Pith. sign in

REVIEW 3 major objections 4 minor 56 references

Quasinormal modes in Kerr spacetime as a 2D Eigenvalue problem

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The Kerr quasinormal-mode spectrum for each azimuthal number $m$ can be computed as a direct eigenvalue problem on a two-dimensional hyperboloidal slice, eliminating root-finding and initial guesses.

desk verdict The 2D eigenvalue formulation is a real step forward, but validation is self-referential and needs an independent benchmark before the method is fully trusted. read the letter →

arxiv 2506.04326 v1 pith:AA7A52RK submitted 2025-06-04 gr-qc hep-th

Jamil Assaad , Rodrigo Panosso Macedo This is my paper
classification gr-qchep-th MSC 83C5783C3565N25 PACS 04.70.-s04.30.-w
keywords quasinormalmodesKerrblackholesTeukolskyequationhyperboloidalcompactificationeigenvalueproblemm-modedecompositionholespectroscopyspheroidalharmonics
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

The paper claims that the quasinormal-mode problem for a Kerr black hole, normally solved by treating the radial and angular equations separately with a coupling separation constant and an initial guess for every mode, can be recast as one genuine eigenvalue problem for each azimuthal number $m$. The key move is to keep the Teukolsky equation as a two-dimensional elliptic PDE on a hyperboloidal slice, so the outgoing and ingoing boundary conditions are built into the coordinates rather than imposed by hand. If that works, an entire tower of overtones and angular harmonics for a fixed $m$ is obtained from a single matrix diagonalization, with no seed values. The paper also uses the complete spectrum to propose a cleaner single-index overtone label $q$, and to show that steep near-horizon gradients found by earlier radial hyperboloidal solvers are coordinate artifacts rather than physical features of extremal Kerr. This matters because quick, assumption-free spectrum construction is the natural input to black hole spectroscopy and to studies of mode stability.

What carries the argument

The load-bearing object is the hyperboloidal $m$-mode Teukolsky operator: after the coordinate transformation (7), the regularity factor (15), and the azimuthal Fourier decomposition (16), the master equation becomes the 2D elliptic PDE (17), written as $s\bar D_{m;\bar\omega}\bar\Phi_{m;\bar\omega}=0$ with spatial operators $L_1$ and $L_2$. Introducing the auxiliary function $\bar\Upsilon_{m;\bar\omega}=s\bar\Phi_{m;\bar\omega}$ converts it into the linear eigenvalue problem $Lu=s u$ of Eq. (20). Discretizing $u$ on a Chebyshev tensor grid renders the spectrum as eigenvalues of a finite matrix, with boundary conditions at null infinity and the horizon enforced geometrically by the hyperboloidal coordinates and by the regularity prefactors in Eq. (15).

What would settle it

Run the solver at several resolutions for a fixed $m$ and compare every filtered eigenvalue against independent high-accuracy continued-fraction values across many spins and overtones: if any retained eigenvalue fails to match a known mode, or any known mode is missing at tolerance $10^{-3}$, the central claim is refuted. A complementary check is to substitute a computed eigenpair back into the original Teukolsky equation and verify that the residual and the boundary behaviour decay at the expected rates.

Watch

Extended reading notes

Core claim

The central discovery is that separating variables is unnecessary for computing Kerr quasinormal modes. Combining the hyperboloidal compactification with a decomposition into azimuthal modes $m$ turns the frequency-domain Teukolsky equation into a two-dimensional linear operator whose eigenvalues are the QNM frequencies, with the auxiliary function $s\bar\Upsilon_{m;\bar\omega}=s\,s\bar\Phi_{m;\bar\omega}$ making the problem first order in the spectral parameter. The paper demonstrates the resulting spectra for gravitational perturbations $s=-2$ across spins from Schwarzschild to $a/M=0.99$, reproduces known values, and shows that both hyperboloidal gauges converge exponentially to the same frequencies. It then exploits the directly available eigenfunctions to show that the near-extremal gradients are gauge-dependent, and to project the angular profile onto both spheroidal and spherical harmonic bases. In the authors' reading, the QNM problem is not two coupled ODE eigenvalue problems but a single 2D spectral problem for each $m$.

