Nov 15, 1992 - The quasinormal-mode frequencies of a Schwarzschild black hole are calculated within an accurate phase-integral analysis. Two different ...
PHYSICAL REVIEW D
Quasinormal
15 NOVEMBER 1992
VOLUME 46, NUMBER 10
modes of a Schwarzschild
black hole: Improved phase-integral
treatment
Nils Andersson* and Staffan Linna:us Department
of Theoretical
Physics, University
of Uppsala,
Thunbergsvagen
3, S-752 38 Uppsala, Sweden
(Received 9 June 1992)
The quasinormal-mode frequencies of a Schwarzschild black hole are calculated within an accurate phase-integral analysis. Two different phase-integral formulas are derived by means of uniform approximations using parabolic Weber functions and Coulomb wave functions, respectively. These formulas are valid when clusters of possibly close-lying transition points in the complex coordinate plane must be considered. By comparison with results of exact phase-amplitude calculations the phase-integral results are proved to be of high accuracy. Conclusively, the improved phase-integral method so far provides the most scient way to determine approximate values for the characteristic frequencies of the lowest-lying, as well as highly damped, quasinormal modes of a Schwarzschild black hole.
PACS number(s): 97.60.Lf, 02.60. + y, 02.70. + d, 04.30.+ x
I. INTRODUCTION After an initial wave burst a small perturbation of a black hole is radiated away in an oscillating manner. The frequencies of these oscillations correspond to the socalled quasinormal modes of the black hole and depend only on the perturbed metric itself. In the case of the spherically symmetric Schwarzschild metric the quasinormal modes can be described as solutions to a single, one-dimensional, differential equation of the second order d2 dz
+R (z)/=0 .
In this picture [1 —3], the analytic function R (z) is given by
—2
R (z)=
2 1 ——
to
2
Z
—V(z) +
2
3
Z3
Z4
(2)
and it is assumed that the perturbation has a time dependence exp( i tot), whe—re co is a complex-valued frequency in units M '. The mass M of the black hole is expressed in geometrized units, such that M=6m/c, and the coordinate z is related to the physical radius r by z =r/M. The quasinormal modes are solutions to the differential equation (1) that satisfy boundary conditions of purely outgoing waves at spatial infinity and waves traveling into the event horizon (located at z =2). The effective potential V(z) describing axial perturbations of a Schwarzschild black hole was initially derived by Regge and Wheeler [4], and can be written
V(z)=
1
2 l(1+1) + 2(1 —s ) —— z
Z2
Z3
(3)
The spin weight s of the perturbing field has the discrete value 0, 1, or 2 for scalar, electromagnetic and gravita-
'Present address: Department of Physics and Astronomy, Uniof Wales, College of Cardiff, Cardiff CF2 3YB, U. K.
versity
tional perturbations, respectively. Integer values of the angular momentum I greater than or equal to the spin weight correspond to radiated waves. We will restrict the present analysis to gravitational perturbations, since, in connection with the continuing efforts to detect gravitational radiation, this is the most interesting case. Hence, l =2 is the lowest value of the angular momentum considered. The physical contents of the equations governing the two different classes of perturbations of a Schwarzschild black hole can be proved equivalent [5,6], cf. also the numerical calculations by Andersson [3]. In effect, we will restrict our investigation of the quasinormal-mode problem to axial perturbations. Furthermore, the intended analysis is more straightforward for the effective potential (3). This is because it does not give rise to as many transition points in the complex coordinate plane as the potential describing the polar perturbations, cf. [1 —3]. It is important to note, however, that there is no apparent reason why a similar investigation of the equation governing the polar perturbations could not be made. Unfortunately, the quasinormal-mode problem is not simple to solve. The desired solutions to the differential equation (1) are exponentially increasing along the real coordinate axis. Hence, given a finite numerical accuracy, the ingoing-wave solution at spatial infinity will be small compared to the unavoidable error introduced in the desired outgoing-wave solution. Likewise, close to the event horizon, the outgoing-wave solution is exponentially small and impossible to distinguish from the numerical error in the solution that corresponds to the imposed boundary condition. For closer description of this complication see Detweiler [7], Leaver [8], or Nollert [9]. In a previous paper, Froman et al. [2] derived a simple phase-integral condition to determine the quasinormalmode frequencies. Their condition is analogous to the generalized Bohr-Sornmerfeld quantization condition for complex energy resonances in quantum mechanics. The same condition had previously been proposed, although without derivation, by Quinn et al. [10]. Unfortunately, the Bohr-Sornmerfeld condition gives poor numerical results for all but the lowest-lying quasinorrnal modes. It
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1992
