Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

Transients in black hole perturbation theory

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This review argues that the dissipative nature of black holes makes the linear perturbation Hamiltonian non-normal, so quasinormal modes are non-orthogonal and produce long-lived or growing transients before modal decay sets in.

desk verdict A clear and useful review of transients from non-normal QNM operators, but the 'arbitrarily long-lived' plateau claim is only demonstrated in a finite-mode truncation, and the review's own toy-model result warns that the energy-norm plateau may not survive the continuum limit. read the letter →

arxiv 2507.16493 v1 pith:YO3VXPTK submitted 2025-07-22 gr-qc

Jérémy Besson , Javier Carballo , Christiana Pantelidou , Benjamin Withers This is my paper · ORCID
classification gr-qc MSC 83C5783C35 PACS 04.70.-s04.30.-w
keywords quasinormalmodesblackholeperturbationtheorynon-normaloperatorstransientspseudospectraringdowntransientgrowthSobolevnorms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review argues that the dissipative nature of black holes makes the linear perturbation Hamiltonian non-normal, so quasinormal modes are neither orthogonal nor complete under standard inner products. A direct consequence is that sums of quasinormal modes can display transient behaviour, either long-lived plateaus or actual growth, before settling into the familiar exponential modal decay. The paper assembles evidence from frequency-domain pseudospectra and from time-domain optimal perturbation constructions in Schwarzschild, Reissner-Nordström anti-de Sitter, and the Pöschl-Teller de Sitter toy model. A sympathetic reader should care because these transients are in principle observable, potentially contaminating black hole spectroscopy and holographic thermalization predictions.

What carries the argument

The central object is the non-normal evolution operator $H$ in $i\partial_\tau u = Hu$ on hyperboloidal slices, whose eigenfunctions are the quasinormal modes $\chi_n$ with complex frequencies $\omega_n$. Non-normality means the quasinormal modes fail orthogonality under the energy inner product $\langle\cdot,\cdot\rangle_E$, so the energy of a superposition is not the sum of individual mode energies: cross-terms appear. The paper uses two complementary tools to expose the consequences: the pseudospectrum $\sigma_\epsilon(H)$ with the Kreiss constant $K(H)$, which bounds possible transient growth, and time-domain optimal perturbations constructed by singular value decomposition of the finite-rank evolution operator, which realise the maximum of the energy growth curve $G(\tau)$. In the Sobolev $H^p$ norms the truncation to a mode subspace is replaced by higher-derivative regularity, shifting the transient peak to shorter times.

What would settle it

Take the Schwarzschild Regge-Wheeler problem and compute the energy growth curve $G(\tau)$ at increasing spectral resolution $N$; if the transient plateau duration continues to scale as $\log N$ without converging as $N\to\infty$, the continuum existence of arbitrarily long-lived linear perturbations is falsified. A complementary check is to evaluate the pseudospectral abscissa $\alpha_\epsilon(H)$ of the untruncated operator and test whether the Kreiss constant $K(H)$ exceeds 1 in the continuum limit.

Watch

Extended reading notes

Core claim

The central claim is that non-normality of the Hamiltonian, caused by energy flowing through the horizon and out to null infinity, is not a technical nuisance but the origin of a class of linear phenomena. Because quasinormal modes are non-orthogonal with respect to the energy inner product, the energy of a multimode perturbation contains cross-terms that can cancel or amplify. These cross-terms sustain energy packets near the future horizon and null infinity for times scaling with the number of modes, or produce genuine transient energy growth when the perturbation can borrow from a coupled field or when measured in higher-derivative Sobolev norms. The review presents the strongest established cases: arbitrarily long-lived perturbations in Schwarzschild, transient superradiance in Reissner-Nordström anti-de Sitter branes, and $H^p$ growth in the de Sitter static patch, all culminating in late-time modal decay.

Load-bearing premise

The load-bearing assumption is that a computation with finitely many quasinormal modes or finitely many grid points faithfully represents the true continuum dynamics, and the evidence for arbitrarily long-lived perturbations would collapse if the plateau duration keeps growing with resolution instead of settling down.

