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arXiv:gr-qc/9903031v1 8 Mar 1999
Quasinormal Modes of Dirty Black Holes
P.T. Leung(1), Y. T. Liu(1), W.-M. Suen(1,2), C. Y. Tam(1)and K. Young(1)
(1)Department of Physics, The Chinese University of Hong Kong, Hong Kong
(2)McDonnell Center for the Space Sciences, Department of Physics, Washington University, St Louis, MO 63130, U S A
(February 7, 2008)
Quasinormal mode (QNM) gravitational radiation from black holes is expected to be observed
in a few years. A perturbative formula is derived for the shifts in both the real and the imaginary
part of the QNM frequencies away from those of an idealized isolated black hole. The formulation
provides a tool for understanding how the astrophysical environment surrounding a black hole, e.g.,
a massive accretion disk, affects the QNM spectrum of gravitational waves. We show, in a simple
model, that the perturbed QNM spectrum can have interesting features.
PACS numbers: 04.30.-w
1. Introduction. The new generation of gravitational
wave observatories (LIGO, VIRGO) will soon be able to
probe black holes in their dynamical interactions with the
astrophysical environment (e.g., matter or another black
hole falling into it). Numerical simulations [1] show that
the gravitational waves emitted in this process will carry
a signature associated with the well-defined quasinormal
mode (QNM) frequencies of the black hole, and thereby
confirm its existence.
A stationary neutral black hole in an otherwise
empty and asymptotically flat spacetime is a Kerr hole
(Schwarzschild hole in the case of zero angular momen-
tum) [2]. Linearized gravitational waves propagating on
the Kerr or Schwarzschild background can be described
by the Klein-Gordon equation [3]:
?∂2
t− ∂2
x+ V (x)?φ(x,t) = 0, (1)
where x is a radial (tortoise) coordinate, φ is the radial
part of a combination of the metric functions representing
the gravitational wave. The potential V (x) describes the
scattering of the gravitational waves by the background
geometry. The outgoing wave boundary condition is ap-
propriate for waves escaping to infinity, and a monochro-
matic solution [φ ∝ exp(−iωt)] is a QNM, with Im ω < 0.
The QNM spectra of Kerr and Schwarzschild black holes
have been extensively studied [3], and provide a template
against which one can try to determine the nature of the
source; for an isolated black hole, the no-hair theorem
[2] implies that the QNM spectrum depends only on the
mass M and the angular momentum J.
However, the black holes that are observed will not be
isolated, but will be situated at the centers of galaxies,
or will be surrounded by accretion disks. Therefore the
observed spectra should not be matched against those of
a pure Kerr or Schwarzschild hole, but to a black hole
perturbed by interactions with its surrounding — a dirty
black hole. So far, the perturbation of black hole QNMs
has attracted little attention, partly because a perturba-
tive formalism for the QNMs of open systems, as opposed
to the normal modes (NMs) of conservative systems, has
not hitherto been available. In this paper we develop
such a formalism, which then opens the way to inferring
the astrophysical environment of the black holes from the
observed signal, beyond M and J.
Two kinds of perturbations are involved here.
the standard black hole perturbation theory [3], (1)
is obtained by linearizing the metric about the Kerr
or Schwarzschild background, and the time-independent
eigenvalue problem (with the outgoing wave boundary
condition) determines the QNM spectrum. The second
type of perturbations are the perturbations that change
the background on which the wave propagates, e.g., by
the presence of an accretion disk; these are often quasi-
static, and hence separable from that of the gravitational
wave perturbation by the time scales involved (in a suit-
able gauge choice).In this paper we focus on time-
independent perturbation of the background, described
by (1) with a potential V (x) = V0(x) + µV1(x), |µ| ≪ 1.
Therefore we are led to study the following eigenvalue
problem in powers of µ:
In
− φ′′(x) + [V0(x) + µV1(x)]φ = ω2φ(2)
While reminiscent of standard textbook problems,
e.g., the usual Rayleigh-Schr¨ odinger perturbation theory
(RSPT), the problem here is fundamentally different: the
outgoing wave condition renders the system physically
nonconservative (energy escapes to infinity) and the as-
sociated operator −d2/dx2+ V (x) non-hermitian; her-
miticity underpins the usual RSPT.