The American Physical Society
NILS ANDERSSON AND STAFFAN LINN&US
4180
was suggested [3] that an approximate analysis considering several, possibly close-lying, transition points would yield more accurate results. The purpose of this paper is to describe such a phase-integral treatment and to justify it by a qualitative discussion of the problem. In this paper we will assume that the function R (z) can be continued into the entire complex coordinate plane. This requires that the effective black-hole potential V(z) can be continued accordingly. This is the case for the of the effective potentials perturbations governing Reissner-Nordstrom, and Kerr black Schwarzschild, holes. In situations where the potential is not analytic (for example, when it is cut off at some large value of the coordinate, or for perturbations of stars, where the interior and exterior of the star must be considered separately) our method must be generalized. We believe that the method can be used to consider also piecewise analytic potentials, albeit different analytic continuations must then be used. Local solutions obtained by considering the different pieces of the potential must be joined together at the points of "discontinuity" on the real coordinate axis. Although it may be hard to find a uniform approximation that is valid across these "matching" points, expressions similar to the ones discussed in this paper may possibly be used to determine the desired local solution for each
piece of the potential.
II. THE PHASE-INTEGRAL APPROXIMATION In the phase-integral method (cf. [11] for references and a recent review) a local solution to (1) is given by a linear combination of the functions
fi z(z)=q
'
(z)exp
=q
'
(z)exp[+iw(z)]
+i jq(z)dz (4)
.
The function q (z) is generated by a so-called base function Q (z) in a way such that
q(z)=g(z)
g
n=0
Yz„
The first yields the (2%+1)th order of approximation. few terms in the expansion are explicitly given by
Yo=1,
(6a)
R(z)
1
2 Q (z)
Q (z)
Y4= —— 2Y', +
d
4
Y6=
16
+g
3/2(
)
d2 dz
g
),
1/2(
(6b)
Y2
dg2
8Y2+12Y2
d Y2 dg22
+10
d Y2
dg
+
d
Y2
(6d)
where we have introduced
d dg
1
d
Q(z} dz
The functions Y2„were also given, although in an alternative way, in the paper by Froman et al. [2]. To some extent, the function Q (z) can be chosen freely (for a discussion see [11]). In the analysis of the quasinormal-mode problem, it is preferable to use Q
(z)=R(z)—
4(z
—2)
This choice ensures that the approximate solutions (4) have the behavior expected of the exact solutions to (1) close to the second-order pole at the event horizon. In the present analysis the phase of Q (z) is chosen such that the function (z), with the time dependence assumed above, corresponds to purely outgoing waves as This implies that Reg (z) must become positive as we approach infinity along the real coordinate axis. It can easily be verified that this solution is exponentially increasing along the real axis.
f,
III. STOKES AND
z~ ~.
ANTI-STOKES LINES
The diSculty to single out an exponentially increasing boundsolution that corresponds to a quasinormal-mode ary condition on the real axis can be avoided if the analysis of the differential equation (1) is continued into the complex coordinate plane. Assuming that z is complex valued the boundary conditions determining a quasinormal mode can be introduced on so-called anti-Stokes lines [1 —3]. In this context, the anti-Stokes lines are curves defined by Im[g(z)dz)=0. As follows from (4), the exponential growth or decay is suppressed on these curves and the approximate solutions correspond to traveling waves (with a constant fiow). Furthermore, a given linear combination of the two approximate functions (4} is preserved as long as the solution is continued along an anti-Stokes line. Hence, the importance of the antiStokes lines in an analysis of the quasinormal-mode problem is obvious. On the other hand, the Stokes lines are defined by Re[g (z)dz] =0. Along such contours the two solutions (4) are either exponentially growing or decreasing. The approximate solution to (1), given by an appropriate linear combination of the two functions (4), cannot be continued through a region of the coordinate plane that contains zeros, or poles, of the function Q (z). In the vicinity of such points the linear combination is expected to change (the so-called Stokes phenomenon). These points are usually called transition points, but in order to avoid confusion we will refer to them as zeros and poles in our discussion below. Let us now brieAy discuss the pattern of anti-Stokes problem. It can easily be lines for the quasinormal-mode verified that three anti-Stokes lines, separated by an angle of 2'/3, emanate from each zero of Q (z). Furthermore, assuming that the imaginary part of the quasinormalmode frequency c~ is negative [1 —3], i.e. , that the black hole is considered as stable against a small perturbation, iv. It then immediately follows we may write ~=p — that, for large values of z, the anti-Stokes lines tend to straight lines with a slope equal to v/p. To analyze how the anti-Stokes lines behave close to the event horizon we
QUASINORMAL MODES OF A SCHWARZSCHILD BLACK.
