Quasinormal modes (QNMs)

Quasinormal modes (QNMs) are the damped linear excitations of black holes, black branes, and related open systems. In AdS/CFT, they are not just gravitational resonances: they are the poles of retarded Green’s functions of the dual finite-temperature quantum field theory. This page is a living dictionary for recognizing, naming, checking, and citing common QNM phenomena.

Convention used on this page. Perturbations scale as $e^{-i\omega t+i k x}$ . Stable decaying modes have $\operatorname{Im}\omega<0$ ; an instability has $\operatorname{Im}\omega>0$ . Some papers use $e^{+i\omega t}$ , which flips this sign convention.

Core AdS/CFT dictionary

For a bulk fluctuation $\Phi(r;\omega,k)$ in an asymptotically AdS black-hole or black-brane background:

Horizon condition: impose an ingoing condition at the future horizon. In ingoing Eddington—Finkelstein coordinates this is usually regularity.
Boundary expansion: near the AdS boundary, $\Phi(r;\omega,k)=A(\omega,k)\,r^{\Delta-d}+B(\omega,k)\,r^{-\Delta}+\cdots ,$ schematically, with $A$ the source coefficient and $B$ the response/vev coefficient. Precise powers and counterterms depend on the field and dimension.
Retarded correlator: up to local contact terms, $G_R(\omega,k)\sim B(\omega,k)/A(\omega,k)$ for a single uncoupled field.
QNM condition: a QNM is an ingoing, source-free solution: $A(\omega_\star,k)=0$ . In coupled systems, replace this by the vanishing of the determinant of the source matrix for gauge-invariant variables.

Thus a QNM frequency $\omega_\star(k)$ is a pole of the boundary retarded correlator. Its residue controls how strongly that excitation appears in a given response function; its imaginary part controls the damping time.

For mixed, double-trace, or alternative quantization, the source-free condition is modified. For example, scalar mixed boundary conditions replace the simple $A=0$ condition by a linear relation between $A$ and $B$ .

Conventions and notation

Symbol	Meaning	Comment
$\omega$	complex frequency	$\operatorname{Re}\omega$ is oscillation; $-\operatorname{Im}\omega$ is damping for the convention above.
$k$ or $q$	spatial momentum	Continuous for planar branes; discrete/angular for compact horizons or global AdS.
$n$	overtone index	Usually $n=0$ is the least damped mode in a fixed family, but labels can swap near collisions.
$\mathfrak w=\omega/(2\pi T)$	dimensionless frequency	Common in holography; some papers use $\omega/T$ , $\omega r_h$ , or $\omega L$ .
$\mathfrak q=k/(2\pi T)$	dimensionless momentum	Always state the convention before comparing numerical values.
$G_R$	retarded Green’s function	QNMs are its poles; spectral functions depend on residues and operator normalization.
$\Omega_H,\Phi_H$	horizon angular velocity and electrostatic potential	Superradiant thresholds involve $\omega_R-m\Omega_H-q\Phi_H$ .

Fast diagnostic table

What you see in the spectrum	Likely interpretation	What to check
Pole tends to $\omega=0$ as $k\to0$	Hydrodynamic/conservation-law mode	Ward identity, Kubo coefficient, small- $k$ expansion.
Two branches collide at complex $k$	Hydrodynamic radius of convergence	Square-root Puiseux behavior and two-sheet analytic continuation.
Pole crosses into $\operatorname{Im}\omega>0$	Linear instability	Gauge constraints, boundary conditions, threshold zero mode.
Nearly degenerate modes exchange identity	Avoided crossing or exceptional point nearby	Eigenfunction overlap, residues, parameter continuation.
Many overtones move under tiny UV changes	Pseudospectral sensitivity	Resolution study, norm choice, contour/pseudospectrum.
Late-time power law instead of exponential	Branch cut contribution	Asymptotic flatness, mass thresholds, extremal limit, contour deformation.
Pole approaches the real axis near extremality	Zero-damped/near-horizon branch	Horizon frequency $m\Omega_H+q\Phi_H$ and near-AdS $_2$ scaling.
Retarded correlator ambiguous at a point	Pole-skipping	Near-horizon equation degeneracy and line-dependent limiting value.
Purely real frequencies at zero temperature without a horizon	Normal modes, not QNMs	Boundary normalizability and absence of dissipation.
Unstable mode appears only with reflecting boundary or mass trap	Superradiant or quasi-bound instability	Amplification condition plus confinement mechanism.