Editorial extensions

If this is right

  • In Schwarzschild, optimal sums of the first $M$ Regge-Wheeler quasinormal modes keep their energy constant for a duration $\sim \log M$ before decaying at the fundamental rate, so linear theory contains arbitrarily long-lived packets for sufficiently large $M$.
  • In the Reissner-Nordström anti-de Sitter black brane, a stable spectrum still allows the scalar-field energy to grow transiently by borrowing from the gauge field through a transient form of superradiance.
  • In Sobolev $H^p$ norms the transient peak grows as $G(\tau_{\max})\sim p$ and occurs at $\tau_{\max}\sim 1/p$, with the peak dominated by the $(p+1)$-th quasinormal-mode pair.
  • Pseudospectral protrusions into the unstable half-plane, quantified by a Kreiss constant $K>1$, predict such growth; for Schwarzschild in the energy norm $w(H)=0$ and $K=1$, consistent with the observed lack of energy growth.

Reading between the lines

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

  • If the transients survive in the continuum limit, gravitational-wave ringdown templates that assume early modal decay could misattribute a portion of the signal, so one could look for plateau-like persistence in numerical relativity waveforms.
  • The non-convergence of the $H^0$ plateau duration with grid size suggests that only higher-derivative norms may capture a true continuum effect, and a fully resolved continuum computation would be the test.
  • The open nonlinear question points to a concrete numerical experiment: evolve the optimal perturbations identified here beyond linear order to see whether transient linear energy becomes a seed for nonlinear instability or turbulence-like behaviour.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The manuscript is a review article on transient (non-modal) phenomena in linear black hole perturbation theory. It argues that the dissipative nature of black hole horizons renders the evolution operator non-normal, so that quasinormal modes (QNMs) are not orthogonal under standard inner products such as the energy product; this non-orthogonality gives rise to transient plateaus (arbitrarily long-lived sums of QNMs) and transient growth as measured in suitable norms. The review covers pseudospectral methods (Kreiss constant, numerical abscissa), time-domain studies of optimal perturbations in a truncated QNM subspace, and higher-derivative Sobolev norms, with examples from Schwarzschild, RN-AdS4, and a Pöschl-Teller toy model. It also corrects a transient-growth result in a previous paper [23].

Significance. If the summarized results are correct, the review highlights a genuine gap in the standard QNM picture: linear perturbations can exhibit long-lived transient behavior even when the mode spectrum is stable. The review is clearly written and brings together recent developments, including several from the authors' own work, with useful figures and a coherent narrative. It also explicitly notes a convergence caveat for the energy-norm plateau in the toy model, which is a useful caution. The correction of [23] is a strong claim that would be impactful if substantiated, but it is currently asserted without the promised calculation.

major comments (3)
  1. [Section 3, H^p discussion] The review claims that [22] demonstrates "arbitrarily long-lived linear black hole perturbations" based on a plateau in GW(τ) whose duration scales as log M within a finite QNM subspace W of dimension M. However, later in the same section, the review states that in the Pöschl-Teller model the energy-norm (H^0) plateau computed on the full grid has duration scaling as log N and is non-convergent as N grows. Since H^0 is the same energy norm, this raises the real possibility that the log M plateau for Schwarzschild is a truncation artifact. The review does not reconcile these observations; it merely says the non-convergence "further motivates the use of H^p norms." To sustain the headline claim of arbitrarily long-lived perturbations, the authors should either provide evidence of convergence (e.g., show that the optimal initial data converge in the full Hilbert space as M increases, or compare against a full-grid evolution) or carefully qualify the claim as a statement about the finite-dimensional truncated system rather than about the continuum evolution. As written, the abstract's "arbitrarily long-lived" is not established.
  2. [Footnote 4] The statement that "(3.19) there can be written as a total derivative" in [23] is a mathematical assertion that is not demonstrated. Since the review explicitly concludes that the reported transient growth in [23] is "incorrect," this correction is a load-bearing claim and should be backed by an explicit calculation or by a reference to a paper that provides one. Without this, the wording should be softened to indicate that the result appears inconsistent or that the total-derivative claim is made without proof.
  3. [Section 3, Eq. (12); Section 4] The review repeatedly states that after the transient plateau the evolution "conforms to modal decay" with the fundamental mode decay rate. This is, however, an artifact of the QNM-sum ansatz (12), which restricts the dynamics to a finite subspace of QNMs. Full asymptotically flat evolutions generally exhibit power-law tails at late times, not pure modal decay. The review does not warn the reader that the claimed modal-decay behavior holds only within the truncation and is not a property of the exact evolution. This should be clarified explicitly, as it bears on the physical interpretation of the plateau phenomenon.
minor comments (3)
  1. [Abstract] The phrase "arbitrarily long-lived sums of quasinormal modes" in the abstract is too strong without the caveat that this holds within a truncated QNM subspace and that the continuum limit is not established; consider qualifying it, e.g., "arbitrarily long-lived within the truncated QNM model."
  2. [Section 2] The discussion of the pseudospectrum and the bounds (6)-(7) would benefit from explicitly stating which norm is used for the operator norm, since the values of w(H) and K(H) depend crucially on the chosen norm (energy, Sobolev, etc.). The reader is left to infer that the energy norm is used for the Schwarzschild result in [20].
  3. [References] There are several formatting inconsistencies in the references, such as "V ," and "Toomani V ," with reversed comma placement; these should be corrected in the final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the review's transients claims rest on prior independent numerical studies and standard linear algebra, not on a result defined into existence.