46
4181
~
~
5
..
l 1
I I I
-2 -2
-1
0
1
2
3
4
5
6
1
B
FIG. 1. The pattern of Stokes (dashed) and anti-Stokes (solid) lines for the quasinormal-mode problem, with parameters I =2 and n =0 (and co according to Table I). The two zeros considered in the previous analysis of Froman et al. [2] are t, and t2
~
use polar coordinates z =2+pe''P. For small values we then have the approximate relation
Q(z)dz
= —2 p
of p
dp+v dy +i —— dp+p dy p
(9) By putting the imaginary part of this differential equal to zero and integrating, we obtain the equation that determines the anti-Stokes lines in the vicinity of the event horizon:
p
(fl/v)f p e
(10)
Obviously, the anti-Stokes lines form logarithmic spirals around the event horizon. As ~Imago~ is increased the spirals get tighter. Conclusively, the boundary condition determining the quasinormal modes are imposed on the anti-Stokes line that emanate from t& toward infinity, and on the antiStokes line that emerges from t2 toward the event horizon, cf. Fig. 1. With the proposed choice of phase for Q(z), the desired solution at infinity is proportional to (z). Furthermore, it can be verified that waves traveling into the event horizon correspond to a solution that 2)'~2 ' . Hence, the boundary condition behaves as (z — at the event horizon corresponds to the nonregular power series solution to (1).
f,
IV. IMPROVED PHASE-INTEGRAL CONDITIONS A. Two possibly close-lying zeros
For quasinormal modes such that ~Imago~ =Redo~ the phase-integral analysis of Froman et al. [2] broke down. The pattern of anti-Stokes lines corresponding to this situation indicates that the two zeros t2 and t3 may well be close lying, ' cf. Fig. 2(a). In fact, since the two zeros are It is important to note that the terms "close lying" and "well separated" do not refer to the geometric distance in the coordinate plane. Rather, they refer to distance in the "phaseintegral" sense, i.e., cannot be immediately inferred from, for example, Fig. 2.
-1
.0 .5
1.0
1.5
2.0 2.5 3.0
15
FIG. 2. (a) The pattern of Stokes (dashed) and anti-Stokes (solid) lines for 1=2 and n =2 (and co according to Table I). The three zeros considered in the derivation of our condition (19) are t2, and t, . (b) The integration contours I &, I &, and I discussed in the derivation of our condition (19). The direction of integration is indicated by arrows. Cuts introduced in order to make q(z) single valued are represented by wavy lines. Solid and dashed curves correspond to integration on different Riemann sheets.
