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Quasinormal modes and holography

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Pavel K. Kovtun, Andrei O. Starinets

High Energy Physics - Theory · Submitted 22 Jun 2005 · 29 references

Pith's verdict: The standard reference for why gauge-invariant quasinormal modes are the poles of thermal correlators; the general argument in §3.1 is a plausibility argument, and the authors say so, but the channel-by-channel work in §4 is solid and the hydrodynamic cross-checks are real.

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arXiv IDhep-th/0506184 (version 2)
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CitationAndrei O. Starinets, Pavel K. Kovtun. "Quasinormal modes and holography." arXiv:hep-th/0506184 (2005). https://pith.science/paper/5RW4PHUG
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The paper

The paper claims

Quasinormal ringing of an asymptotically AdS black brane, computed for the right gauge-invariant variables, is exactly the set of poles of the dual hot plasma's retarded Green's functions.

The reading

The paper asks a purely gravitational question with a field-theory answer: if you want the quasinormal spectrum of metric and gauge-field fluctuations around an asymptotically anti-de Sitter black brane to coincide with the poles of the retarded correlators in the holographically dual finite-temperature theory, which variables and which boundary conditions should you use? The authors' answer is that one must work with gauge-invariant combinations of the fluctuations, impose the incoming-wave condition at the horizon and a Dirichlet condition on those gauge-invariant variables at the boundary. The reason this works is visible in the structure of the on-shell boundary action: written in terms of a gauge-invariant variable $Z$, it collapses to a term proportional to $Z'(r)Z(r)$, so the retarded correlator is a ratio $B/A$ of the two connection coefficients of the wave equation, and its poles are precisely the zeros of $A$, which is the same statement as the Dirichlet condition. The authors then carry this out explicitly for the five-dimensional AdS-Schwarzschild black brane, obtaining the poles of the R-symmetry current correlators and of all three channels of the stress-energy correlator in strongly coupled $N=4$ supersymmetric Yang-Mills theory at large $N_c$, and recovering the known hydrodynamic transport coefficients in the low-momentum limit. What a reader should care about is that a classical general-relativity eigenvalue problem, with no field theory calculation at all, is claimed to encode the full singularity structure of strongly coupled thermal correlators.

What carries the argument

The load-bearing device is the pair (gauge-invariant variable, connection coefficients). Because linearized diffeomorphisms and gauge transformations make the raw metric and vector components ambiguous, the authors build combinations that are invariant under $h_{mu nu} -> h_{mu nu} - nabla_mu xi_nu - nabla_nu xi_mu$ and $A_mu -> A_mu - partial_mu lambda$, in the spirit of gauge-invariant cosmological perturbation theory. Each such variable $Z$ satisfies a single second-order ODE; expanding its incoming-wave solution in the two local Frobenius solutions at the boundary gives $Z = A phi_1 + B phi_2$. The on-shell quadratic boundary action, once rewritten in terms of $Z$, reduces to $F(omega,q) Z' Z$ plus contact terms, so the Lorentzian AdS/CFT prescription gives a retarded correlator proportional to $B/A$. The zeros of $A$ are therefore the poles of the correlator, and $A=0$ is the Dirichlet condition $Z(boundary)=0$. This is what converts a general-relativity eigenvalue problem into a statement about singularities of thermal Green's functions, and it also explains which channel maps to which field-theory function: $E_z$ to $Pi_L$, $E_alpha$ to $Pi_T$, $Z_1$ to $G_1$, $Z_2$ to $G_2$, $Z_3$ to $G_3$. The numerical spectra are then obtained from a Frobenius series around the horizon, with the Dirichlet condition becoming a transcendental equation on the series coefficients.

