accounted for in an accurate analysis.

Using formulae that are valid when all transition points can be considered as lying well away from each other, Andersson et al. (1992) has recently derived a phase- integral condition that is reasonably accurate for highly damped modes. This new formula can be considered as the first correction to the formula of Froman et al.

(1992). Moreover, by using uniform approximations, Andersson & Linnseus (1992) have derived formulae that remain valid also when some of the transition points are close. In all these investigations the accuracy of the phase-integral results was inferred from comparison with phase-amplitude calculations. The phase-amplitude Proc. R. Soc. Lond. A (1993)

429



method generates reliable results for, at least, the first fifty modes and each value of 1. The only factor restricting its use seems to be computing time (see the concluding remarks by Andersson & Linnaeus (1992)).

In the present paper the use of the phase-amplitude normal-mode condition in the context of Reissner-Nordstrom black holes is discussed. For a closer description of the phase-amplitude formulae we refer to the previous paper (Andersson 1992).

In the phase-amplitude method it is assumed that the general solution to (6) is a linear combination of the two functions

+ = q 2(r) exp {?i (r)dr}. (12)

The differential equation (6) is then replaced by the nonlinear equation

d2q 3 (dq2+2-- 0 =, v =1,2, (13)

2q dr2 4q2 \dr) + - = 2

and numerical difficulties that often appear in a direct integration of (6) may, to a

large extent, be avoided. That is, the numerically determined function q(r) is smooth and non-oscillatory provided that the integration of the nonlinear equation (13) is started from accurate initial conditions (Andersson 1991). In the present investi-

gation analytic expressions for q(r) given by the phase-integral approximation (Fr6man & Froman 1991) were used to initiate the integration of (13).

In the normal-mode problem, the requirement that the black hole be stable against small perturbations (with our choice of time-dependence) implies that the imaginary part of ( is negative. This means, however, that the desired normal-mode solutions to (6) (corresponding to outgoing waves at infinity and ingoing waves crossing the event horizon, r+) are exponentially increasing as infinity, or the horizon, is

approached along the real coordinate axis. This gives rise to numerical difficulties; - to identity a normal mode we must single out an exponentially small solution from the error in an exponentially large one (see Chandrasekhar & Detweiler (1975) or Nollert (1986) for further discussion of this point). Needless to say, this is not an easy task. To avoid this difficulty the boundary conditions determining a normal mode

may be introduced on so-called anti-Stokes lines in the complex coordinate plane (for a comprehensive discussion see Anderson et al. (1992b)). The (semiclassical) Stokes and anti-Stokes lines are contours along which the quantity [R+]ldr is purely imaginary and purely real, respectively. From each zero of R+ emanate three Stokes and three anti-Stokes line, see figure 1. The pattern of Stokes and anti-Stokes lines is a useful tool in analysing any second-order differential equation of the form (6).

Let us briefly return to the numerical integration of (13). As long as it is continued

along an anti-Stokes line the function q(r) remains nicely behaving. It can not, however, be continued through the vicinity of a transition point (mainly due to the so-called Stokes phenomenon). If this is attempted oscillations will appear, thus

leading to probable numerical inaccuracies (Andersson 1991). To avoid this problem, two solutions, as integrated from different regions of the coordinate plane, should be matched at the transition point (or any point relatively close to it).

In an investigation of the normal-mode problem we introduce three integration contours: A1, A and A3 as in figure 1. The contour A, corresponds to an anti-Stokes line that extends towards the event horizon, r+, from the innermost of the two transition points considered, t,. The contour A3 extends from the other transition


430 N. Andersson



Reissner-Nordstrom black holes

5 A3

4- t

3

2

i , .= - ....? A2--. 1

--1 0 1 2 3 4

Figure 1. The pattern of Stokes (dashed) and anti-Stokes (solid) lines for the function R- and a frequency corresponding to n = 1 and e = 0.5 in table 2. The transition points considered in the analysis are t1 and t2. Integration contours are indicated by Ax, A2 and A3. The event horizon is r+.