t„
almost connected by a Stokes line, the situation is similar to that of a potential barrier. This could explain the failure of the previous phase-integral analysis [2] where the zero t3 was not considered at all. Formulas describing two, possibly close-lying, zeros provide a powerful tool for solving problems close to the top of a potential barrier in quantum mechanics. For real problems, the phase-integral formulas were initially given by Froman, Froman, Myhrman, and Paulsson [12]. The approach has since been generalized to complex potential barriers by Froman and Lundborg [13], and the resulting formulas were successfully used by one of the present authors to calculate phase shifts for a heavy-ion optical potential [14]. Recently, similar formulas were also used in complex angular momentum theory [15]. It should, of course, be mentioned that, in the first order of approximation the corresponding formula is well known from WKB theory and has been used by several authors, see for example [16,17]. With the proposed choice of phase [and with cuts introduced to make q(z) single valued as in Fig. 2(b)], the desired solution close to the event horizon is proportional to f2(z). Using the notations of Froman, Froman, and Lundborg [18], the corresponding solution on the antiStokes line that emerges from t2 toward t, is given by
P(z) -q '~ (z)cos w2(z)
a,
——lna,
+—
where is a Stokes constant. It is important to note that q(z) is singular at a zero of Q (z) in the higher orders of approximation. The integral w (z) must therefore be determined via integration along a contour that encircles the relevant zero. In effect, as indicated by the sub-
NILS ANDERSSON AND STAFFAN LINN&US
4182
script, the integration determining w2(z) is performed along the contour I 2 of Fig. 2(b). A uniform treatment of the two zeros tz and t3, i.e., using the comparison equation method to locally map the solution tj'i(z) in a one-to-one manner onto the solution for a parabolic barrier, yields an analytical expression for the Stokes constant [13]: —( 1+e —2ma)1/2e —2icr (12)
on the anti-Stokes lines that almost connect t, and t, . The integration determining ur, (z) is performed along the contour I, of Fig. 2(b). If P(z) is to represent a continuous solution to (1), the two solutions (11) and (17) must be the same, except for a constant factor. This leads to the condition
q(z)dz
+ 1 }th order
In the (2N
of approximation
we have
a
(13)
A
Q(z) Y2„dz .
(14)
The integration contour I is depicted in Fig. 2(b). Furthermore, the quantity cr is defined by o
1 I (1/2+i a) — = ——. + 1 alnao+ ln . —
I (1/2
4i
ia)
2
g
o2„,
(15)
with
(16a} 1
1
2 CT
0'
4 ao 6
48 ao
7 + 5760
(16c} ao
a2a4
7
2
ao
1 920
31
1
ao (16d)
80640 ao
The branch of the logarithm in (15) must be chosen such that reduces to i in the limiting case when
la,
l
»1.
a,
It is straightforward to show [19] that the desired solution at infinity, which is represented by (z) on the appropriate anti-Stokes line, corresponds to a solution
f,
td((z)
-
q
'
(z)cos w, (z)
+—
(17)
2A different notation has been used for these formulas. Note, for example, that, in the work by Amaha and Thylwe [15] the corresponding formulas have a different appearance. To a large extent this is due to a difference in the definition (14). Regarding the different conventions of summation, e.g. , in (15), we have used notation consistent with Y2„, see (5), throughout the paper. We are aware that, from an esthetic point of view, it is rather unfortunate that the (2n +1)th-order contribution to the approximation is denoted by an index 2n. Nevertheless, we do not find it advisable to change the notation Y2„ that is well established in phase-integral theory.
q (z }dz
l —— ln( 1+e
4
}
—cr = ( n + — ')n 2
.
(19)
«
»1.
B. Two For
zeros and a second-order pole
when quasinor mal modes, calculations show that the two zeros t2 and t3 again become separated, i.e. , laol Consequently, the condition (19) derived in the previous section should not essentially improve the analysis of [2]. Nevertheless, compared to reliable numerical results [3,8,20] the phase-integral results are poor. From the pattern of anti-Stokes lines, see Fig. 3(a), it follows that the anti-Stokes line along which the solution to (1) is formally traced from the event horizon forms a spiral that passes close to each of the zeros t2 and t3 several times. Physically, the breakdown of the previous phase-integral treatment may then be explained as a failure to account for the wave reflection expected each time the solution is traced through the region close to a zero of Q (z). The only way to account for all such possible reflections in Fig. 3(a) would be to consider not only the zeros t~ and I 3 but also the second-order pole at the event horizon as a cluster. The phase-integral formulas describing such a cluster, consisting of two zeros and a second-order pole, can be derived by locally mapping (1) onto a Coulomb wave equation. This was done by Froman, Froman, and Linnaeus [21] who derived the phase-integral representation of the regular power-series solution. As already pointed out, the solution that satisfies the condition at the event horizon can be ingoing-wave identified as the nonregular power series solution. This solution cannot be obtained directly from the comparison equation treatment in [21]. Fortunately, an asymptotic formula for the nonregular solution can be derived by means of the F-matrix method [18,19). After some calcullmcul
1
(18)
—,
This is the improved phase-integral condition corresponding to two, possibly close-lying, and one wellseparated zero. It is appropriate when the pattern of anti-Stokes lines is similar to that in Fig. 2(a). Consequently, one would expect (19) to improve the numerical results of the analysis in [2] for quasinormal modes such that llmcol & IRecol. It is important to note that the formula (19) reduces to the Bohr-Sommerfeld condition [2, 10] in the limit when This is the case when llmcol lRecol, i.e., for laol the lowest-lying modes.