full rationale

This paper is a review, not a first-principles derivation, so the circularity test is whether any of its organizing claims are true by construction or forced by self-citation. The chain 'non-normal H -> non-orthogonal QNMs -> cross-terms in Eq. (10) -> transient phenomena' is elementary linear algebra plus standard non-normal operator theory (Kreiss constants, pseudospectral bounds), and is presented with external references [15,16] and explicit formulas, not with a loaded definition. The quantitative headline results (the log M plateau in Schwarzschild, transient superradiance in RN-AdS4, Hp-norm growth in the Pöschl-Teller model) are reported from the authors' own prior papers [22,29,26]. That is heavy self-citation, but under the stated rules self-citation is not itself circularity: those prior works are independent, falsifiable numerical studies, and the review does not invoke an unverified uniqueness theorem or import an ansatz solely through a citation. The plateau construction is also transparent: G_W(τ) is explicitly the maximum energy over the truncated QNM subspace W, and the optimal perturbation is defined by an SVD at a target time, so the existence of a perturbation that retains energy until τ* is a computed consequence of the stated optimization, not a hidden fit. The review's own caveat that the H0 (energy-norm) plateau in the Pöschl-Teller model scales as log N and is non-convergent is a genuine limitation on the continuum-lifetime claim, but it is a correctness/scientific-convergence concern, not a circularity. No equation in the text reduces to its own input, and no fitted parameter is relabeled as a prediction. Score 2 reflects the density of self-citations for the showcased phenomena, not any identified circular step.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The ledger captures the two regulators the results depend on (M and p) plus the main domain assumptions about the energy inner product and the probe limit. No new entities are introduced.

free parameters (2)
  • M, dimension of QNM subspace W = 10, 19, 39 (Fig. 1); 10 (Fig. 2)
    Truncation parameter chosen by hand; plateau duration scales as log M and growth peaks scale with M within the tested range.
  • p, order of Sobolev norm = p = 25 in Fig. 2; variable elsewhere
    Regulator for regularity; transient growth peak behaves as G ~ p at time ~1/p; physical interpretation left open in the review.
assumptions (4)
  • standard math Standard pseudospectral bound theorems: Kreiss constant lower bound (6) and numerical abscissa upper bound (7).
    Used in Section 2 to connect pseudospectrum protrusions to transient growth; cited from Trefethen and Embree [15] without proof.
  • domain assumption The hyperboloidal foliation and energy density (3) supply a positive definite energy inner product (4).
    Section 1; all energy-norm statements and the non-orthogonality of QNMs are formulated with respect to this product.
  • domain assumption Truncating to the first M QNMs gives a valid finite-dimensional model of the transient dynamics.
    Section 3; used to define GW and optimal perturbations; exact only inside W, with convergence to the continuum not proven for the plateau effect.
  • domain assumption In the q to infinity probe limit, the total energy splits as E = E_psi + E_F with no metric backreaction.
    Section 3; needed for the interpretation of scalar energy growth as borrowing from the gauge field energy bath in the RN-AdS4 example.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Transients in black hole perturbation theory." pith.science (2026). https://pith.science/paper/YO3VXPTK

@misc{pith2026250716493,
  author       = {Pith},
  title        = {Pith review of: Transients in black hole perturbation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YO3VXPTK}},
  note         = {Machine review of arXiv:2507.16493}
}
read the original abstract

Black hole quasinormal modes arise as eigenmodes of a non-normal Hamiltonian and consequently they do not obey orthogonality relations with respect to commonly used inner products, for example, the energy inner product. A direct consequence of this is the appearance of transient phenomena. This review summarises current developments on the topic, both in frequency- and time-domain. In particular, we discuss the appearance of i) transient plateaus: arbitrarily long-lived sums of quasinormal modes, corresponding to localised energy packets near the future horizon; ii) transient growth, with the latter either appearing in the vicinity of black hole phase transitions or in the context of higher-derivative Sobolev norms.