Dictionary of QNM phenomena

Index

AdS/CFT and boundary data: AdS/CFT pole dictionary, Boundary conditions, Gauge-invariant channel, Normal modes vs QNMs, Pole and residue, Spectral function.

Hydrodynamics and holographic plasma: Diffusive / hydrodynamic mode, Hydrodynamic series, Branch point in momentum, Non-hydrodynamic mode, Pole-skipping, Pseudo-hydrodynamic mode, Transport coefficient.

Spectral geometry: Algebraically special frequencies, Asymptotic spacing, Avoided crossing, Exceptional point, Isospectrality, Mode mixing, Mode trajectory, Pole pattern, Pole motion, Pseudospectrum.

Time-domain response and asymptotics: Branch cut, Late-time tail, Quasinormal expansion, Ringdown phases, Spectral gap, Universality by asymptotics.

Instabilities and special limits: Eikonal / photon-sphere limit, Extremal / zero-damped branch, Instability, Near-AdS $_2$ branch cut, Quasi-bound states, Superradiance, Zero mode.

AdS/CFT pole dictionary

What: In Lorentzian holography, QNMs of a bulk black hole or black brane are poles of retarded correlators in the boundary theory.
Bulk statement: Ingoing at the horizon plus source-free at the AdS boundary.
Boundary statement: Poles of $G_R(\omega,k)$ govern near-equilibrium relaxation of the dual plasma or thermal state.
Caveat: Gauge fields, metric perturbations, and coupled systems require gauge-invariant master fields or a source matrix; individual gauge-dependent components can mislead.
See also: Boundary conditions, Gauge-invariant channel, Pole and residue.

Algebraically special / symmetry-protected frequencies

What: Special frequencies fixed by symmetry, factorization, or hidden supersymmetric/Darboux structure of the perturbation equations. They can yield exact, often purely imaginary, solutions.
Why it matters: They anchor the spectrum and can mark transitions between parity sectors, total-transmission modes, or special scattering behavior.
Identify by: Closed-form solution, factorized master equation, or persistence under deformations that preserve the protecting symmetry.
Caveat: A frequency being simple-looking is not enough; check whether it obeys the same QNM boundary conditions as the rest of the spectrum.
See also: Isospectrality, Mode mixing, Exceptional point.

Asymptotic / high-overtone spacing

What: At large overtone number $n$ , many QNM spectra organize into regular ladders or asymptotic curves in the complex $\omega$ plane.
Why it matters: High- $n$ behavior probes monodromy, horizon data, boundary conditions, and the analytic structure of the radial equation.
Identify by: Large- $|\operatorname{Im}\omega|$ sequences with approximately regular spacing or approach to an asymptotic line/curve.
Caveat: The detailed spacing is not universal. It depends on asymptotics, dimension, spin, charge, rotation, and boundary condition.
See also: Overtones, Pole pattern, Universality by asymptotics.

Avoided crossing / level repulsion

What: As a control parameter changes, two modes approach but avoid an exact eigenvalue crossing and exchange character.
Why it matters: Reveals non-Hermitian spectral geometry and explains why mode labels become ambiguous in parameter scans.
Identify by: Smooth exchange of eigenfunction content, residues, or boundary operator overlap while eigenvalues veer away from each other.
Caveat: In a one-parameter real scan, an avoided crossing may be the visible trace of an exceptional point in a larger complexified parameter space.
See also: Exceptional point, Mode trajectory, Mode mixing.

