Quasinormal Modes, Resonances, and Pseudospectra

Quasinormal frequencies describe black-hole resonances, but frequencies alone do not determine a waveform. This chapter connects three questions: which radiative boundary problem defines the resonances, which computations reliably locate them, and how their sensitivity and excitation affect a measured response. The recurring spacetime example is a linear massless field outside an asymptotically flat Schwarzschild black hole. Kerr extensions require their own angular and radial conditions. Detailed source modeling and detector inference belong to Gravitational Radiation and Gravitational-Wave Astronomy.

Helpful background. Black-Hole Geometry and Mechanics supplies horizons and photon orbits; PDEs and Green Functions for Gravitation supplies resolvents and transforms; Formulations of General Relativity supplies perturbative variables; Computational Relativity supplies residual and convergence analysis.

To follow a resonance through to an observation, continue from the definitions here to spectral quasinormal-mode solvers for numerical convergence and residual tests, quasinormal-mode waveform expansion for finite-window signal modeling, and ringdown likelihoods and start-time dependence for inference from detector data. These are successive questions, with different preparation and checks: locating a pole, constructing an excited signal, and identifying its content in noise. Choose the stage relevant to your calculation and follow its background notes.

Resonances of the Schwarzschild exterior

Take a fixed four-dimensional Schwarzschild vacuum with mass $M\gt0$, and a massless test scalar obeying $\Box\Phi=0$, without backreaction. For one spherical harmonic $Y_{\ell m}$, with integer $\ell\ge0$ and $-\ell\le m\le\ell$, write

$$\Phi=e^{-i\omega t}Y_{\ell m}(\theta,\phi)\frac{X_\ell(r)}{r}, \qquad f(r)=1-\frac{2GM}{r}.$$

The tortoise coordinate satisfies $dr_*/dr=f^{-1}$ on $r\gt2GM$. It sends the future-horizon end to $r_*\to-\infty$ and the distant end to $r_*\to+\infty$. Substitution gives

$$\begin{aligned} \mathcal L_\omega X_\ell &=\left[\frac{d^2}{dr_*^2}+\omega^2-V_\ell(r)\right]X_\ell=0,\\ V_\ell(r)&=f(r)\left[\frac{\ell(\ell+1)}{r^2}+\frac{2GM}{r^3}\right]. \end{aligned}$$

Thus $V_\ell$ and $\omega^2$ both have dimensions of inverse length squared. This scalar potential is not the spin-two gravitational potential used in the numerical example below.

Radiation conditions and damping

First use real $\omega\gt0$. The horizon-ingoing solution $u_H$ behaves as $e^{-i\omega r_*}$ at the horizon; the infinity-outgoing solution $u_\infty$ behaves as $e^{+i\omega r_*}$ at infinity. Their full phases depend on advanced time $t+r_*$ and retarded time $t-r_*$, respectively. These are conditions at different ends, not statements that either solution has only one wave everywhere. See Berti–Cardoso–Starinets 2009, §3.1, author-manuscript pp. 17–18, Eqs. (27), (30), PDF.

With a prime denoting $d/dr_*$, the real-frequency current gives an immediate check:

$$\mathcal J[X]=\frac{X^*X'-XX'^*}{2i}, \qquad \mathcal J[Ae^{\pm i\omega r_*}]=\pm\omega|A|^2.$$

Continue the retarded endpoint solutions in frequency on a specified analytic sheet. A damped resonance has $\omega=\omega_R-i\gamma$, $\gamma\gt0$: its time factor decays as $e^{-\gamma t}$, but its radial radiation factors grow at both ends of a Schwarzschild-time slice:

$$\begin{aligned} |e^{-i\omega r_*}|&=e^{-\gamma r_*}\longrightarrow\infty && (r_*\to-\infty),\\ |e^{+i\omega r_*}|&=e^{+\gamma r_*}\longrightarrow\infty && (r_*\to+\infty). \end{aligned}$$

This does not make a causal, finite-data evolution divergent: a single resonant profile is not the entire response at all positions and times. Nor is $\mathcal J$ a positive norm. For complex frequency and this real potential, the ODE gives

$$\mathcal J'=-2\operatorname{Re}\omega\operatorname{Im}\omega\,|X|^2.$$

It is generally not conserved; a purely imaginary frequency is an exception, not a restoration of a positive QNM norm.