Figures

Figures reproduced from arXiv: 2507.16493 by the authors.

Figure 1
Figure 1. Energy growth curves and optimal perturbation for Schwarzschild s = 2, l = 2 Regge-Wheeler perturbations (figure taken from [22]). Top: GW for various M = dim(W) (solid curves), and the energy of an optimal perturbation of M = 39 QNMs with τ∗ = 8.5 (red-dash). Bottom: Modulus (left) and energy density (right) of the optimal perturbation in the conformal diagram of Schwarzschild. The energy is initially localised at … view at source ↗
Figure 2
Figure 2. Left: Optimal perturbation and energy growth curve GW (τ ) (black dash) for complex charged scalar QNMs of the RN-AdS4 black brane with M = 10. Eψ (solid black curve) is shown to transiently grow before modally decaying at asymptotic time. The additional energy is borrowed from the energy bath EF via a transient form of superradiance, as can be seen from the first correction to EF , E (2) F (solid blue curve). The e… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Parity violating spectral dynamics of black holes in dynamical Chern-Simons gravity

    gr-qc 2025-11 conditional novelty 6.0 of 10

    In dynamical Chern-Simons gravity, an environmental potential bump reshapes black hole quasinormal-mode spectra, producing branch reconnections, a delayed overtaking instability, and scalar-mode-dominated ringdown tha...

Reference graph

Works this paper leans on

38 extracted references · 14 canonical work pages · cited by 1 Pith paper

  1. [22]

    Transient dynamics of quasinormal mode sums

    Carballo J, Withers B. Transient dynamics of quasinormal mode sums. JHEP 10 (2024) 084. doi:10.1007/JHEP10(2024)084

    work page doi:10.1007/jhep10(2024)084 2024

  2. [26]

    Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach

    Besson J, Jaramillo JL. Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach. arXiv:2412.02793v2 (2025)

    arXiv 2025

  3. [29]

    Non-modal effects in black hole perturbation theory: Transient Superradiance

    Carballo J, Pantelidou C, Withers B. Non-modal effects in black hole perturbation theory: Transient Superradiance. arXiv:2503.05871 (2025)

    arXiv 2025

  4. [23]

    The pseudospectrum and transient of Kaluza–Klein black holes in Einstein–Gauss–Bonnet gravity

    Chen JN, Wu LB, Guo ZK. The pseudospectrum and transient of Kaluza–Klein black holes in Einstein–Gauss–Bonnet gravity. Class. Quant. Grav. 41 (2024) 235015. doi:10.1088/1361-6382/ ad89a1. 9 Transients in black hole perturbation theory

    doi:10.1088/1361-6382/ 2024

  5. [1]

    Quasinormal modes of black holes and black branes.Class

    Berti E, Cardoso V , Starinets AO. Quasinormal modes of black holes and black branes.Class. Quant. Grav. 26 (2009) 163001. doi:10.1088/0264-9381/26/16/163001

    doi:10.1088/0264-9381/26/16/163001 2009

  6. [2]

    Quasinormal modes of black holes: From astrophysics to string theory

    Konoplya RA, Zhidenko A. Quasinormal modes of black holes: From astrophysics to string theory. Rev. Mod. Phys. 83 (2011) 793–836. doi:10.1103/RevModPhys.83.793. 8 Transients in black hole perturbation theory

    doi:10.1103/revmodphys.83.793 2011

  7. [3]

    Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole

    Lagos M, Hui L. Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole. Phys. Rev. D 107 (2023) 044040. doi:10.1103/PhysRevD.107.044040

    doi:10.1103/physrevd.107.044040 2023

  8. [4]

    Thermal three-point functions from holographic Schwinger-Keldysh contours

    Pantelidou C, Withers B. Thermal three-point functions from holographic Schwinger-Keldysh contours. JHEP 04 (2023) 050. doi:10.1007/JHEP04(2023)050

    work page doi:10.1007/jhep04(2023)050 2023

Show all 38 references
  1. [5]