point, t2, towards infinity, while A2 connects the two points in an appropriate way (see Andersson (1992) for a thorough discussion). With these definitions the phase- amplitude condition determining a normal mode of the black hole can be written (Andersson 1992)

q2(r)dr = n7+1- 02, n = 0, 1,2,..., (14) tl

where n is a non-negative integer labelling the modes. The integration in (14) is to be performed along the contour A2. Introducing subscripts on the function q(r) to indicate on which of the integration contours it is numerically determined, the two connection phases, 01 and 02, are given by

q2 tan 01 = iql- I( 1 dql 1 dq t 2 41 dr dg,

(2 dr15)

and q2 tan 02= iq3 -( d3 q2d) r=t2. (16) 243 dr 2 dr

Note that it is necessary to introduce branch cuts to make 01 and 02 single valued (Andersson 1992).

In practise, the integration determining q2 was initiated in the point where the contour A2 crosses the real coordinate axis and then continued towards the two transition points. As initial value or the integration the first order phase-integral expression was used (see Fr6man & Froman 1991). The integrations determining q1 and q3 were initiated well away from the transition points using the ninth order

phase-integral expressions for q(r) and continued towards the transition points along the contours A1 and A3, respectively.

If the condition (14) is to generate reliable numerical results, the integration contours must be appropriately chosen. For a closer description of this point the

analysis of the Schwarzschild problem should be consulted (Andersson 1992). It is worth notice, however, that for high charges (when e is close to unity), as well as for

highly damped modes (Andersson & Linnaeus 1992), the pattern of anti-Stokes lines is more complicated than that depicted in figure 1. As IIm owl increases the anti-Stokes line A1 curls up in a tight spiral around the singularity at the horizon. Furthermore,


431



Table 1. Phase-amplitude results for Reissner-Nordstrom black holes

(Normal-mode frequencies corresponding to ~1 and I = 2.)

n = 0 n = n=2

0.00 0.4575955117-0.0950044258i 0.4365423857-0.290710143 li 0.401 1867339-0.5015873463i 0.10 0.4589262905-0.0950971924i 0.4379399045-0.290968941 li 0.4026929218-0.5019553450i 0.20 0.4629651787-0.0953734357i 0.4421817049-0.2917382330i 0.4072649027-0.503045097 5i 0.30 0.4698649734-0.0958263904i 0.4494296264-0.2929946417i 0.4150786183-0.5048095705i 0.40 0.4799259641-0.0964420852i 0.4600023145-0.2946901132i 0.4264815358-0.5071522320i 0.50 0.4936747435-0.0971923698i 0.4744594015-0.296729159 li 0.442085 2695-0.509 883 8186i 0.60 0.5120109157-0.0980166295i 0.4937572262-0.2989103228i 0.4629347140-0.5126143658i 0.70 0.5365077467-0.0987714133i 0.5195640989-0.3007667467i 0.4908422024-0.514464137 li 0.80 0.5701302340-0.0990690623i 0.5549927979-0.3010592253i 0.5291206161-0.5131543583i 0.90 0.6193976120-0.0975828697i 0.6066256953-0.2956085375i 0.5842171532-0.501 1593325i 0.99 0.6927520172-0.0886422048i 0.6786593043-0.2675038674i 0.6509530580-0.451 180391 8i

Table 2. Phase-amplitude results for Reissner-Nordstrom black holes

(Normal-mode frequencies corresponding to 2- and I = 2.)

e n=0 n=l n=2

0.00 0.3736716845-0.0889623154i 0.3467109968-0.2739148753i 0.3010534546-0.4782769832i 0.10 0.3739323754-0.0889905326i 0.3469810326-0.2739949524i 0.3013347991-0.4783938105i 0.20 0.3747444264-0.0890747863i 0.3478260318-0.2742328378i 0.302222 152 7-0.478 736 3303i 0.30 0.3761967826-0.0892129455i 0.3493500816-0.2746184890i 0.3038454329-0.4792750883i 0.40 0.3784368879-0.0893981137i 0.3517275307-0.2751243038i 0.3064243084-0.4799404498i 0.50 0.3816771513-0.0896123790i 0.3552129875-0.2756843767i 0.3102827686-0.4805820369i 0.60 0.3862174693-0.0898136748i 0.3601702905-0.2761500473i 0.3158848208-0.480879419 li 0.70 0.3924984909-0.0899044263i 0.3671333898-0.2761821833i 0.3238936124-0.4801213227i 0.80 0.4012171853-0.0896432298i 0.3769040058-0.2749433745i 0.3351715065-0.4765638872i 0.90 0.4135705032-0.0883330123i 0.3904569087-0.2700150438i 0.3495943473-0.4652176230i 0.99 0.4292967934-0.0842659442i 0.4035200546-0.2570142492i 0.3539125461-0.4435776104i