Wronskian zeros and Green-function poles

Fix the radial source normalization by $\mathcal L_\omega g=\delta(r_*-r_*')$. Away from endpoint-basis singularities, set $x=r_*$ and define

$$\begin{aligned} W(\omega)&=u_Hu_\infty'-u_\infty u_H',\\ g(x,x';\omega)&=\frac{u_H(x_<,\omega)u_\infty(x_>,\omega)}{W(\omega)}, \end{aligned}$$

where $x_<=\min(x,x')$ and $x_>=\max(x,x')$. The ODE makes $W$ independent of $x$; differentiating the two branches gives the required right-minus-left jump $\partial_xg|_+-\partial_xg|_-=1$.

If the endpoint solutions are nonzero and holomorphic near an isolated simple zero of $W$, they become linearly dependent there: one solution obeys both radiative conditions. This produces a pole of the continued Green operator. A singular basis normalization, a threshold, or a cancellation must instead be examined in the full kernel; a source/observer projection can also hide an operator pole. Compare Leaver 1986, §II.B, author-version p. 7, Eqs. (20)–(23), PDF.

Multiplying either endpoint solution by a nonvanishing holomorphic frequency factor multiplies numerator and denominator together. It changes neither this fixed Green kernel nor its residues. Coefficients assigned to separately normalized mode profiles can change. The mode Green-function derivation develops the jump and cancellation tests; Leaver explicitly checks normalization independence in Leaver 1986, §II.B, author-version p. 9, Eq. (27) and following paragraph, PDF.

Check your preparation

Use these questions to choose a short repair route, not to delay entering the chapter.

Can you explain this?	Working answer	If not, start here
Why is the horizon wave ingoing?	Its full phase depends on the regular advanced time, and its real-frequency current points toward decreasing tortoise radius.	Black-hole geometry and scattering bases.
Why does a pole not specify its amplitude?	A response includes the Green operator acting on data and then an observation. Either projection may suppress a pole.	PDE preparation and the worked example below.
Why can a small matrix residual mislead?	It tests the chosen finite matrix, not the accuracy of its differential operator, domain, or endpoint conditions.	Computational relativity, then the collocation page.

If the difficulty is identifying the field being evolved, first review perturbative formulations. If it is the prompt–pole–cut split, use inverse transforms and causal response.

Routes through the chapter

The pages occur in the following order:

1. Quasinormal Boundary Conditions and Resonances develops endpoint domains, analytic continuation, flux signs, and Schwarzschild frequency-pair symmetry.
2. Frobenius Series and Continued Fractions connects endpoint-compatible series to minimal recurrences, with pinned downloadable benchmarks and Schwarzschild/Kerr comparisons.
3. WKB and Geometric-Optics Spectra develops barrier-top approximations, the exact Pöschl–Teller comparison, and the large-angular-number limit at fixed overtone.
4. Spectral Collocation and Direct Eigenvalue Methods constructs finite eigenvalue pencils and tests differential residuals and spectral pollution. Its downloadable Kerr radial calculation fixes the angular eigenvalue; it is not a coupled Kerr angular–radial spectrum.
5. Pseudospectra and Nonnormal Sensitivity reproduces a finite nonnormal model in two norms and explains a continuum-validation checklist. It does not locally reproduce Schwarzschild pseudospectra with two independent discretizations.
6. Completeness, Excitation, and Ringdown Content examines contour expansions, source projections, and finite-window fits. Its assessment of black-hole expansion results is literature-based, not a local black-hole time-evolution reproduction.

Analytic route: 1 → 3 → 6. Learn what defines a resonance, why a potential peak can approximate some frequencies, and why this local approximation does not supply prompt propagation or a long-range tail. You should finish able to state both a WKB regime and what it omits.

Computational route: 1 → 2 → 4, with 3 as an analytic check. Compare methods with different failure mechanisms. Vary recurrence depth and precision separately from collocation resolution and coordinate map. You should finish able to distinguish a converged matrix eigenvalue from evidence for a continuum resonance.

Stability-and-response route: 1 → 5 → 6. Fix the evolution operator, its admissible states, and the norm before interpreting sensitivity; then fix the source and observation. You should finish able to distinguish eigenvalue migration, state amplification, and a changed waveform.