    The Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma

    Policastro G, Son DT, Starinets AO. The Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma. Phys. Rev. Lett. 87 (2001) 081601. doi:10.1103/PhysRevLett.87.081601

    2001 doi

  2. [6]

    Quasinormal modes and holography

    Kovtun PK, Starinets AO. Quasinormal modes and holography. Phys. Rev. D 72 (2005) 086009. doi:10.1103/PhysRevD.72.086009

    2005 doi

  3. [7]

    Agnostic black hole spectroscopy: Quasinormal mode content of numerical relativity waveforms and limits of validity of linear perturbation theory

    Baibhav V , Cheung MHY , Berti E, Cardoso V , Carullo G, Cotesta R, et al. Agnostic black hole spectroscopy: Quasinormal mode content of numerical relativity waveforms and limits of validity of linear perturbation theory. Phys. Rev. D 108 (2023) 104020. doi:10.1103/PhysRevD.108.104020

    2023 doi

  4. [8]

    Dynamical Origin of Black Hole Radiance

    York JW Jr. Dynamical Origin of Black Hole Radiance. Phys. Rev. D 28 (1983) 2929. doi:10.1103/ PhysRevD.28.2929

    1983

  5. [9]

    Analysis of linear waves near the Cauchy horizon of cosmological black holes

    Hintz P, Vasy A. Analysis of linear waves near the Cauchy horizon of cosmological black holes. J. Math. Phys. 58 (2017) 081509. doi:10.1063/1.4996575

    2017 doi

  6. [10]

    Quasinormal quantization in de Sitter spacetime

    Jafferis DL, Lupsasca A, Lysov V , Ng GS, Strominger A. Quasinormal quantization in de Sitter spacetime. JHEP 01 (2015) 004. doi:10.1007/JHEP01(2015)004

    2015 doi

  7. [11]

    Conserved currents for a Kerr black hole and orthogonality of quasinormal modes

    Green SR, Hollands S, Sberna L, Toomani V , Zimmerman P. Conserved currents for a Kerr black hole and orthogonality of quasinormal modes. Phys. Rev. D 107 (2023) 064030. doi:10.1103/PhysRevD. 107.064030

    2023 doi

  8. [12]

    A radial scalar product for Kerr quasinormal modes

    London LT. A radial scalar product for Kerr quasinormal modes. arXiv:2312.17678 (2023)

    2023

  9. [13]

    QNM orthogonality relations for AdS black holes.arXiv:2505.04696 (2025)

    Arnaudo P, Carballo J, Withers B. QNM orthogonality relations for AdS black holes.arXiv:2505.04696 (2025)

    2025

  10. [14]

    Pseudospectra in non-Hermitian quantum mechanics

    Krejcirik D, Siegl P, Tater M, Viola J. Pseudospectra in non-Hermitian quantum mechanics. J. Math. Phys. 56 (2015) 103513. doi:10.1063/1.4934378

    2015 doi

  11. [15]

    Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University Press) (2005)

    Trefethen L, Embree M. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University Press) (2005)

    2005

  12. [16]

    Energy scales and black hole pseudospectra: the structural role of the scalar product

    Gasperin E, Jaramillo JL. Energy scales and black hole pseudospectra: the structural role of the scalar product. Class. Quant. Grav. 39 (2022) 115010. doi:10.1088/1361-6382/ac5054

    2022 doi

  13. [17]

    Pseudospectrum and Black Hole Quasinormal Mode Instability

    Jaramillo JL, Panosso Macedo R, Al Sheikh L. Pseudospectrum and Black Hole Quasinormal Mode Instability. Phys. Rev. X 11 (2021) 031003. doi:10.1103/PhysRevX.11.031003

    2021 doi

  14. [18]

    Quantifying excitations of quasinormal mode systems

    Nollert HP, Price RH. Quantifying excitations of quasinormal mode systems. J. Math. Phys. 40 (1999) 980–1010. doi:10.1063/1.532698

    1999 doi

  15. [19]

    About the significance of quasinormal modes of black holes

    Nollert HP. About the significance of quasinormal modes of black holes. Phys. Rev. D 53 (1996) 4397–4402. doi:10.1103/PhysRevD.53.4397

    1996 doi

  16. [20]