(16b)
48 ao
= n+ ' rr,
&
where we have defined
f
i
where the contour A encircles the transition points t, and t2 in the positive direction. Inserting the expression (12) for a we obtain the Anal formula
n=0
a2„=
——Ina
A
&.
damped »highly lRecol, numerical
»l.
QUASINORMAL MODES OF A SCHWARZSCHILD BLACK.
lations, described in the Appendix, we obtain the solution on the anti-Stokes line that emerges from t2 toward t, , see Fig. 3(a): g(z)-q '~ (z)sin[wz(z) —As] . (20)
..
4183
The integration determining wz(z) is performed along the contour I z of Fig. 3(b). When Q (z} is chosen according to (8), the quantity b, s, in the (2%+1)th order of approximation, is given by
I
1 I (1/2 —/+ill) 1 bs= ——. ln +— ( 2i I (1/2 g—i.— 2i g)
.
g— +i71)ln[
1 —— . . —— g— . ( g izi)in[ +iso] p
2
~
. —— —g —idio] + p m.
4
g
hz„, 2n
(21)
I
I is to encircle the pole at the event horizon as well as the two zeros t2 and t3 in the positive direction, cf. Fig. 3(b), and the quantity g is defined by
n=
g
n=0
(22)
nz.
i Re— s . z=2 Q
and we have defined
f hp=
„
The contour
where
Q(z)Yz„dz .
(23)
The contributions
b, z„are
—gp,
(25a}
" 24(+„z 1
4
'9O
(25b)
1
'9o'9z
1
2
(2+ qz
24
norizn4
(2+ ~2
+ 241
0
'9o riz
7
((2+ gz)2
(2+ ~2)2
(25c)
(p+r/z)z
2gg0
(4' no)(4nz+ — &4} (
(30' no)no(3k'-no)none 24 (gz+ z)3 1
7
960
(g
6g rlo+—halo)zlz
(g +ri
)
31 + 40320
(5$ —10$ rio+rlo)rlo (gz+rizo)5
(25d)
The two solutions (20) and (17) immediately lead to a condition determining a continuous quasinormal-mode solution z ck
kg= 7l+ —7T,
(26}
which is expected to be considerably more accurate than the condition (19), at least when Imago~ && Redo ~. Note that when the asymptotic expansion for the I condifunctions in (21) is valid, our quasinormal-mode tion (26) reduces to the phase-integral condition derived in [2]. When zl has a large imaginary part this is not the case, however. ~
a)
~
V. NUMERICAL RESULTS
In Table I we compare quasinormal-mode frequencies obtained from the phase-integral condition (26) in the seventh order of approximation with exact results for gravitational perturbations and l =2. The exact results were calculated using the phase-amplitude method [1,3]. The results obtained by the different methods of Leaver [8] and Nollert and Schmidt [20] agree with the phaseamplitude results to the accuracy quoted by each author. In Fig. 4 results obtained from the phase-integral conditions (19) and (26) of this paper are compared with results obtained from the Bohr-Sommerfeld condition [2, 10], as well as the results given by Leaver [8]. From Fig. 4 it follows that, for the lowest-lying modes,
b) 2
0
FIG. 3. (a) The pattern of Stokes (dashed) and anti-Stokes (solid) lines for I =2 and n =20 (and co according to Table I). The three zeros considered as relevant in the derivation of our condition (26) are t2, and t3. (b) The integration contours I I, 12, and I „discussed in the derivation of our condition (26). The direction of integration is indicated by arrows. Cuts introduced in order to make q(z) single valued are represented by wavy lines. Solid and dashed curves correspond to integration on difterent Riemann sheets.