Same poles, different observed signals

A two-component model makes the last distinction calculable. It extends the finite matrix on the pseudospectrum page; it is not a discretized black-hole field. Choose a time scale $T_0\gt0$, dimensionless time $q=t/T_0$, frequency $\nu=T_0\omega$, and coupling $K\gt0$. Let

$$\frac{du}{dq}=A_Ku+b\,s(q), \qquad y=c^\dagger u, \qquad A_K=\begin{pmatrix}-1&K\\0&-2\end{pmatrix}.$$

Here $u\in\mathbb C^2$ is the state, $b$ specifies where the scalar source $s$ acts, and $c^\dagger$ specifies the measured component. The dagger is conjugate transpose. Assume zero prehistory, and transform using $e^{+i\nu q}$. The frequency-domain transfer function from source to output is

$$\begin{aligned} H_{cb}(\nu)&=c^\dagger R(\nu)b,\\ R(\nu)&=(-i\nu I-A_K)^{-1}\\ &=\begin{pmatrix} \dfrac{1}{1-i\nu}&\dfrac{K}{(1-i\nu)(2-i\nu)}\\[5pt] 0&\dfrac{1}{2-i\nu} \end{pmatrix}. \end{aligned}$$

The operator poles are always $\nu=-i,-2i$, or $\omega=-i/T_0,-2i/T_0$. They do not depend on $K$. To see their excitation directly, drive component 2 with a unit impulse, $b=e_2$ and $s(q)=\delta(q)$, where $e_1,e_2$ are the coordinate unit vectors. Integrating across zero gives $u(0^+)=e_2$. Subsequently,

$$u_2=e^{-2q}, \qquad u_1'+u_1=Ke^{-2q}, \qquad u_1=K(e^{-q}-e^{-2q}).$$

The prime in this finite model denotes $d/dq$. Observing component 1 therefore gives the causal impulse response

$$h_{12}(q)=\Theta(q)K(e^{-q}-e^{-2q}), \qquad H_{12}(\nu)=\frac{K}{(1-i\nu)(2-i\nu)}.$$

$\Theta$ is the Heaviside step function. The positive first-order impulse normalization here differs from the second-order wave kernel above. A general source produces the causal convolution of $h_{12}$ with $s$.

For $q\gt0$, differentiating $h_{12}$ gives $K(-e^{-q}+2e^{-2q})$. The output rises from zero, reaches $K/4$ at $q=\ln2$, and then decays. This is the maximum of one observed component, not the maximum of the full evolution norm. Changing only the source or observation also changes which poles appear:

Source and observation	Transfer function	Impulse output for $q\gt0$
Drive 2, observe 1	$K/[(1-i\nu)(2-i\nu)]$	$K(e^{-q}-e^{-2q})$
Drive 1, observe 1	$1/(1-i\nu)$	$e^{-q}$
Drive 2, observe 2	$1/(2-i\nu)$	$e^{-2q}$

The operator has two poles in every row, but the last two measured transfers each contain only one. A missing exponential is not necessarily a missing resonance. Conversely, a larger output need not mean that any pole moved.

Where the norm enters

For the Euclidean induced operator norm, unit vectors obey

$$|c^\dagger R(\nu)b|\le\lVert R(\nu)\rVert_2, \qquad \lVert R(0)\rVert_2\ge|H_{12}(0)|=\frac K2.$$

The resolvent norm optimizes over forcing directions; a specified source need not realize that optimum. In the time-generator plane $z=-i\nu$, the $\varepsilon$-pseudospectrum consists of the eigenvalues and the points where $\lVert(zI-A_K)^{-1}\rVert\gt\varepsilon^{-1}$, for a declared norm and $\varepsilon\gt0$. It measures sensitivity to arbitrary additive operator perturbations bounded in that induced norm; a physically restricted perturbation class can give a smaller set.

The existing norm-comparison figure shows how the contours change while the eigenvalues remain fixed. Changing state coordinates also transforms $b$ and $c$: leaving the physical source and observation fixed preserves $H_{cb}$. A state-norm comparison is not permission to rescale a measured signal without explanation.

The finite model has no spatial travel time or branch cut. For a compact Schwarzschild pulse, moving the initial support leaves the background operator fixed but generally changes source overlaps and arrival times; the full response can also contain prompt and cut contributions. See Leaver 1986, §III, author-version pp. 16–17, Eqs. (47)–(53), PDF. A finite sum of two exponentials here is not a black-hole completeness theorem.

Distinctions that control conclusions

Endpoint regularity is a derived condition. Factoring radiation phases or choosing slices that reach future null infinity can make the desired field regular. The admissible function class and endpoint equations must still reproduce the continued retarded problem; smoothness alone need not uniquely select it. A zero-value wall at finite radius generally defines a different spectrum.