The difficulty can be seen in several guises if one
tries naively to transcribe the usual formulas.
first-order shift cannot be given by the usual formula
?φ0|µV1|φ0?/?φ0|φ0?, in obvious notation — the usual in-
ner product leads to ?φ0|φ0? =
QNM φ0extends over all space (and indeed grows expo-
nentially at infinity). Higher-order shifts are even more
problematic, since the usual RSPT formula involves a
The
?∞
−∞dxφ∗
0φ0= ∞ since a
1
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sum over intermediate eigenstates, but now the unper-
turbed eigenstates do not in general form a complete ba-
sis [4], at least not in the case of black holes.
2. Formulation. Our formulation generalizes the loga-
rithmic perturbation theory (LPT) [5] to QNM systems;
LPT has the property that it does not require a complete
set of eigenstates. Attention is focussed on the logarith-
mic derivative f(x) = φ′(x)/φ(x). From (2)
f′(x) + f2(x) − [V0(x) + µV1(x)] + ω2= 0(3)
For any ω, we define two solutions f±(ω,x) by the bound-
ary conditions f±(ω,x) → ±iω as x → ±∞. At an eigen-
value ω, f+(ω,x) = f−(ω,x).
For many cases of interest, V (x) = V0(x) + µV1(x) is
nontrivial only in a finite domain (L−,L+), and is rel-
atively simple in the asymptotic regions (−∞,L−) and
(L+,∞). In particular, we assume that the asymptotic
regions can be solved with the outgoing wave conditions
to give the logarithmic derivatives D±(ω) = f±(ω,L±).
We then expand all quantities in powers of µ: f ≡
f0+ g = f0+ µg1+ µ2g2+ ···; ω = ω0+ µω1+ ···;
D±= D±0+ µD±1+ ···.
While the details of the derivation will be given else-
where, the central result for the nth order shift is
ωn=?φ0|Vn|φ0?
2ω0?φ0|φ0?
(4)
in which we have introduced the suggestive notation
?φ0|Vn|φ0? =
?L+
L−
Vn(x)φ2
0(x)dx
− ∆+nφ2
0(L+) + ∆−nφ2
0(L−) (5)
?φ0|φ0? =
?L+
L−
1
2ω0
φ2
0(x)dx
+
?D′
+0φ2
0(L+) − D′
−0φ2
0(L−)?
(6)
Here V1is the perturbing potential in (2), and for n > 1,
Vn(x) = −?n−1
φ2
0(x)gn(x) =
?ωnD′
+
L−
i=1[gi(x)gn−i(x) + ωiωn−i], with
−0(ω0) + ∆−n
?φ2
0(L−)
?x
dy [Vn(y) − 2ω0ωn]φ2
0(y)(7)
Here ∆±nis the nth-order part of D±(ω)−D±0(ω0); ex-
plicitly ∆±1 = D±1, ∆±2 = D±2+ ω1D′
etc., where all D±nand their derivatives are understood
to be evaluated at the unperturbed frequency ω0. These
results express the nth order correction to the eigenvalue
in quadrature in terms of lower-order quantities. One
can hence in principle obtain the corrections to any or-
der. Similar to LPT for conservative systems, a sum over
intermediate states is not needed.
±1+1
2ω2
1D′′
±0
3. Properties of the Perturbed Spectrum For Open Sys-
tems in General. The result in (4) has been written in
a way formally similar to the hermitian case. The fac-
tor 2ω0occurs because the eigenvalue is ω2rather than
ω. The numerator and the denominator in (4) are sep-
arately independent of L±, so that they can be given
physical interpretations as a generalized matrix element
and a generalized norm respectively.
The generalized norm has some unusual properties
[6,7]. (a) It involves φ2
complex. (b) It involves surface terms at x = L±, though
the value of the entire expression is independent of the
choice of L±. Thus, it is not a norm in the strict sense,
but rather a useful bilinear map. Nevertheless, in cases
where the system parameters can be tuned so that the
leakage of the wavefunction approaches zero (e.g., V0(x)
contains tall barriers on both sides), the generalized norm
does reduce to the usual (real and positive-definite) norm
for a NM.
It is useful to define a function H(x) for each QNM
which depends only on the original unperturbed system
0rather than |φ0|2, and is in general
δω
δ(µV1(x))≡ H(x) =
φ0(x)2
2ω0?φ0|φ0?