Load-bearing premise

The load-bearing premise is that the eigenvalues of the discretized hyperboloidal operator that survive the tolerance filter are the actual quasinormal modes of the Teukolsky equation with the correct boundary conditions, rather than numerical artifacts of the discretization.

Editorial extensions

If this is right

  • For a fixed $m$, one matrix eigenvalue problem returns many QNMs at once, so constructing the Kerr spectrum no longer needs a root finder with a good initial guess.
  • The single overtone index $q$, ordered by decay rate, replaces the four-way labels and exposes the prograde/retrograde ordering directly.
  • Radial-fixing and Cauchy-horizon-fixing gauges give the same frequencies with the same exponential convergence, so the gauge choice does not affect spectral accuracy.
  • Steep near-horizon gradients in extremal-Kerr eigenfunctions are slicing artifacts, not physical features, resolving a question raised by earlier radial computations.
  • QNM eigenfunctions can be projected onto either spin-weighted spheroidal or spherical harmonics, easing direct comparison with gravitational-wave templates.

Reading between the lines

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

  • Beyond the paper, the same no-seed eigenvalue setup should make pseudospectrum and QNM-instability analyses in Kerr substantially cheaper, because the full eigensystem is available in one solve rather than mode by mode.
  • The $q$-index ordering by decay rate could serve as a standard for ringdown comparisons, removing the ambiguity in which one overtone label $n$ covers four distinct modes.
  • Because near-horizon gradient formation is gauge-dependent, any physical claim tied to eigenfunction steepness near extremality, such as an instability of the horizon, should be checked in a gauge-invariant quantity rather than in a fixed slicing.
  • A direct extension is to Kerr-Newman or other non-separable backgrounds, where the same 2D diagonalization could replace the Newton-Raphson root searches currently used.
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 / 4 minor

Summary. The paper develops a numerical method for computing quasinormal modes (QNMs) of the Kerr spacetime by formulating the frequency-domain Teukolsky equation as a two-dimensional eigenvalue problem. The authors combine a hyperboloidal compactification with an azimuthal m-mode decomposition, obtaining an operator whose eigenvalues directly yield QNM frequencies without a separation constant or root-finding. They present the discretized operator, a filtering procedure to remove spurious eigenvalues, convergence tests for two hyperboloidal gauges, a proposed re-labelling of overtones by a single index q, and studies of the angular and radial structure of the eigenfunctions. The central advertised results are that the method extracts the Kerr QNM spectrum directly as eigenvalues, that both gauges perform comparably, and that strong near-horizon gradients seen in one gauge are coordinate artefacts.

Significance. If the central claim is correct, the paper provides a practical and conceptually clean route to Kerr QNM spectra: for each azimuthal number m, all overtones and angular harmonics are obtained simultaneously as eigenvalues of a single discretized operator, with no need for initial guesses. This would be a useful tool for QNM stability studies, pseudospectrum computations, and mode-excitation problems, and the paper's demonstration of equivalent performance between two hyperboloidal gauges is a valuable practical message. The m-mode projection onto both spheroidal and spherical harmonic bases is also potentially useful for gravitational-wave data analysis. However, the paper's validation is entirely self-referential: the convergence study compares the solver to its own high-resolution results, and the only independent anchor is a qualitative Schwarzschild plot. Because the correctness of the extracted spectrum is the load-bearing claim, the lack of a direct quantitative comparison with established Kerr QNM values leaves the central result not fully established.