Boundary conditions

What: QNMs are defined by dissipative boundary conditions: ingoing at the event horizon plus an outer condition appropriate to the asymptotic region.
Flat asymptotics: Outgoing at null infinity; the retarded response typically contains QNM poles plus branch cuts.
AdS asymptotics: Normalizable/source-free at the conformal boundary. For non-extremal finite-temperature AdS black holes/branes, one usually obtains a discrete set of poles at fixed $k$ .
dS asymptotics: Outgoing/regular at the cosmological horizon; the spectrum between horizons is typically discrete.
Caveat: Extremal AdS horizons, mass thresholds, and mixed boundary conditions can introduce branch cuts or modified pole conditions.
See also: AdS/CFT pole dictionary, Branch cut, Near-AdS $_2$ branch cut.

Branch cut / continuous spectrum

What: A non-meromorphic part of the Green’s function, often associated with an accumulation of modes, a continuum, threshold behavior, or long-range propagation.
Why it matters: Branch cuts produce non-exponential late-time response, commonly power-law tails in asymptotically flat black holes.
Identify by: Non-isolated singularity in the complex $\omega$ plane, failure of a pure pole sum, or a contour integral contribution in inverse Laplace/Fourier transforms.
AdS nuance: Non-extremal finite-temperature AdS black holes are usually pole-dominated at fixed $k$ , but extremal or zero-temperature IR geometries can develop branch cuts.
See also: Late-time tail, Quasinormal expansion, Near-AdS $_2$ branch cut.

Branch point in momentum / hydrodynamic breakdown

What: In momentum-dependent spectra, a hydrodynamic branch and a gapped branch can collide at complex $k_\star$ , producing a branch point of $\omega(k)$ .
Why it matters: The nearest such singularity sets the radius of convergence of the hydrodynamic dispersion relation around $k=0$ .
Identify by: Analytic continuation in complex $k$ , a pole collision, and local Puiseux behavior $\omega(k)=\omega_\star+c\,(k-k_\star)^{1/2}+\cdots .$
Caveat: Beyond the radius of convergence the Taylor series fails; the pole may still be continued on another sheet of the spectral curve.
See also: Diffusive / hydrodynamic mode, Exceptional point, Hydrodynamic series.

Diffusive / hydrodynamic mode

What: A pole dictated by conservation laws, with $\omega(k)\to0$ as $k\to0$ .
Examples: Charge diffusion $\omega=-iD k^2+\cdots$ , shear diffusion $\omega=-iD_\eta k^2+\cdots$ , sound $\omega=\pm c_s k-i\Gamma k^2+\cdots$ .
Where: Planar horizons and translationally invariant plasmas; also hydrodynamic modes on compact spaces after replacing $k$ by angular eigenvalues.
Identify by: Ward identities, vanishing frequency at $k=0$ , and overlap with conserved densities or currents.
Caveat: Explicit symmetry breaking, momentum relaxation, finite volume, or weakly gauged symmetries can gap or pin the would-be hydrodynamic pole.
See also: Hydrodynamic series, Transport coefficient, Pseudo-hydrodynamic mode.

Eikonal / photon-sphere limit

What: In high-angular-momentum limits, QNM frequencies are often controlled by unstable null geodesics. Schematically, $\operatorname{Re}\omega \sim \ell\,\Omega_c, \qquad \operatorname{Im}\omega \sim -(n+\tfrac12)\lambda_c,$ where $\Omega_c$ is an orbital frequency and $\lambda_c$ a Lyapunov exponent of the null orbit.
Why it matters: Connects wave dynamics with geometric optics and light-ring physics.
Caveat: The simplest photon-sphere formula is most direct for asymptotically flat compact black holes. In AdS, boundary reflection and black-brane kinematics can modify the interpretation.
See also: Asymptotic spacing, Pole pattern.