(8)
Both the magnitude and the phase of H(x) are well de-
fined and physically significant. The magnitude implies
that we can now give a precise meaning to the normaliza-
tion of a QNM, even though the wavefunction diverges
at infinity. The phase of H determines the phase of the
first-order shift ω1for a real and positive localized per-
turbation V1(x). The phase is intriguing because it has
no counterpart for a hermitian system — in that case,
H(x) must be real and non-negative.
The functions H(x) are then convenient objects for dis-
cussing the effect of any perturbation on the QNMs of a
given system. We next present some properties of H(x)
for the Schwarzschild black hole.
4. General Properties of the Perturbed Spectrum of
a Schwarzschild Hole. Waves propagating on the exact
Schwarzschild background geometry is described by (1)
with the potential [3]
VSc(M,x) =
?
1 −2M
r
??l(l + 1)
r2
+ (1 − s2)2M
r3
?
(9)
with x = r + 2M ln(r/2M − 1), where s is the spin of
the field (s = 2 for gravitational waves).
In Fig. 1 we plot for the s = 0,l = 1 case the func-
tions H(x), which depends only on the unperturbed po-
tential. The diagrams refer to the lowest QNMs (labeled
as j = 0,1,···,5). We note that for a localized perturba-
tion µV1(x) = µδ(x−x1), the frequency shift ω1is given
by H(x1), and therefore can be read out directly from the
figure. Both Re H(x) and Im H(x) alternate in sign as
(−1)jnear the event horizon. The patterns are different
for different values of x1and not simple, demonstrating
2
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that a localized perturbation will push the QNMs along
different directions in the complex frequency plane, gen-
erating a rich pattern of frequency shifts (in contrast to
shifts all of the same phase in the case of the NMs of a
conservative system). This implies much better prospects
for extracting information about the perturbing potential
from the observed shifts.
1.01.52.0
-0.03
0.00
0.03
j=4
1.01.52.0
-0.10
0.00
0.10
j=2
1.01.52.0
-0.20
0.00
0.20
j=0
1.0 1.52.0
-0.02
0.00
0.02
j=5
1.01.52.0
-0.03
0.00
0.03
j=3
1.01.5 2.0
-0.10
0.00
0.10
j=1
Fig.1 Graph of Re
H(x)e−2γ√
?
H(x)e−2γ√
1+(x/2M)2?
(solid line)
and Im
?
1+(x/2M)2?
(dashed line) vs r/2M, where
γ=Im(−2Mω)
The richness of the pattern could be diluted if the
perturbation has a spatial extent ∆x large compared
to the typical wavelength of oscillation of H(x), λ ≈
2π/|Re ω0| ≈ a few M. Next we discuss a model problem
with an effective potential which extends over an infinite
range of x.
5. Perturbed Spectrum of a Schwarzschild Black Hole
in a Model Problem. Consider a Schwarzschild hole sur-
rounded by a static shell of matter. Denote the total mass
of the system as measured at infinity (ADM mass) by Mo,
and the mass of the black hole as measured by its horizon
surface area by Ma. The perturbation is characterized by
µ ≡ (Mo−Ma)/Maand the circumferential radius r = rs
where the shell is placed. For scalar wave (s = 0), both
the unperturbed potential V0 and the perturbation V1
can be given in terms of VScin (9): V0(x) = VSc(Mo,x);
µV1(x) = κδ(x−xs)+(β/α)VSc(Ma,x)−VSc(Mo,x), for
x < xs, and V1= 0 for x > xs, where xsis the tortoise
coordinate at rs. The constants κ,α and β are given
by Mo,µ and rsin some complicated expressions. This
perturbation consists of a δ-function at the shell, plus a
contribution inside the shell extending all the way to the
horizon (x → −∞, r → 2Ma). There is no perturbation
outside the shell; in terms of the ADM mass, the outside
metric is exactly that of a Schwarzschild hole with Mo.
For x < 0, the full potential V = V0+ µV1 can be
expressed as a sum of exponentials, for which (2) with
the outgoing wave boundary condition can be integrated
analytically, thus giving the log derivatives D−, whereas
the log derivative D+is trivial because the perturbation
vanishes outside the shell. The details of the treatment
of exponential potentials will be given elsewhere [8].
We first demonstrate the convergence of the perturba-
tion results. Fig. 2 shows the magnitude of the error
in the frequencies in the 0th, 1st and 2nd order results
versus µ, for l = 1 scalar waves, compared to the exact
numerical results (which can be obtained by brute force
in this simple case [9].) The error of the nth order re-
sult goes as µn+1, as it should. Detailed estimation of
the error and the large n behavior of the perturbation
expansion will be given elsewhere.