Extended reading notes

Core claim

The central claim is that quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically AdS spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite-temperature field theory, provided the quasinormal problem is posed for gauge-invariant quantities obeying an incoming-wave condition at the horizon and a Dirichlet condition at the boundary. For a black brane with translationally invariant horizon, the authors construct these variables channel by channel: the electric fields $E_z = qA_t + omega A_z$ and $E_alpha = omega A_alpha$ for the Maxwell field, and the combinations $Z_1$, $Z_2$, $Z_3$ for the shear, sound, and scalar channels of the metric. Each obeys a second-order ordinary differential equation whose solution near the boundary takes the form $A(1+...)+B u^2+...$, so that the retarded correlator is proportional to $B/A$ up to contact terms, and the poles of the correlator are the zeros of $A$; setting $A=0$ is exactly the Dirichlet boundary condition. Applied to the near-horizon geometry of $N_c$ black three-branes, this yields the poles of the R-current correlators $Pi_T$ and $Pi_L$ and of the three stress-tensor functions $G_1$, $G_2$, $G_3$, with numerical tables of the lowest five non-hydrodynamic frequencies at spatial momentum $q/2pi T = 1$ and, in the hydrodynamic regime, the charge diffusion constant $D_R = 1/2pi T$, the shear damping $gamma_eta = 1/4pi T$, and the sound speed $v_s = 1/sqrt{3}$ with attenuation $Gamma_s = 1/6pi T$.

Load-bearing premise

The general argument assumes that for any such background the on-shell quadratic boundary action, written in terms of a single gauge-invariant variable, reduces to one term of the form $F(omega,q) Z'(r) Z(r)$ plus contact terms with no surviving cross terms among fluctuation components; the paper verifies this channel by channel in the five-dimensional black-brane case but asserts it in general.

Editorial extensions

If this is right

  • Thermal correlation functions of conserved currents and the stress-energy tensor in strongly coupled $N=4$ Yang-Mills at large $N_c$ can be read off from a classical boundary value problem, with the singularity positions tabulated rather than computed diagram by diagram.
  • The hydrodynamic limit drops out automatically: the lowest purely imaginary poles reproduce charge diffusion with $D_R = 1/2pi T$, transverse momentum diffusion with $gamma_eta = 1/4pi T$, and the sound mode $omega = pm q/sqrt{3} - i q^2/6pi T$.
  • The absence of a hydrodynamic pole in the transverse electric and scalar gravitational channels is a prediction, not an assumption: those channels simply have no Dirichlet solution with $omega$ small.
  • Because the identification is channel by channel, a measured or lattice-computed set of poles in a strongly coupled plasma could in principle be matched against a black-brane spectrum channel by channel.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the identification holds beyond the brane case, then the analytic structure of finite-temperature correlators at infinite $N_c$ is controlled by the singular points of a Fuchsian ODE, which suggests the meromorphy the authors leave open could be attacked by classifying those singular points rather than by field-theory methods.
  • The same gauge-invariant-variable bookkeeping should transfer to rotating or charged black branes, where the extra background fields mix channels; the interesting question is whether the on-shell action still collapses to a single $Z'Z$ term or whether cross terms survive and spoil the $B/A$ form.
  • Since the Dirichlet condition is what selects the poles, a natural test is to impose a different boundary condition (a Robin condition, say) and check that the resulting spectrum matches a correlator computed with a correspondingly modified double-trace coupling in the dual theory.
  • The paper's own remark that $alpha'$ corrections might reveal new poles suggests a concrete programme: compute the leading string corrections to the wave equations and see whether new zeros of $A$ appear at frequencies scaling with the 't Hooft coupling.
Original abstract

Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically AdS spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite temperature field theory. The quasinormal modes are defined for gauge-invariant quantities which obey incoming-wave boundary condition at the horizon and Dirichlet condition at the boundary. As an application, we explicitly find poles of retarded correlation functions of R-symmetry currents and the energy-momentum tensor in strongly coupled finite temperature N=4 supersymmetric SU(Nc) Yang-Mills theory in the limit of large Nc.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,064 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.33 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically AdS spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite temperature field theory. The quasinormal modes are defined for gauge-invariant quantities which obey incoming-wave boundary condition at the horizon and Dirichlet condition at the boundary. As an application, we explicitly find poles of retarded correlation functions of R-symmetry currents and the energy-momentum tensor in strongly coupled finite temperature N=4 supersymmetric SU(Nc) Yang-Mills theory in the limit of large Nc.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically AdS spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite temperature field theory. background 0.70 0.15 0.70 asserted 0.96
s1 The quasinormal modes are defined for gauge-invariant quantities which obey incoming-wave boundary condition at the horizon and Dirichlet condition at the boundary. method 0.56 0.12 0.07 asserted 1.00
s2 As an application, we explicitly find poles of retarded correlation functions of R-symmetry currents and the energy-momentum tensor in strongly coupled finite temperature N=4 supersymmetric SU(Nc) Yang-Mills theory in the limit of large Nc. contribution 0.62 0.95 0.24 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.93.