Exceptional point / pole collision

What: Two or more modes coalesce as eigenvalues and eigenvectors, producing a defective non-Hermitian eigenvalue.
Why it matters: Generates square-root branch structure, large residues, mode identity exchange, and enhanced numerical sensitivity.
Identify by: Coincident roots plus a rank-deficient eigenvector problem; local behavior $\omega_\pm(\lambda)=\omega_\star\pm c\sqrt{\lambda-\lambda_\star}+\cdots$ .
Caveat: Degenerate eigenvalues are not automatically exceptional; verify defectiveness, not only equality of frequencies.
See also: Avoided crossing, Branch point in momentum, Pseudospectrum.

Extremal / zero-damped branch

What: Families of modes whose damping vanishes, $\operatorname{Im}\omega\to0^-$ , as an extremal limit is approached.
Typical locking: For rotating or charged horizons, $\operatorname{Re}\omega$ often approaches a horizon frequency such as $m\Omega_H+q\Phi_H$ .
Why it matters: Controls very long-lived near-extremal response and often signals singular behavior of the strict extremal limit.
Caveat: Zero-damped modes can coexist with damped modes; which family dominates depends on quantum numbers, excitation, and the order of limits.
See also: Near-AdS $_2$ branch cut, Superradiance, Spectral gap.

Gauge-invariant channel

What: A decoupled or coupled set of perturbations organized by gauge-invariant combinations and symmetry channels: scalar/sound, vector/shear, tensor, charge, helicity, etc.
Why it matters: In gravity and gauge fields, QNMs of gauge-dependent variables can include pure-gauge artifacts; physical poles belong to gauge-invariant correlators.
Identify by: Residual gauge transformations leave the master variable unchanged; constraints are satisfied; the same pole appears in the appropriate boundary correlator.
Caveat: At finite density, magnetic field, rotation, anisotropy, or broken translations, channels that were decoupled can mix.
See also: Mode mixing, AdS/CFT pole dictionary, Numerical and reporting checklist.

Hydrodynamic series / gradient expansion

What: The small- $k$ expansion of hydrodynamic pole dispersion relations and correlators.
Examples: $\omega_\text{diff}(k)=-iDk^2-iD_4k^4+\cdots,$ $\omega_\text{sound}(k)=\pm c_s k-i\Gamma k^2\pm c_3k^3+\cdots .$
Why it matters: Coefficients encode transport and effective theory data.
Limit: The series converges only up to the nearest singularity in complex momentum, often a level crossing with a non-hydrodynamic QNM.
See also: Branch point in momentum, Transport coefficient, Non-hydrodynamic mode.

Instability ( $\operatorname{Im}\omega>0$ )

What: A pole in the upper half of the $\omega$ plane signals exponential growth for the time convention used here.
Mechanisms: Superradiance plus confinement; Gregory—Laflamme-type long-wavelength instabilities; scalar condensation; near-horizon AdS $_2$ BF-bound violation; thermodynamic or elastic instabilities in branes.
Identify by: A mode crossing through $\operatorname{Im}\omega=0$ , usually through a zero mode at the threshold.
Caveat: Check gauge artifacts and boundary conditions carefully; spurious unstable modes are common in under-resolved coupled systems.
See also: Zero mode, Superradiance, Near-AdS $_2$ branch cut.

Isospectrality

What: Distinct perturbation sectors have identical QNM spectra, often due to a Darboux, supersymmetric, or Chandrasekhar-type transformation.
Why it matters: Strongly constrains the spectrum and can provide checks on computations.
Fragility: Rotation, charge, higher curvature terms, nontrivial matter, anisotropy, dimension changes, or boundary-condition changes can lift isospectrality.
Caveat: Equality of a few low modes is not proof; compare full families, asymptotics, and boundary conditions.
See also: Algebraically special frequencies, Mode mixing.