-3.0-2.5-2.0-1.5-1.0-0.5
log10µ
-8.0
-6.0
-4.0
-2.0
0.0
log10(2Mae)
Fig. 2
the 0th (circles), 1st (squares) and 2nd order (triangles) per-
turbation for l = 1, s = 0, j = 1, and rs = 2.52Ma, vs the
size of the perturbation µ.
The magnitude of the error e in the frequencies of
We next study the dependence on the parameters of
the shell. Fig. 3a shows the trajectories of the lowest
damping QNMs (j = 0,1,···,6) for different rs for the
case of l = 1 scalar wave with µ = 0.01 based on ex-
act numerical calculation. We note the rich features of
the perturbed spectra. As rs changes, the QNMs ex-
ecute complicated trajectories on the complex ω plane,
with the higher-order modes moving more rapidly as rs
varies. This behavior is readily understood from the per-
turbation formula (4), which gives
ω1∼ e2iω0xs/x2
s
forxs/2Ma≫ 1.(10)
With Im ω0< 0, the QNMs move away from the unper-
turbed positions in an exponential fashion as xsincreases,
and the higher-order modes (−Im 2Maω0 ≫ 1) move
with higher speed.
More intriguingly there are complicated fine structures
in these trajectories. Fig. 3b shows the fine details of the
trajectory of the j = 0 mode. The results obtained by
direct numerical integration and by the 1st order pertur-
bation formula are shown. For larger xs, the trajectory
shows a spiral structure, which can be explained from the
first-order perturbation formula:
3
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ω1=
?xs
−∞
H(x)V1(x)dx + κ(xs)H(xs).(11)
The asymptotic behavior of H(x) is H(x) ∼ e2iω0x, so
for large xs, (′= d/dxs)
ω′
1∼ [V1(xs) + κ′(xs) + 2iω0κ(xs)]e2iω0xs.(12)
The exponential factor e2iω0xsgives the spirial structure.
0.00.30.6 0.91.2
Re(2M0ω)
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Im(2M0ω)
j=0
j=1
j=2
j=3
j=4
j=5
j=6
0.5800.5850.590
Re(2Moω)
0.5950.600
-0.206
-0.202
-0.198
-0.194
-0.190
Im(2Moω)
Fig. 3a (the upper graph) The trajectory of the lowest QNMs
of l = 1 scalar waves for µ = 0.01 and rs/Mavarying from 2.22
to 60. The circles are QNMs of a bare Schwarzschild hole with
mass Mo; the squares are the QNMs for rs = 2.22Ma (The
dominant energy condition is violated when rs < 2.22Ma);
the triangles show the positions of QNMs at rs/Ma = 6 to 60
in intervals of 6. Fig. 3b (the lower graph) shows the detail
of the trajectory of the j = 0 mode based on exact (solid line)
and 1st order (dashed line) calculation.
6. Conclusion. We have developed a formulation for
the perturbation of QNMs, in close parallel to the famil-
iar perturbation theory, which is directly applicable to
black holes. With QNM gravitational wave signals from
black holes to be detected soon, and many black holes
expected to be perturbed by their astrophysical environ-
ments, e.g., accretion disks, this formulation will be of
interest to gravitational wave astronomy.
Although the QNMs of any system can in principle be
obtained through brute force numerical integration, per-
turbation formulas are often more revealing. We note the
usefulness of perturbation theory in conventional conser-
vative systems,e.g., in quantum mechanics. Moreover,
the numerical integration of QNM spectrum is much
more difficult than for NM system.
In summary, we raise the importance of studying the
QNMs of dirty black holes, and have developed a per-
turbation formulation for this purpose. The formulation
opens the way to extracting rich information from grav-
itational wave signals from black hole events, and leads
the way to study of the inverse problem. We show in
a simple example that the perturbed spectrum shows
interesting features, which can be understood with the
perturbation formula.
This work is supported in part by the Hong Kong Re-
search Grants Council grant 452/95P, and the US NSF
grant PHY 96-00507. WMS also wants to thank the sup-
port of the Institute of Mathematical Science of The Chi-
nese University of Hong Kong. We thank C. K. Au for
discussions about LPT.
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