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sources read
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Quasinormal spectrum of electric field fluctuations in the plane of complex w ≡ ω/2πT , shown for spatial momentum q ≡ q/2πT = 1. Quasinormal frequencies on the left are defined by equation (4.5a) for the transverse electric field, and coincide with the poles of ΠT (ω, q), as explained in the text. Quasinormal frequencies on the right are defined by equation (4.5b) for the longitudinal electric field, and coincide with the poles of ΠL(ω, q). As q decreases, all poles stay at a finite distance from the real axis, except for the one marked with a large dot. This pole is purely…
Figure 2. Quasinormal spectrum of gravitational fluctuations in the scalar channel, shown in the plane of complex w ≡ ω/2πT , for spatial momentum q ≡ q/2πT = 1. The quasinormal frequencies coincide with poles of G3(ω, q), as explained in the text. As q → 0, all poles stay at a finite distance from the real axis, as one expects from the absence of hydrodynamic singularities in G3(ω, q). Proceeding as in Section 4.1, we obtain the correlator13 Gxy,xy = − π 2N2 c T 4B(3) 2A(3) . (4.21) Comparing this expression to the general result (2.34), we find G3(ω, q) = − π 2N2 c T 4 B(3)(ω, q) A(3)(ω, q)…
Figure 3. Quasinormal spectrum of gravitational fluctuations in the shear channel, shown in the plane of complex w ≡ ω/2πT , for spatial momentum q ≡ q/2πT = 1. The quasinormal frequencies coincide with poles of G1(ω, q), as explained in the text. As q decreases, all poles stay at a finite distance away from the real axis, except for the one marked with a large dot. This pole is purely imaginary and approaches the origin in the limit q → 0. The presence of this special quasinormal frequency is a manifestation of the diffusive relaxation of transverse momentum density fluctuations in the dual…
Figure 4. Quasinormal spectrum of gravitational fluctuations in the sound channel, shown in the plane of complex w ≡ ω/2πT , for spatial momentum q ≡ q/2πT = 1. The quasinormal frequencies coincide with the poles of G2(ω, q), as explained in the text. As q decreases, all poles stay at a finite distance away from the real axis, except for the ones marked with large dots, which approach the origin as q → 0 (see Appendix B for the corresponding dispersion curves). Such behavior of the lowest quasinormal frequencies is a manifestation of oscillatory relaxation of longitudinal momentum density (as…
Figure 5. Real and imaginary parts of the lowest quasinormal (sound wave) frequency as a function of spatial momentum. Light curves correspond to the analytic approximation (4.43) for small q. Dashed line is w = q. As q grows, the dispersion curve enters the region where ∂Re (w)/∂q > 1. This, however, happens for q ∼ 1, where the corresponding singularity of the energy-momentum correlation function no longer has the interpretation of a sound wave. poles. It may indeed be true that real-time thermal correlation functions of gauge-invariant operators are meromorphic in complex frequency plane;…
Figure 6. Real and imaginary parts of three lowest quasinormal frequencies as function of spatial momentum. The curves for which w→0 as q→0 correspond to hydrodynamic sound mode in the dual finite temperature N =4 SYM theory. behavior of the lowest (hydrodynamic) frequency which is absent for Eα and Z3. For Ez and Z1, hydrodynamic frequencies are purely imaginary (given by Eqs. (4.16) and (4.32) for small ω and q), and presumably move off to infinity as q becomes large. For Z2, the hydrodynamic frequency has both real and imaginary parts (given by Eq. (4.44) for small ω and q), and eventually…

Figures reproduced from arXiv:hep-th/0506184 by the authors.