Late-time tail

What: The post-ringdown decay after the dominant QNM window.
Flat asymptotics: Long-range potentials and branch cuts often generate power-law tails.
AdS/dS non-extremal cases: When the relevant spectrum is discrete and gapped, the slowest pole gives exponential late-time decay.
Variants: Massive fields, thresholds, compact dimensions, and extremal horizons can produce oscillatory or anomalous tails.
See also: Branch cut, Ringdown phases, Quasinormal expansion.

Mode mixing / coupled sectors

What: Rotation, charge density, magnetic fields, anisotropy, axions, lattices, higher-derivative terms, or broken translations can mix perturbations that are decoupled in simpler backgrounds.
Why it matters: Hybrid modes can carry several operator overlaps, leading to avoided crossings and changing which pole dominates a given correlator.
Identify by: A coupled source matrix, mixed eigenvectors, and residues that redistribute between operators as parameters vary.
See also: Gauge-invariant channel, Avoided crossing, Isospectrality.

Mode trajectory / parameter flow

What: The path of a pole $\omega(\lambda)$ in the complex plane as a parameter $\lambda$ changes: spin, charge, temperature, momentum, coupling, magnetic field, lattice strength, etc.
Why it matters: Reveals monotonic trends, spirals, collisions, instabilities, and changes in the dominant relaxation channel.
Best practice: Track eigenfunctions, residues, and analytic continuation, not just sorted numerical eigenvalues.
See also: Pole motion, Avoided crossing, Exceptional point.

Near-AdS $_2$ branch cut / IR critical spectrum

What: Extremal charged black holes and branes often develop an AdS $_2$ throat. At strictly zero temperature, the IR Green’s function can have branch-cut behavior; at small nonzero temperature it resolves into a dense tower of poles.
Why it matters: Controls low-frequency response, non-Fermi-liquid scaling in holographic matter, near-horizon instabilities, and the singular nature of the extremal limit.
Identify by: Frequencies scaling with $T$ , accumulation near the origin or a superradiant threshold, and IR exponents set by effective AdS $_2$ masses and charges.
Caveat: The branch cut is an IR statement; UV AdS boundary conditions determine how it appears in the full boundary correlator.
See also: Extremal / zero-damped branch, Branch cut, Instability.

Non-hydrodynamic / gapped mode

What: A pole with $\omega(k\to0)$ finite and nonzero, not fixed by a conservation law.
Why it matters: These modes determine transient relaxation beyond hydrodynamics and often set the obstruction to hydrodynamic convergence.
Identify by: Finite damping at $k=0$ , weak coupling to conserved densities at small $k$ , and no Ward-identity protection.
In holography: The lowest non-hydrodynamic modes often set the equilibration time after the hydrodynamic regime has been subtracted.
See also: Hydrodynamic series, Branch point in momentum, Spectral gap.

Normal modes vs QNMs

Normal modes: Real frequencies in a conservative system, such as pure global AdS without a horizon.
QNMs: Complex frequencies in a dissipative or open system, such as an AdS black hole, where energy can fall through the horizon.
Why it matters: In holography, adding a horizon turns reversible oscillations into thermal relaxation; in the boundary theory this is the difference between isolated finite-volume dynamics and dissipative finite-temperature response.
Caveat: Small AdS black holes can display long-lived modes close to normal modes, but the horizon still makes them quasinormal.
See also: Boundary conditions, Quasi-bound states.

Overtones / radial index $n$

What: For fixed angular, momentum, and channel labels, the overtone index orders modes within a family. The fundamental mode is often the least damped member.
Why it matters: Higher overtones contribute at earlier times and can affect fits, transients, and spectral stability.
Caveat: Near collisions, branch points, or avoided crossings, the label $n$ can cease to be globally meaningful.
See also: Asymptotic spacing, Pseudospectrum, Spectral gap.

