3.1 Schwarzschild black holes

The study of perturbations of Schwarzschild black holes assumes a small perturbation hμν on a static spherically symmetric background metric
2 0 μ ν v(r) 2 λ(r) 2 2( 2 2 2) ds = gμνdx dx = − e dt + e dr + r d𝜃 + sin 𝜃dϕ , (17 )
with the perturbed metric having the form
gμν = g0μν + hμν, (18 )
which leads to a variation of the Einstein equations i.e.
δG μν = 4π δTμν. (19 )
By assuming a decomposition into tensor spherical harmonics for each hμν of the form
∑ χ ℓm (r,t) χ(t,r,𝜃,ϕ) = ----r----Yℓm(𝜃,ϕ), (20 ) ℓm
the perturbation problem is reduced to a single wave equation, for the function χ ℓm(r,t) (which is a combination of the various components of h μν). It should be pointed out that Equation (20View Equation) is an expansion for scalar quantities only. From the 10 independent components of the hμν only htt, htr, and hrr transform as scalars under rotations. The ht𝜃, htϕ, hr𝜃, and hrϕ transform as components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components h 𝜃𝜃, h𝜃ϕ, and hϕϕ transform as components of a 2 × 2 tensor and can be expanded in a series of tensor spherical harmonics (see [202Jump To The Next Citation Point212Jump To The Next Citation Point152Jump To The Next Citation Point] for details). There are two classes of vector spherical harmonics (polar and axial) which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between the two families is their parity. Under the parity operator π a spherical harmonic with index ℓ transforms as (− 1)ℓ, the polar class of perturbations transform under parity in the same way, as (− 1)ℓ, and the axial perturbations as (− 1)ℓ+11. Finally, since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.

The radial component of a perturbation outside the event horizon satisfies the following wave equation,

2 ( 2 ) -∂-χ + − ∂---+ V (r) χ = 0, (21 ) ∂t2 ℓ ∂r2∗ ℓ ℓ
where r∗ is the “tortoise” radial coordinate defined by
r∗ = r + 2M log (r ∕2M − 1), (22 )
and M is the mass of the black hole.

For “axial” perturbations

[ ] ( 2M ) ℓ(ℓ + 1) 2σM V ℓ(r) = 1 − ---- ---2----+ ---3- (23 ) r r r
is the effective potential or (as it is known in the literature) Regge–Wheeler potential [173], which is a single potential barrier with a peak around r = 3M, which is the location of the unstable photon orbit. The form (23View Equation) is true even if we consider scalar or electromagnetic test fields as perturbations. The parameter σ takes the values 1 for scalar perturbations, 0 for electromagnetic perturbations, and –3 for gravitational perturbations and can be expressed as σ = 1 − s2, where s = 0,1,2 is the spin of the perturbing field.

For “polar” perturbations the effective potential was derived by Zerilli [212] and has the form

( 2M ) 2n2(n + 1 )r3 + 6n2M r2 + 18nM 2r + 18M 3 Vℓ(r) = 1 − ---- ---------------3-----------2--------------, (24 ) r r (nr + 3M )
2n = (ℓ − 1 )(ℓ + 2). (25 )
Chandrasekhar [54] has shown that one can transform the Equation (21View Equation) for “axial” modes to the corresponding one for “polar” modes via a transformation involving differential operations. It can also be shown that both forms are connected to the Bardeen–Press [38] perturbation equation derived via the Newman–Penrose formalism. The potential Vℓ(r∗) decays exponentially near the horizon, r∗ → − ∞, and as − 2 r∗ for r∗ → + ∞.

From the form of Equation (21View Equation) it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in Section 2.

Kay and Wald [117Jump To The Next Citation Point] have shown that solutions with data of compact support are bounded. Hence we know that the time independent Green function G(s,r ,r′) ∗ ∗ is analytic for Re (s) > 0. The essential difficulty is now to obtain the solutions f± (cf. Equation (10View Equation)) of the equation

s2ˆχ − χˆ′′ + V χˆ= 0, (26 )
(prime denotes differentiation with respect to r∗) which satisfy for real, positive s:
f+ ∼ e−sr∗ for r∗ → ∞, f − ∼ e+r∗x for r∗ → − ∞. (27 )
To determine the quasi-normal modes we need the analytic continuations of these functions.