major comments (3)
  1. [Sec. 4.2, Eq. (59)] The convergence error ϵ in Eq. (59) is computed relative to ω_Ref obtained from the same solver at N_ref = 55. This is a self-consistency check and cannot detect a systematic error in the boundary-condition encoding or in the filtering procedure. The only independent anchor is the qualitative Schwarzschild benchmark in Fig. 2, which is not quantified. Since the paper's central claim is that the filtered eigenvalues are the physical Kerr QNM spectrum, please add a quantitative comparison against independent published values—for example Leaver's continued-fraction results or the high-accuracy data of Cook and Zalutskiy—for representative modes, for spins including a/M = 0.8, 0.9, and 0.99, and for both prograde and retrograde branches.
  2. [Sec. 4, filtering criterion with TOL ≈ 1e-3] The filtering procedure retains eigenvalues satisfying |1 − ω_low/ω_high| < TOL, but the paper reports no sensitivity analysis with respect to TOL or the truncation pair (n_low, n_high), nor a demonstration that spurious eigenvalues near the branch cut are always discarded. The concern is most acute precisely where the paper itself notes slow convergence, namely retrograde modes as a/M approaches 1 and spurious eigenvalues cluster near the branch cut. Without a robustness study, the filter could in principle admit or discard the wrong eigenvalues. Please quantify the dependence of the retained spectrum on TOL and resolution, and cross-check the filtered values against independent Kerr QNM data.
  3. [Sec. 4.3.1 and Fig. 6] The conclusion that the near-horizon gradients in the radial-fixing gauge are coordinate artefacts is inferred from comparing eigenfunctions in the radial-fixing and Cauchy-horizon-fixing gauges. These gauges use different radial coordinate choices (ρ_o = 0 versus ρ_o = κ²), so the same physical field is represented by different functions of σ; the absence of steep gradients in one gauge does not by itself prove that the gradients are unphysical. Please compare an invariant quantity, such as the original Teukolsky master function or a suitably normalized covariant quantity, or explicitly justify why the comparison of the two gauges is conclusive.
minor comments (4)
  1. [Sec. 2.1.1, before Eq. (21)] The text states "For the radial fixing gauge (σc = κ2)", but Eq. (8) with ρ_o = 0 gives σ_c = κ^{-2}, and the factor (1 − κ²σ) in Eq. (22) is consistent with σ_c = κ^{-2}; please correct this typo.
  2. [Throughout] There are several typographical errors that should be corrected, including "respecvelty", "Teulkolsky", "straightfoward", "the the", and "discretized operador".
  3. [Fig. 4 caption] The horizontal axis is described as the "total size" of the discretized operator; please clarify that it is the total number of grid points n_total = n1 × n2, since the truncation is parameterized by N1 = 5N2 = N.
  4. [Sec. 4.1, Eq. (56)] The proposed ordering by decay rate is natural, but the statement that even q corresponds to prograde and odd q to retrograde modes requires the caveat, already partly given, that for m = 0 the q = 0 and q = 1 modes are degenerate in frequency; please make this caveat explicit in the definition of the ordering.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is a direct spectral reformulation of the Teukolsky equation; no circular reduction.

full rationale

The paper's derivation chain is explicit and algebraic: the Teukolsky equation (Eq. 14) is transformed via hyperboloidal coordinates (Eq. 7), the master function is regularized (Eq. 15), an m-mode Fourier ansatz is introduced (Eq. 16), and the resulting 2D operator (Eq. 20) is discretized with Chebyshev collocation (Eqs. 35-50). The QNM frequencies are obtained as eigenvalues of this discretized operator, not as fitted parameters or as quantities defined in terms of the target spectrum. The hyperboloidal framework is cited from prior work, including [27] and [41], but the relevant coordinate transformations and operator coefficients are stated in full, so the argument does not reduce to an unverified self-citation or an imported uniqueness theorem. The main caveat is that the convergence study in Sec. 4.2 defines the reference values omega_Ref using the same solver at Nref = 55, so the reported errors measure self-convergence rather than agreement with an independent calculation; the Schwarzschild benchmark (Fig. 2) is also only qualitative. These are accuracy and robustness limitations, not circularity, because the eigenvalues themselves are computed independently of any externally imposed values. The tolerance-based filtering (TOL ~ 1e-3) selects eigenvalues that are stable under resolution changes, but it does not construct or define the eigenvalues. No load-bearing step in the derivation is equivalent to its own input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted physical parameters or new entities. Its central numerical results are eigenvalues of a discretized operator. The main assumptions are standard domain assumptions about the Teukolsky equation and the hyperboloidal framework, plus an ad hoc numerical filter for identifying physical modes.