A small residual answers a limited question. Matrix residuals test a matrix. Independent differential residuals, endpoint tests, precision changes, and discretization comparisons are needed to test its continuum interpretation. Agreement with one exact fundamental mode need not test every derivative coefficient or overtone.

Spectral sensitivity and excitation are different. A large resolvent, motion of eigenvalues under a perturbation, growth of the state norm $\lVert e^{qA_K}\rVert$, and growth of one output are related but unequal quantities. Specify the domain, norm, perturbation class, source, and observer before drawing a physical conclusion.

A useful fit is not completeness. Several damped exponentials may approximate one signal over a chosen interval without providing a global expansion, a unique overtone identification, or source-independent amplitudes. The two final pages state the scope of the research results they discuss.

Chapter review

Diagnose a claimed black-hole instability

A calculation labels a candidate as the Schwarzschild gravitational $\ell=2$, $n=0$ fundamental and reports $GM\omega=0.37367+0.08896i$. It imposed $X=0$ at a finite outer wall, found agreement at three resolutions in one coordinate map, drew a Euclidean-matrix pseudospectrum, and fitted six damped exponentials to a waveform. Which conclusions about instability, sensitivity, and completeness are justified?

Solution

The quoted sign gives time growth under the adopted Fourier convention, but it does not establish a physical instability: the boundary problem itself is wrong for asymptotically flat outgoing radiation. The familiar positive-real fundamental is

$$GM\omega_{\ell=2,n=0}\simeq0.37367-0.08896i.$$

This is the gravitational, not scalar, value. Leaver 1986, author-version p. 12, Table 1, PDF lists $s=-0.17792-0.74734i$ in units $2GM=1$; converting $s=-i(2GM)\omega$ gives the displayed result. The table’s $m=2$ denotes spin, not azimuthal number. The Schwarzschild scalar equation also pairs $\omega$ with $-\omega^*$ by complex conjugation; it does not pair a damped frequency with its growing conjugate while keeping the same endpoints.

Replace the wall by a justified outgoing realization and verify both endpoint conditions. Then vary precision, resolution and map, test the eigenfunction in the original differential equation, and compare an independent method. Three matrix resolutions alone do not establish that continuum problem.

The matrix pseudospectrum describes its chosen matrix norm. To claim continuum sensitivity, specify the generator and domain, approximate the intended scalar product, and test resolvent convergence with independent discretizations. To claim physical amplification, also specify the allowed perturbations or forcing and compare the actual evolution or observed transfer.

Finally, the six-exponential fit establishes at most finite-window approximation quality. Test the fit interval, held-out data, prompt/tail contamination and amplitude stability. Neither completeness nor unique excitation of five overtones follows from the number of fitted terms.

Change coordinates without changing the experiment

In the finite example take $S=\operatorname{diag}(K,1)$ and write $u=Sv$. Find the new generator, source, and observation for driving component 2 and observing the original component 1. Does the transfer function lose its factor $K$?

Solution

The transformed quantities are

$$\begin{aligned} A'&=S^{-1}A_KS=\begin{pmatrix}-1&1\\0&-2\end{pmatrix},\\ b'&=S^{-1}e_2=e_2, \qquad c'^\dagger=e_1^\dagger S=(K,0). \end{aligned}$$

Because $R'=S^{-1}RS$, the product $c'^\dagger R'b'=c^\dagger Rb$ is unchanged. The factor $K$ moved into the observation; measuring $v_1$ instead gives $u_1/K$, a different output when $K\ne1$. Likewise, the weighted norm $\lVert u\rVert_S=\lVert S^{-1}u\rVert_2$ differs from $\lVert u\rVert_2$ when $K\ne1$, even though the eigenvalues agree. At $K=1$, $S=I$ and the coordinate change is trivial. Neither norm is automatically a physical black-hole energy.

For the next step, choose boundary conditions to define a spectrum, continued fractions to compute one, or excitation and ringdown content to interpret a signal. The subsequent late-time chapter develops what pole sums omit at late times.

References

• Berti, E., Cardoso, V., and Starinets, A. O. (2009). “Quasinormal modes of black holes and black branes.” Classical and Quantum Gravity 26, 163001. DOI. Open PDF, arXiv:0905.2975v2.
• Leaver, E. W. (1986). “Spectral decomposition of the perturbation response of the Schwarzschild geometry.” Physical Review D 34, 384–408. DOI. Open PDF, author version with separate pagination.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Physical Meaning of Quasinormal Pseudospectra