t„
NILS ANDERSSON AND STAFFAN LINN&US
4184
TABLE I. Quasinormal-mode frequencies of a Schwarzschild black hole corresponding to gravitational perturbations and 1=2. The improved phase-integral results are calculated from the condition (26) in the seventh order of approximation. Results corresponding to n = 8 have been omitted since this mode corresponds to a purely imaginary frequency [8,22], a case that cannot be investigated within the present analysis. Improved
phase-integral
method
0.373 70 —0. 088 957i 0.346 703 —0. 273 921i 0.301 048-0. 478 272i 0.251 507-0. 705 140i 0.207 523-0. 946 837i 0. 169 31-1.195 602i 0. 133 27 —1.447 908i 0.092 84-1.703 840 8i 0.063 29 —2. 302 651i 0.076 58-2. 560 833i 0.087 42 —5. 081 48i 0.082 62 —7. 588 70i 0.078 65 —10.093 02i 0.075 60 —12. 596 01i
0 1
2 3
4 5
6 7
9 10 20 30 40 50
as well as when ~1m'~ = ~Redo~, the condition (19) generates quasinorrnal-mode frequencies at least a factor of 10 more accurate than those obtained from the BohrSommerfeld formula [2, 10]. As expected, the condition (19) hardly improves the accuracy of the previous treatment when ~1m'~~)) ~Redo~. Meanwhile, the condition (26) generate frequencies with an accuracy of at least four decimal places for all modes. Naturally, one should not be surprised that the condition (26) generates the most accurate approximate results (for all but the lowest-lying mode) since it is derived under more general assumptions than any of the other phase-integral conditions considered.
V
-1 -2
~&ay~
d
~
""
"&.------0-----0------Q (?""""t? ""
0 c5
0 bQ
O
4
6
5
-6 W
1
I
s
I
10
20
30
40
quasinormal-mode
index
50
n
FIG. 4. The relative numerical accuracy obtained by different methods. The absolute difference between each result and the, presumably exact, phase-amplitude result is plotted as a function of the quasinormal-mode index n. Results obtained from the Bohr-Sommerfeld condition discussed in [2, 10] are denoted by unfilled circles, our condition (19) yields results denoted by unfilled squares while calculations using (26) are denoted by unfilled triangles. The results of Leaver [8] are denoted by solid circles.
Exact phase-amplitude
method
0.373 671 684 —0.088 962 315i 0.346 710 997 —0. 273 914 875i 0.301 053 455 —0.478 276 983i 0.251 504 962 —0. 705 148 202i 0.207 514 580 —0. 946 844 891i 0. 169 299 403 —1. 195 608 054i 0. 133 252 340 —1.447 910 626i 0.092 822 336-1.703 841 172i 0.063 263 505 —2. 302 644 765i 0.076 553 463 —2. 560 826 617i 0.087 394 388 —5. 081 467 099i 0.082 598 482 —7. 588 682 777i 0.078 631 090 —10.092 999 986i 0.075 580 032 —12. 595 990 808i
It is important to note that the results obtained using the phase-integral condition (26} do not agree with the results of Leaver [8] for the highly damped modes, see Fig. 4. However, it can be verified that the numerical results for ru obtained from (26) "converge" for all values of n as the order of approximation is increased. Moreover, the pattern of Stokes and anti-Stokes lines remains similar to Fig. 3(a} as n increases. Hence, there is no apparent reason why the present treatment should break down for the modes. The damped approximate highly quasinormal-mode frequencies also agree perfectly with the exact phase-amplitude calculations, cf. Table I and Fig. 4. Consequently, we consider the phase-integral results as reliable. It should also be noted that the calculations of Leaver [8] were made using a computer that restricted the obtainable accuracy to seven decimal places. Using double precision Leaver has recently recalculated his results. He states [private communication] that the frequency values for highly damped modes changed in the fifth significant figure. In fact, it seems as if Leaver's new results may agree better with the frequencies determined from our condition (26). Let us also point out that the "wiggles" in the curve representing Leaver's results (for the lowest ten modes) follow from the division by two that is necessary to express the results in units of M It was concluded by Leaver [8] that there exists an solutions for each infinite number of quasinormal-mode value of 1. He stated that, for large values of n, "the normal-mode frequencies lie evenly spaced along an asymptote parallel to the imaginary axis. In the case when l =2 the characteristic frequencies should, accord~ =+0.075 the value to Leaver, approach ing —i( —,'n+0. 10) as n is increased. In a recent investigation, Quinn et al. [10] found that the real part of quasinormal-mode frequencies obtained from the BohrSommerfeld condition approaches zero asymptotically. However, as already mentioned above, the analysis of
"
46
QUASINORMAL MODES OF A SCHWARZSCHILD BLACK.