As the horizon (r∗ → ∞) is a regular singular point of (26View Equation), a representation of f− (r∗,s) as a converging series exists. For M = 1 2 it reads:

∞∑ f − (r,s) = (r − 1 )s an(s)(r − 1)n. (28 ) n=0
The series converges for all complex s and |r − 1| < 1 [162]. (The analytic extension of f− is investigated in [115Jump To The Next Citation Point].) The result is that f− has an extension to the complex s plane with poles only at negative real integers. The representation of f+ is more complicated: Because infinity is a singular point no power series expansion like (28View Equation) exists. A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas [115Jump To The Next Citation Point], see also [159Jump To The Next Citation Point]. It turns out that the continuation of f+ has a branch cut Re (s) ≤ 0 due to the decay r− 2 for large r [115].

The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot [34Jump To The Next Citation Point]. Here the existence of an infinite number of quasi-normal modes is demonstrated. Truncating the potential (23View Equation) to make it of compact support leads to the estimate (16View Equation).

The decay of solutions in time is not exponential because of the weak decay of the potential for large r. At late times, the quasi-normal oscillations are swamped by the radiative tail [166Jump To The Next Citation Point167Jump To The Next Citation Point]. This tail radiation is of interest in its own right since it originates on the background spacetime. The first authoritative study of nearly spherical collapse, exhibiting radiative tails, was performed by Price [166167].

Studying the behavior of a massless scalar field propagating on a fixed Schwarzschild background, he showed that the field dies off with the power-law tail,

−(2ℓ+P +1) χ (r,t) ∼ t , (29 )
at late times, where P = 1 if the field is initially static, and P = 2 otherwise. This behavior has been seen in various calculations, for example the gravitational collapse simulations by Cunningham, Price and Moncrief [666768]. Today it is apparent in any simulation involving evolutions of various fields on a black hole background including Schwarzschild, Reissner–Nordström [106Jump To The Next Citation Point], and Kerr [132Jump To The Next Citation Point133Jump To The Next Citation Point]. It has also been observed in simulations of axial oscillations of neutron stars [18Jump To The Next Citation Point], and should also be present for polar oscillations. Leaver [136Jump To The Next Citation Point] has studied in detail these tails and associated this power low tail with the branch-cut integral along the negative imaginary ω axis in the complex ω plane. His suggestion that there will be radiative tails observable at + 𝒥 and + ℋ has been verified by Gundlach, Price, and Pullin [106]. Similar results were arrived at recently by Ching et al. [62] in a more extensive study of the late time behavior. In a nonlinear study Gundlach, Price, and Pullin [107] have shown that tails develop even when the collapsing field fails to produce a black hole. Finally, for a study of tails in the presence of a cosmological constant refer to [49], while for a recent study, using analytic methods, of the late-time tails of linear scalar fields outside Schwarzschild and Kerr black holes refer to [3637].

Using the properties of the waves at the horizon and infinity given in Equation (27View Equation) one can search for the quasi-normal mode frequencies since practically the whole problem has been reduced to a boundary value problem with s = iω being the complex eigenvalue. The procedure and techniques used to solve the problem will be discussed later in Section 6, but it is worth mentioning here a simple approach to calculate the QNM frequencies proposed by Schutz and Will [180Jump To The Next Citation Point]. The approach is based on the standard WKB treatment of wave scattering on the peak of the potential barrier, and it can be easily shown that the complex frequency can be estimated from the relation

( ) [ 2 ]1∕2 2 1- d-V-ℓ(r0) (M ωn ) = V ℓ(r0) − i n + 2 − 2 dr2∗ , (30 )
where r0 is the peak of the potential barrier. For ℓ = 2 and n = 0 (the fundamental mode) the complex frequency is M ω ≈ (0.37,− 0.09), which for a 10M ⊙ black hole corresponds to a frequency of 1.2 kHz and damping time of 0.55 ms. A few more QNM frequencies for ℓ = 2,3 and 4 are listed in Table 1.