all transition points tend to cluster around this horizon for high charges. As e increases the outer and inner horizons also move together, as is obvious from (4). It is easily realized that the determination of the desired integration contour requires a great deal of care for these cases, and the application of (14) is not at all as straightforward as for the first few modes and moderate charges. In fact, because of the changes in the structure of the problem, it would be surprising if (14) could be used to consider the limiting case.when e goes to unity at all.

3. Numerical results and conclusions

Numerical calculations using the condition (14) have been performed for 1 > 2. For these cases both electromagnetic and gravitational radiation can be emitted. Calculations for the first five modes for = 2-5 were made (Andersson 1991). A sample of results (for the first few modes of the lowest multipole) are given in tables 1 and 2. Calculations were made for both axial and polar perturbations in order to confirm the accuracy of the results. Modes corresponding to the different kind of perturbations should, theoretically, be equivalent (Chandrasekhar 1979, 1980). Hence, a comparison of results obtained for the different potentials V+ and Vy effectively ensures the reliability of the method used.

The results in the tables should, of course, be compared with results from other


N. Andersson 432



Reissner-Nordstroim black holes

approaches to the normal-mode problem. A numerical technique developed by Chandrasekhar & Detweiler (1975) for the Schwarzschild black hole has been used by Gunter (1980, 1981) in the Reissner-Nordstrom case (see also Chandrasekhar 1983). The same numerical approach, together with a higher-order WKB treatment (Schutz & Will 1985; Iyer & Will 1987; Iyer 1987) was used by Kokkotas & Schutz (1988). These calculations can not really be expected to yield accurate numerical results for but the lowest lying modes of the Reissner-Nordstrom black hole, however. This is because they give poor results for highly damped modes already in the uncharged Schwarzschild case (Andersson 1992). Nevertheless, it must not be forgotten that the WKB results of Kokkotas & Schutz agree with the results in our tables to a few parts in 10-.

An approximate formula determining normal-mode frequencies can be obtained from a relation between the modes and the bound states of the inverted black-hole

potential (Ferrari & Mashoon 1984). Unfortunately, this approach also gives poor numerical results. This is probably because only a very crude approximation to the black hole potential is used in the actual calculations. It is possible, however, that this approach - together with a numerical method such as (say) the one discussed in this paper - may lead to interesting results. Can some of the problems involved in a direct analysis of the differential equation (6) be avoided? Such an investigation should be encouraged.

The continued fraction method developed by Leaver (and used by him to determine accurate normal-mode frequencies for the Schwarzschild black hole

(Leaver 1985)) seems to be reliable also in the Reissner-Nordstrom case (Leaver 1990, 1991). The phase-amplitude results of the present paper are in good agreement with Leaver's calculations and improve his results with roughly five significant digits.

It is interesting to study the qualitative change in the characteristic frequency of each mode as e is increased. Kokkotas & Schutz (1988) suggested that the real part of the normal-mode frequency increases monotonically with e and that Im 01 attains a maximum value for e w 0.7 or 0.8. The phase-amplitude results unveil further details. As the normal-mode index, n, increases the maximum in the imaginary part occurs for smaller values of e (for n = 3 and 4 it appears for e w 0.5). For the

eigenfunctions T+ and all but the two lowest lying modes it seems as if both the real and the imaginary part of the frequency attains extreme values as e increases. Furthermore, this first extremum is followed by several tiny 'wiggles', see figures 2 and 3. These new extrema appear for lower charges for the higher modes. In fact, for but the lowest lying modes the qualitative behaviour of the real and imaginary part with increasing charge is similar in this respect. For the eigenfunctions j+ our results

suggest that the real part of o is monotonically increasing with e for all modes, however. This is in accordance with the conclusion of Kokkotas & Schutz (1988).