Froman et al. [2] implied that the Bohr-Sommerfeld formula was not reliable for highly damped modes. On the other hand, our condition (26) should not break down as Moreover, an unmentioned quality of (Imago~ increases. the phase-integral of the analysis is the uniqueness quasinormal-mode index n. Hence, given only a reasonable initial guess for the frequency, we xnay do calculations for any value of n without actually knowing the preceding frequencies. We have done some trial calculations to investigate the asymptotic behavior of the quasinormal-mode frequencies for very large ~Imago~. We found that ~Imago~ increased almost linearly, in good Meanwhile, for agreement with Leaver's assumption. n =1000 we obtain Redo=0. 051, which is considerably smaller than the asymptotic value estimated by Leaver. In fact, our calculations suggest that the frequency does approach a limiting value which is slightly smaller than 0.045. It is important to remember that, as Im~ increases according to the approximate formula, the real part of co remains small and the numerical uncertainty in our analysis may increase. In effect, our results cannot be considered as conclusive and the asymptotic behavior will be analyzed further elsewhere. Finally, it is worth mentioning the computational time the demanded for determining damped highly quasinormal-mode frequencies by means of the, presumably exact, phase-amplitude method [1,3]. For low-lying modes, the computational time necessary for obtaining frequencies with a high accuracy is reasonable (roughly 2 min for each result in our table), while for the highly damped modes the phase-amplitude method must be considered as quite inefficient. Even though only a few iterations in the complex frequency plane (using Miiller's method) are needed, a determination of one of the last modes in our table takes several hours. This is due to the increasing number of turns in the spiral on which the solution is numerically traced from the event horizon, cf. [3]. Since each phase-integral result is determined in less than 2 min we consider the use of our phase-integral condition (26) as the most eScient method, demanding a the reasonable numerical to calculate accuracy, quasinormal-mode frequencies of a Schwarzschild black hole. In fact, it is conceivable that the phase-integral quantities could be expressed in terms of elliptic integrals. Should this be done, the computational time required for numerically determining the frequencies will be decreased
..
4185
All other transition points are situated well away from this cluster. For the sake of simplicity, we consider the relevant second-order pole to be situated at the origin. Although the actual quasinormal-mode problem deviates strongly from this simplified picture (cf. Fig. 5), the formulas discussed below are still valid. Let us first consider the exact solutions close to the singularity at the origin (corresponding to the event horizon of the black hole). The regular and the singular power-series solutions of a differential equation, such as (1), can be written
(Al) and
(A2)
We have defined g according to
g= [ —,' —lim[z R (z)]]', Re('~0 . z~o
(A3)
Anti-Stokes lines emerge from the considered cluster in two opposite directions, as shown in Fig. 5. We denote by zo a point on an anti-Stokes line emerging from one of the two considered zeros. The same zero will be used as reference when determining the integral w(z) below. The phase of Q(z) is chosen such that w(zo) becomes positive. If we continue, in the positive direction, from zo along a semicircle around the origin we arrive at a point approximately situated on an anti-Stokes line emerging from the other zero. In this point, which we denote w(z) will be negative. Finally, zz is a point obtained by moving from zo one complete turn, in the positive direction, around the whole cluster. In the phase-integral analysis below, we will assume that the same approximations are valid on all parallel anti-Stokes lines emerging from the cluster. This assumption holds if the cluster is sufficiently tight. In the phase-integral representation, the regular solution can be written [21]
z„
I I
further.
I
I I
ACKNOWLEDGMENTS
We would like to thank Dr. E. W. Leaver for discussing his recent results with us. One of us (N. A. ) also acknowledges travel grants from the Royal Society of London and the Royal Swedish Academy of Sciences. Furthermore, we are grateful to Professor B. F. Schutz and
I
I
Zo
Dr. M. E. Araujo for constructive criticism of a draft for
this paper.
APPENDIX: THE PHASE-INTEGRAL REPRESENTATION OF THE NONREGULAR POWER-SERIES SOLUTION
In this appendix we consider an idealized case where the function R (z) has a second-order pole and two zeros.
FIG. 5. The pattern of Stokes (dashed) and anti-Stokes (solid) lines for the simplified situation discussed in the Appendix. The two zeros are denoted by tl and t2, while special points discussed in the analysis are zp z&, and z&.