References
How later papers used it

326 later papers cite it

Evidence: substantial. 311 of the citing papers share no author with it, and 102 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 676 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 17 of 92 papers with this label · 2 by the same authors

By the same authors

  • Constructed the Frobenius series solution at u=1 and truncated the series at a sufficiently high order, as in this paper.

    On the convergence of the gradient expansion in hydrodynamics (2019) (same authors) · 0.96 · paraphrased: the licence does not allow quoting

  • Found the lowest quasinormal frequency by imposing the Dirichlet condition at the boundary, as in this paper.

    Zero Sound from Holography (2008) (same authors) · 0.97 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 7 papers with this label
Compares with it 1 of 22 papers with this label

  • Verified that its condition k^2 = 4iω is equivalent to ω̃ = -i k̃^2/2, exactly the dispersion relation of shear hydrodynamic quasinormal modes derived in this paper using a different coordinate system.

    Coupled Oscillator Model for Nonlinear Gravitational Perturbations (2015) · 0.99 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 3 papers with this label

  • Deduced from the behavior of this paper's sound dispersion curve an asymptotic behavior of the viscosity function, which supports its understanding that the function falls at large momenta.

    Improved Hydrodynamics from the AdS/CFT (2009) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

306 arXiv papers cite this in 648 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 348 sentences without a confident label

Contests it 1 paper
  • A closely related investigation of the longitudinal vector modes and a massless minimally coupled scalar appeared in [ref] and [ref] , respectively.[There is an erroneous claim in [ref] that the results for the transverse vector modes should be precisely the same as for that for the massless minimally coupled scalar field given in [ref] .] The relevant equation of motion for the transverse vector modes $E_T$ in $AdS_5$ can be found in [ref] and can be written as ∂_ρ(ρ^3f∂_ρE_T) +8ρ ff̃(^2f̃^2f^2-^2) E_T=0 , where $f$ and $f̃$ are defined above in ffff.

    The fast life of holographic mesons · Beyond $$ · contest 0.90

Builds on it 2 papers
Uses its method or tool 90 papers
Uses its data or values 7 papers
Compares with it 20 papers
Cites it as support 3 papers
  • Finally we would like to note that from the behavior of the sound dispersion curve [ref] one can deduce the following asymptotic behavior of the viscosity function η(k^2∼^2 → ∞) → i2 ω , which supports our understanding that it is a falling function at large momenta.

    Improved Hydrodynamics from the AdS/CFT · On quasinormal modes and analytic structure of the viscosity function · 0.97

  • What is really significant here is that — from the quasinormal-mode perspective of brane hydrodynamics [ref] — the pole in the correlator assigns a clear physical credence to Eq. ([ref]) as the spectrum for the dissipative modes of the black brane.

    The sound damping constant for generalized theories of gravity · Sound mode conformal hydrodynamics for Einstein's gravity · 0.96

  • This singularity is not undesirable; rather, it can be viewed as a well-motivated expectation from the quasinormal-mode perspective of brane hydrodynamics [ref] .

    The shear diffusion coefficient for generalized theories of gravity · Section · 0.96

Cites it as background 106 papers
  • For gravitational perturbations the QNM spectrum was first obtained in the infinite coupling limit in [ref] and the diffusion poles in the hydrodynamic limit at finite coupling were worked out in [ref] .

    Holographic thermalization in N=4 Super Yang-Mills theory at finite coupling · Quasinormal mode spectrum · 0.96

  • In particular, thermodynamic and hydrodynamic behavior are encoded in black hole geometries and their quasi-normal modes (QNMs) [ref] .

    Holographic D-brane constructions with dynamical gauge fields · Introduction · 0.85

  • Recently, a useful picture for the unified description of these two facets of black holes has started to emerge [ref] , inspired in part by a proposal for computing real-time observables in holography [ref] .[Various aspects of the problem have been discussed extensively in the literature, cf., [ref] for computing real-time observables in holography, [ref] for analysis of quasinormal modes, and [ref] for construction of effective actions for hydrodynamics in field theory.

    An effective description of momentum diffusion in a charged plasma from holography · Introduction · 0.97

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