Pole and residue / excitation factor

What: Near an isolated simple pole, $G_R(\omega,k)=\frac{\mathcal R_\star(k)}{\omega-\omega_\star(k)}+\text{regular}.$ The residue $\mathcal R_\star$ controls the pole’s weight in the chosen operator channel.
Why it matters: The least damped pole is not always the most visible pole; a small residue or poor source overlap can suppress it.
Nuance: Near exceptional points or nearly defective eigenvalues, residues can become large, cancel between nearby poles, or depend sensitively on normalization.
See also: Spectral function, Quasinormal expansion, Pseudospectrum.

Pole pattern / ladders, clustering, alignment

What: The global organization of many poles: ladders, arcs, spirals, clustering near thresholds, or alignment toward a branch cut.
Why it matters: Pole patterns diagnose potential barriers, near-horizon throats, boundary reflection, and analytic structure.
Identify by: Plotting several families over a wide range of $n$ , $k$ , and control parameters, with convergence checks for every branch.
See also: Asymptotic spacing, Near-AdS $_2$ branch cut, Universality by asymptotics.

Pole motion / global spectral flow

What: Collective behavior of poles as a physical parameter changes.
Why it matters: Reveals phase transitions, mode rearrangements, spectral topology, and onset of instabilities.
Identify by: Continuation of the full spectrum rather than isolated root-finding at each parameter value.
See also: Mode trajectory, Exceptional point, Instability.

Pole-skipping / holographic special points

What: Points $(\omega_\star,k_\star)$ where the near-horizon ingoing solution is not uniquely fixed and the boundary retarded Green’s function becomes direction-dependent in the $(\omega,k)$ plane.
Chaos point: In many Einstein-gravity holographic systems, the energy-density correlator has a special point at $\omega=i\lambda_L$ and $k=i\lambda_L/v_B$ , where $\lambda_L$ is the Lyapunov exponent and $v_B$ the butterfly velocity.
Why it matters: Provides a bridge between linear response, horizon equations, and many-body chaos.
Caveat: Pole-skipping is not simply “a missing pole.” It is a numerator-and-denominator degeneracy; the value of $G_R$ depends on the path by which the point is approached. Higher pole-skipping points may exist but need not encode the leading chaos data.
See also: Hydrodynamic series, Diffusive / hydrodynamic mode, AdS/CFT pole dictionary.

Pseudospectrum / non-normal sensitivity

What: For a non-self-adjoint QNM operator $L$ , the $\epsilon$ -pseudospectrum consists of frequencies where $(L-\omega)^{-1}$ is large, even if $\omega$ is not an exact eigenvalue.
Why it matters: QNM spectra, especially overtones, can be highly sensitive to small perturbations of the operator, potential, boundary conditions, or numerical discretization.
Identify by: Large resolvent norm, broad pseudospectral contours, migration of modes under small controlled perturbations, or strong transient growth despite modal stability.
Caveat: The physical conclusion depends on the norm and on which perturbations are allowed; state both.
See also: Pole and residue, Exceptional point, Numerical and reporting checklist.

Pseudo-hydrodynamic / weakly broken mode

What: A would-be hydrodynamic pole that acquires a small gap because the associated conservation law or symmetry is weakly broken.
Examples: Momentum relaxation from axions/lattices, pinned charge-density waves, weak explicit symmetry breaking, or finite-volume effects.
Typical form: Instead of $\omega\to0$ , one may find $\omega=-i\Gamma_0-iDk^2+\cdots$ with small relaxation rate $\Gamma_0$ .
Why it matters: Distinguishes true hydrodynamics from long-lived transient dynamics in holographic transport.
See also: Diffusive / hydrodynamic mode, Non-hydrodynamic mode, Transport coefficient.

Quasi-bound states

What: Long-lived resonances trapped by a mass term, angular barrier, mirror, AdS boundary, double-barrier structure, or external potential.
Why it matters: Frequencies can approach thresholds with exponentially small damping and may mimic stable particles or normal modes.
Risk: If the mode satisfies a superradiant amplification condition, confinement can turn it into an instability.
See also: Superradiance, Spectral gap, Normal modes vs QNMs.