Is there a plausible explanation for these 'wiggles' as the charge is close to unity ? Could it be that they indicate that the phase-amplitude calculations should not be trusted when e is close to its limiting value ? These questions are not trivial, and I will

only attempt to answer the second one; it was clear that the method was numerically stable even for the highest charges considered (e = 0.999). The convergence of the

root-searching routine in the complex (o-plane (based on Miller's method) was very rapid. Only two or three iterations were needed (given a one-digit initial guess for the

frequency). The numerical integration of (13) reliably generated functions accurate to at least one part in 10-14 (using a variable-step Adams method), which should be


433



0.25

. 0.24

0.23

E 0.22.

0.21

0.20 0 0.2 0.4 0.6 0.8 1

charge

Figure 2. The behaviour of the normal-mode frequencies for the eigenfunction T- as the charge is increased. The real part of o is plotted as a function of e for I = 2 and n = 4.

-0.89.

o -0.90.

f -0.91.

E -0.92.

-0.93. c ' -0.94

-0.95

0 0.2 0.4 0.6 0.8 1

charge

Figure 3. The behaviour of the normal-mode frequencies for the eigenfunction 7- as the charge in increased. The imaginary part of o is plotted as a function of e for I = 2 and n = 4.

more than sufficient. Moreover, the comparison of calculations for axial and polar perturbations implied that the results were accurate. All the intrinsic error tests indicate that the method is reliable. There may still be some cause for concern, however. Maybe the phase-integral expression used to initiate the integration close to the event horizon is not as accurate when the outer and the inner horizon are close to each other. If that is the case, our solution may not correspond to purely ingoing waves falling across the horizon. It could also be that a new feature of the problem (such as a third transition point that must be considered) enters as the charge is increased. These questions must be investigated further if the qualitative behaviour of the normal-mode frequencies as the charge is close to its limiting value is to be established beyond any doubt.

An interesting clue is provided by the numerical data that Leaver used to generate his figure 1 (Leaver 1990). That figure indicates the 'wiggles' discussed above.

Unfortunately, Leaver's figure is so small that any details remain unclear. The numerical results of Leaver are in good agreement with those of the present analysis for e < 0.99, however. That is, his results agree with those used to generate our

figures 2 and 3 (except for e = 0.999). Hence, it does not seem as if the 'wiggles' arise because of defects in the phase-amplitude analysis. Proc. R. Soc. Lond. A (1993)

N. Andersson 434



Reissner-Nordstr6m black holes

For higher charges than 0.99, our analysis is probably not reliable because the transition points 'cluster' close to the two horizons. Unfortunately, one may also doubt the reliability of Leaver's results in the limiting case. For n = 4 and e 6 0.999 it seems as if some very peculiar 'jumps' occur in the frequency. That is; for a very small change in e the frequency changes significantly. Would not a smooth change in the frequency be more expected? Nevertheless, Leaver's results provide some

interesting suggestions. Also for the fundamental mode is there a maximum in the real part of the frequency. This maximum occurs for e 0.9998. For all modes

'wiggles' appear in both the real and the imaginary part for charges larger than 0.999.

In spite of the open question regarding its reliability in the limit of extremely high charge it must be concluded that the phase-amplitude formula discussed in this

paper is very accurate. The numerical results generated from it agree perfectly with the most accurate results obtained previously (Leaver 1990). This means that normal-mode frequencies for a non-rotating black hole, with a physically realistic net

charge, can surely be considered as known. It also seems as if the normal-mode problem for Kerr black holes may be investigated using the phase-amplitude method

(although it may fail to provide results in the extreme Kerr limit). Interestingly, the method could also be adapted to study perturbations of relativistic stars. In those problems numerical integration is a necessity, since the equation of state for the stellar interior is analytically unknown (see for example Kokkotas & Schutz (1992)). A combination of numerical integration for the interior matched to a phase- amplitude investigation of the exterior may prove to be a very powerful method for stars.

I am grateful to Dr E. Leaver for sending me some of his unpublished results regarding this problem.

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Received 28 October 1992; accepted 9 December 1992


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