Table 1: The first four QNM frequencies (ωM) of the Schwarzschild black hole for ℓ = 2, 3, and 4 [135Jump To The Next Citation Point]. The frequencies are given in geometrical units and for conversion into kHz one should multiply by 2π(5142 Hz ) × (M ⊙ ∕M ).
n ℓ = 2 ℓ = 3 ℓ = 4
0 0.37367 –0.08896 i 0.59944 –0.09270 i 0.80918 –0.09416 i
1 0.34671 –0.27391 i 0.58264 –0.28130 i 0.79663 –0.28443 i
2 0.30105 –0.47828 i 0.55168 –0.47909 i 0.77271 –0.47991 i
3 0.25150 –0.70514 i 0.51196 –0.69034 i 0.73984 –0.68392 i

Figure 2View Image shows some of the modes of the Schwarzschild black hole. The number of modes for each harmonic index ℓ is infinite, as was mathematically proven by Bachelot and Motet-Bachelot [34Jump To The Next Citation Point]. This was also implied in an earlier work by Ferrari and Mashhoon [85Jump To The Next Citation Point], and it has been seen in the numerical calculations in [25Jump To The Next Citation Point157Jump To The Next Citation Point]. It can be also seen that the imaginary part of the frequency grows very quickly. This means that the higher modes do not contribute significantly in the emitted gravitational wave signal, and this is also true for the higher ℓ modes (octapole etc.).

View Image

Figure 2: The spectrum of QNM for a Schwarzschild black-hole, for ℓ = 2 (diamonds) and ℓ = 3 (crosses) [25Jump To The Next Citation Point]. The 9th mode for ℓ = 2 and the 41st for ℓ = 3 are “special”, i.e. the real part of the frequency is zero (s = iω).

As is apparent in Figure 2View Image that there is a special purely imaginary QNM frequency. The existence of “algebraically special” solutions for perturbations of Schwarzschild, Reissner–Nordström and Kerr black holes were first pointed out by Chandrasekhar [57]. It is still questionable whether these frequencies should be considered as QNMs [137Jump To The Next Citation Point] and there is a suggestion that the potential might become transparent for these frequencies [11]. For a more detailed discussion refer to [144].

As a final comment we should mention that as the order of the modes increases the real part of the frequency remains constant, while the imaginary part increases proportionally to the order of the mode. Nollert [157Jump To The Next Citation Point] derived the following approximate formula for the asymptotic behavior of QNMs of a Schwarzschild black hole,

[ ] γ1 1 γ1 M ωn ≈ 0.0437 + --------1∕2 + ...− i − -(2n + 1) + --------1∕2 + ... , (31 ) (2n + 1 ) 8 (2n + 1)
where γ1 = 0.343, 0.7545 and 2.81 for ℓ = 2, 3 and 6, correspondingly, and n → ∞. The above relation was later verified in [10] and [143].

For large values of ℓ the distribution of QNMs is given by [16486Jump To The Next Citation Point85Jump To The Next Citation Point113Jump To The Next Citation Point]

√ -- 1 ( 1) 3 3M ωn ≈ ℓ + -− i n + -- . (32 ) 2 2
For a mathematical proof refer to [39].

The perturbations of Reissner–Nordström black holes, due to the spherical symmetry of the solution, follow the footsteps of the analysis that we have presented in this section. Most of the work was done during the seventies by Zerilli [213], Moncrief [153154] and later by Chandrasekhar and Xanthopoulos [55Jump To The Next Citation Point209]. For an extensive discussion refer to [56]. We have again wave equations of the form (21View Equation), one for each parity with potentials which are like (23View Equation) and (24View Equation) plus extra terms which relate to the charge of the black hole. An interesting feature of the charged black holes is that any perturbation of the gravitational (electromagnetic) field will also induce electromagnetic (gravitational) perturbations. In other words, any perturbation of the Reissner–Nordström spacetime will produce both electromagnetic and gravitational radiation. Again it has been shown that the solutions for the odd parity oscillations can be deduced from the solutions for even parity oscillations and vice versa [55]. The QNM frequencies of the Reissner–Nordström black hole have been calculated by Gunter [108Jump To The Next Citation Point], Kokkotas, and Schutz [129Jump To The Next Citation Point], Leaver [137Jump To The Next Citation Point], Andersson [9], and lately for the nearly extreme case by Andersson and Onozawa [26].

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