Quasinormal expansion and completeness caveat

What: A time-domain response is often approximated by prompt response plus a sum over QNM residues plus tail/branch-cut contributions.
Why it matters: QNM sums are powerful but not automatically complete bases. Their validity depends on the contour deformation, initial data, asymptotics, and time window.
AdS/dS: In many non-extremal, discrete-spectrum settings, the late-time response is dominated by the least damped pole.
Flat or extremal cases: Branch cuts and arcs can dominate at very late times.
See also: Ringdown phases, Late-time tail, Branch cut.

Ringdown phases / prompt to QNM to tail

Prompt: Direct propagation and early response, strongly dependent on the source and initial data.
QNM window: Exponentially damped oscillations governed by poles and residues.
Tail: Branch-cut, threshold, or asymptotic contribution; in purely discrete gapped systems this is replaced by the least damped pole at late times.
Why it matters: Fitting a QNM too early can confuse prompt response with overtones; fitting too late can confuse ringdown with tails or numerical noise.
See also: Quasinormal expansion, Late-time tail, Spectral gap.

Spectral function

What: The spectral density is typically $\rho(\omega,k)=-2\operatorname{Im}G_R(\omega,k)$ , up to convention.
Why it matters: QNM poles shape peaks, widths, transport limits, and threshold behavior in real-frequency observables.
Caveat: A pole far from the real axis may be important analytically but barely visible in $\rho$ ; conversely, a broad feature in $\rho$ need not correspond to a single isolated pole.
See also: Pole and residue, Transport coefficient, AdS/CFT pole dictionary.

Spectral gap

What: The smallest positive damping rate among relevant decaying modes, $\Delta_\text{QNM}=\min_\star[-\operatorname{Im}\omega_\star].$
Why it matters: Sets the longest exponential relaxation time in a discrete, stable spectrum.
Hydrodynamic caveat: Conserved quantities close the gap as $k\to0$ . In plasma applications one often quotes the non-hydrodynamic gap, or the gap at fixed $k$ .
Extremal caveat: Zero-damped modes or IR branch cuts can make the gap vanish.
See also: Non-hydrodynamic mode, Extremal / zero-damped branch, Hydrodynamic series.

Superradiance

What: Wave amplification by extracting rotational or electrostatic energy from a horizon. A common threshold is $\omega_R < m\Omega_H+q\Phi_H,$ with details depending on field, charge, and convention.
Outcomes: Without confinement, superradiance reduces damping or amplifies scattering. With confinement, such as AdS boundary conditions, a mirror, or a massive-field trap, it can produce an instability.
Identify by: Energy flux sign at the horizon and a pole crossing associated with the superradiant threshold.
See also: Instability, Quasi-bound states, Extremal / zero-damped branch.

Transport coefficient from QNMs

What: Hydrodynamic pole coefficients encode transport: diffusion constants, sound speed, sound attenuation, conductivities, viscosities, and relaxation times.
How to extract: Fit the small- $k$ dispersion relation, or compute the same coefficient from a Kubo formula and use the agreement as a check.
Examples: $D$ from $\omega=-iDk^2+\cdots$ ; $c_s$ and $\Gamma$ from sound poles.
Caveat: Contact terms affect correlators but not pole locations; normalization affects residues and Kubo coefficients.
See also: Diffusive / hydrodynamic mode, Hydrodynamic series, Spectral function.

Universality by asymptotics

Asymptotically flat: QNM poles plus branch cuts; power-law tails are common at very late times.
Asymptotically AdS: Boundary normalizability/source-free condition; finite-temperature non-extremal black holes/branes typically yield discrete pole spectra at fixed $k$ ; planar branes support hydrodynamic modes.
Asymptotically dS: Boundary conditions at black-hole and cosmological horizons; discrete spectra and exponential decay in many non-extremal settings.
Extremal/zero-temperature warning: Branch cuts and mode accumulation can modify the simple discrete-pole picture.
See also: Boundary conditions, Late-time tail, Near-AdS $_2$ branch cut.