4186
NILS ANDERSSON AND STAFFAN LINN&US
P„(zp) = Aiiq
'
(zp)sin[w(zp)
—Ail
],
(A4)
where w(zp) is a contour integral, analogous to w2(z) in (20), and the phase b, ii is defined by 1 I (1/2+/+i g) — 1 —— ln AR = — + ((+i 21 )in[/+i
2i
I (1/2+ g i ri)
2i
—— + ii))in[( —ihip] —— (g —
g b2„. n=0
alp]
= a, (z)f, (z)+ a2(z) f2 (z), P'(z) = a, (z)f (z)+ a, (z)f 2 (z) . ',
(A5)
(A6a) (A6b)
At some distance from all transition points, and along an anti-Stokes line, the coefficients Q, and Q2 tend to constants. Their value at different points in the complex coordinate plane are related by a 2 X 2 matrix called the F matrix in a way such that a(zb ) = F(zb, z, )a(z,
),
(A7)
where a(z) is a column vector consisting of the values of a, and a2 at the point z. From (A7) follows that the F matrix must satisfy the composition property
F(z„z, ) = F(z„zb )F(zb z, )
.
(Z2)
Meanwhile, havior
the phase-integral
f, (z2)=e
The quantities 62„are explicitly given by Eqs. (25a) —(25d). As already mentioned above, any exact solution g(z) of the diff'erential equation (1) can be expressed as a linear combination of the two approximate phase-integral functions (4). Thus, defining the coefficients determining the desired linear combination [18,19] we generally have i)'j(z)
= e—' f~(zp), e '"'~ps(zp) . ps(z2) = — 1 ii
(A9)
In order to satisfy the quasinormal-mode boundary condition, we need to find a phase-integral representation of the nonregular solution. The coefficients determining the singular solution can be expressed in terms of the coefficients of the regular solution. One of the necessary equations is provided by the Wronskian relation
0z
6
A itis = —2g,
(A10)
which implies that
a 1R 2s
a2Ra is
(A 1 1)
To derive a second relation between the coefficients, we must investigate how the solution is expected to change as the coordinate z moves one turn around the cluster. From the series expansion (Al) and (A2) it follows that
(A 14)
),
(A15)
27TQQ
2ni g
zp )
2 77YJ
e
Q2S
—2vri g
1R
(A16)
Q 2R
2 7TYf
Q1S
F(z2
Q 1S
2 7T QQ
(A17) 2S
These equations can be solved to determine the unknown F-matrix elements. In particular, we have
F„(Z2 Zp) =
" (a lRa2se
le
2' lse (A18)
On the other hand, phase-integral estimates of the Fmatrix elements between points on two adjacent antiStokes lines emerging from the same zero of Q (z) [18] yield (after neglecting error terms) that the left-hand member of this equation equals unity. By solving the linear system of (All) and (A18), and inserting (A9), we obtain the expressions
Q
1S
=
gR
e 2wg+ e 2wig — e 2ni g — e 2n.i
e
2g
e
g
(A19a)
e2nl+e —2~if
ig
e 2@i g — e
(A19b)
—2mi g
Consequently, the nonregular power-series the phase-integral representation
ijis(zp)= Asq
e
2)
'"f (z
F(z2, zp)
If (a, a, a2ii
~R
solutions (4) have the be-
where ri is defined by (22). The following relations must then hold exactly:
Q 2S
a1R
(A13)
"~f, (zp),
f'(z )=e
(A8)
) and (a, s, a2s) are the coefficients determining the phase-integral representation of the regular and the singular power series solutions in the point zo well away from the cluster, we readily obtain, from (A4),
(A12)
'~
(zp)sin[w(zp)
—bs],
solution
has
(A20)
where
s
R
—— ln 2)
e2vrg+e2ni(
e 2ng+
(A21)
—27Ii g
Using the reAection formula for the I function, the expression for hs can be written according to (21). We prefer to express 5s in this alternative way, rather than using (A21) directly, since (21) is obtained immediately from (A5) if g is formally replaced by — It should be noted that the exceptional case when 2$ equals an integer (when only one power-series solution exists) is not considered in this paper. Presumably, this situation corresponds to the algebraically special modes discussed by Chandrasekhar [22]. Since the method proposed in this paper cannot be used to determine this mode, the corresponding value is omitted from Table I.
j.
QUASINORMAL MODES OF A SCHWARZSCHILD BLACK.
46
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~
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