    Pseudospectrum and binary black hole merger transients

    Jaramillo JL. Pseudospectrum and binary black hole merger transients. Class. Quant. Grav. 39 (2022) 217002. doi:10.1088/1361-6382/ac8ddc

    2022 doi

  17. [21]

    Pseudospectrum of horizonless compact objects: A bootstrap instability mechanism

    Boyanov V , Destounis K, Panosso Macedo R, Cardoso V , Jaramillo JL. Pseudospectrum of horizonless compact objects: A bootstrap instability mechanism. Phys. Rev. D 107 (2023) 064012. doi:10.1103/ PhysRevD.107.064012

    2023

  18. [24]

    Structural aspects of the anti–de sitter black hole pseudospectrum

    Boyanov V , Cardoso V , Destounis K, Jaramillo JL, Macedo RP. Structural aspects of the anti–de sitter black hole pseudospectrum. Phys. Rev. D 109 (2024) 064068. doi:10.1103/PhysRevD.109.064068

    2024 doi

  19. [25]

    (in) stability of de sitter quasinormal mode spectra

    Warnick C. (in) stability of de sitter quasinormal mode spectra. arXiv:2407.19850 (2024)

    2024 arXiv

  20. [27]

    On quasinormal modes of asymptotically anti-de Sitter black holes

    Warnick CM. On quasinormal modes of asymptotically anti-de Sitter black holes. Commun. Math. Phys. 333 (2015) 959–1035. doi:10.1007/s00220-014-2171-1

    2015 doi

  21. [28]

    A toy model of hyperboloidal approach to quasinormal modes

    Bizo´n P, Chmaj T, Mach P. A toy model of hyperboloidal approach to quasinormal modes. Acta Phys. Polon. B 51 (2020) 1007. doi:10.5506/APhysPolB.51.1007

    2020 doi

  22. [30]

    Pseudospectra of the Orr-Sommerfeld Operator

    Reddy SC, Schmid PJ, Henningson DS. Pseudospectra of the Orr-Sommerfeld Operator. SIAM Journal on Applied Mathematics 53 (1993) 15–47

    1993

  23. [31]

    Energy growth of three-dimensional disturbances in plane Poiseuille flow

    Gustavsson LH. Energy growth of three-dimensional disturbances in plane Poiseuille flow. Journal of Fluid Mechanics 224 (1991) 241–260. doi:10.1017/S002211209100174X

    1991 doi

  24. [32]

    A mechanism for bypass transition from localized disturbances in wall-bounded shear flows

    Henningson DS, Lundbladh A, Johansson A V . A mechanism for bypass transition from localized disturbances in wall-bounded shear flows. Journal of Fluid Mechanics 250 (1993) 169–207. doi:10. 1017/S0022112093001429

    1993

  25. [33]

    Three-dimensional optimal perturbations in viscous shear flow

    Butler KM, Farrell BF. Three-dimensional optimal perturbations in viscous shear flow. Physics of Fluids A: Fluid Dynamics 4 (1992) 1637–1650. doi:10.1063/1.858386

    1992 doi

  26. [34]

    Energy growth in viscous channel flows

    Reddy SC, Henningson DS. Energy growth in viscous channel flows. Journal of Fluid Mechanics 252 (1993) 209–238. doi:10.1017/S0022112093003738

    1993 doi

  27. [35]

    Hydrodynamic stability without eigenvalues

    Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA. Hydrodynamic stability without eigenvalues. Science 261 (1993) 578–584. doi:10.1126/science.261.5121.578

    1993 doi

  28. [36]

    Breaking an Abelian gauge symmetry near a black hole horizon

    Gubser SS. Breaking an Abelian gauge symmetry near a black hole horizon. Phys. Rev. D 78 (2008) 065034. doi:10.1103/PhysRevD.78.065034

    2008 doi

  29. [37]

    Building a Holographic Superconductor

    Hartnoll SA, Herzog CP, Horowitz GT. Building a Holographic Superconductor. Phys. Rev. Lett. 101 (2008) 031601. doi:10.1103/PhysRevLett.101.031601

    2008 doi

  30. [38]

    Holographic Superconductors

    Hartnoll SA, Herzog CP, Horowitz GT. Holographic Superconductors. JHEP 12 (2008) 015. doi:10. 1088/1126-6708/2008/12/015. 10

    2008

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.