Zero mode / marginal mode

What: A stationary perturbation, usually $\omega=0$ , at the threshold of an instability or bifurcation to a new branch of solutions.
Why it matters: Zero modes mark phase boundaries: onset of scalar hair, Gregory—Laflamme transitions, superradiant clouds, or symmetry-breaking phases.
Identify by: A normalizable static solution satisfying all constraints and boundary conditions.
Caveat: A gauge transformation can look like a zero mode; verify gauge-invariant content.
See also: Instability, Superradiance, Exceptional point.

Numerical and reporting checklist

Use this checklist when adding data, plots, or claims to the site.

State conventions: time dependence, Fourier transform, units, and dimensionless variables.
State the background: metric, dimension, horizon topology, thermodynamic ensemble, charge/spin, scalar mass, and boundary condition.
State the channel: scalar/vector/tensor, sound/shear, helicity, gauge-invariant variable, and operator dual.
State the QNM condition: horizon condition and boundary source-free or mixed condition.
Check convergence: grid/order refinement, independent method if possible, and residual of the differential equation plus constraints.
Track branches: use eigenfunction overlap, residues, or analytic continuation; do not label modes only by sorting $\operatorname{Im}\omega$ .
Report residues when possible: pole visibility depends on excitation factor, not only damping rate.
Flag spurious modes: remove modes that drift with cutoff, violate constraints, depend on gauge choice, or fail boundary residual tests.
For hydrodynamics: verify Ward identities and compare pole-derived coefficients with Kubo formulae.
For instabilities: locate the threshold, identify the zero mode, and verify that the unstable mode is not pure gauge.

Canonical references

These are good starting points for readers who want to go beyond the dictionary.

Berti, Cardoso & Starinets (2009), Quasinormal modes of black holes and black branes. Broad review covering black-hole and black-brane QNMs, including holographic applications. arXiv:0905.2975
Horowitz & Hubeny (2000), Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium. Classic AdS/CFT interpretation of AdS black-hole QNMs as thermalization timescales. arXiv:hep-th/9909056
Son & Starinets (2002), Minkowski-space correlators in AdS/CFT. Lorentzian prescription for retarded correlators and the QNM-pole relation. arXiv:hep-th/0205051
Kovtun & Starinets (2005), Quasinormal modes and holography. Detailed identification of AdS QNMs with finite-temperature retarded Green’s-function poles. arXiv:hep-th/0506184
Grozdanov, Kovtun, Starinets & Tadić (2019), Convergence of the Gradient Expansion in Hydrodynamics. Hydrodynamic convergence and complex-momentum QNM level crossings. arXiv:1904.01018
Blake, Davison, Grozdanov & Liu (2018), Many-body chaos and energy dynamics in holography. Pole-skipping and its connection to chaos data. arXiv:1809.01169
Faulkner, Liu, McGreevy & Vegh (2009/2011), Emergent quantum criticality, Fermi surfaces, and AdS $_2$ . Near-horizon AdS $_2$ physics and IR critical Green’s functions at finite density. arXiv:0907.2694
Jaramillo, Panosso Macedo & Al Sheikh (2021), Pseudospectrum and black hole QNM (in)stability. Modern pseudospectrum view of spectral sensitivity. arXiv:2004.06434
Jansen (2017), Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for QNMs. Practical spectral-method computation and package reference. arXiv:1709.09178
Leaver (1986), Spectral decomposition of the perturbation response of the Schwarzschild geometry. Classic discussion of QNM sums and branch-cut contributions.
Konoplya & Zhidenko (2011), Quasinormal modes of black holes: from astrophysics to string theory. Review of methods, stability, tails, and holographic interpretations. Rev. Mod. Phys. 83, 793

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