free parameters (1)
  • Filter tolerance TOL = 10^-3
    Chosen by hand to separate physical QNMs from spurious eigenvalues. It affects which eigenvalues are retained but not the converged QNM values themselves.
assumptions (4)
  • domain assumption The Teukolsky equation (Eq. 14) correctly describes linear perturbations of the Kerr spacetime.
    This is the standard master equation of black hole perturbation theory, used throughout the paper as the starting point.
  • domain assumption The hyperboloidal coordinate transformation (Eq. 7) and the minimal gauge conditions from Ref. [41] yield a regular PDE whose eigenvalues correspond to QNMs with the correct boundary conditions.
    The transformation maps infinity and the horizon to finite coordinate surfaces; this is assumed to enforce outgoing/ingoing behavior automatically, as established in prior work.
  • domain assumption The regularity factors (1+x)^{delta1/2}(1-x)^{delta2/2} in Eq. (15) are necessary and sufficient for regular eigenfunctions on the axis.
    This factor follows from standard results (Refs. [27,41]) and is adopted without re-derivation.
  • ad hoc to paper The convergence filter with TOL ~ 1e-3 correctly distinguishes physical QNMs from spurious eigenvalues of the discretized operator.
    The paper introduces this filter (Sec. 4) to discard non-converging eigenvalues; its validity is assumed based on the observed convergence behavior.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quasinormal modes in Kerr spacetime as a 2D Eigenvalue problem." pith.science (2026). https://pith.science/paper/AA7A52RK

@misc{pith2026250604326,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes in Kerr spacetime as a 2D Eigenvalue problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AA7A52RK}},
  note         = {Machine review of arXiv:2506.04326}
}
abstract

We revisit the computation of quasinormal modes (QNMs) of the Kerr black hole using a numerical approach exploiting a representation of the Teukolsky equation as a $2D$ elliptic partial differential equation. By combining the hyperboloidal framework with a $m$-mode decomposition, we recast the QNM problem into a genuine eigenvalue problem for each azimuthal mode. This formulation enables the simultaneous extraction of multiple QNMs, traditionally labelled by overtone number $n$ and angular index $\ell$, without requiring prior assumptions about their structure. We advocate for a simplified notation in which each overtone is uniquely labelled by a single index $q$, thereby avoiding the conventional but artificial distinction between regular and mirror modes. We compare two distinct hyperboloidal gauges-radial fixing and Cauchy horizon fixing-and demonstrate that, despite their different geometric properties and behaviour in the extremal limit, they yield numerical values for the QNM spectra with comparable accuracy and exponential convergence. Moreover, we show that strong gradients observed near the horizon in the extremal Kerr regime are coordinate artefacts of specific slicing rather than physical features. Finally, we investigate the angular structure of the QNM eigenfunctions and show that the $m$-mode approach allows flexible projection onto both spin-weighted spheroidal and spherical harmonic bases. These results underscore the robustness and versatility of the hyperboloidal $m$-mode method as a foundation for future studies of QNM stability, pseudospectra, and mode excitation in gravitational wave astronomy.

Figures

Figures reproduced from arXiv: 2506.04326 by the authors.

Figure 1
Figure 1. Carter-Penrose diagram for the Kerr spacetime, illustrating hyperboloidal slices in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. QNM spectrum for spin s = −2 in the sector m = 2, obtained using the hyperboloidal m-mode eigenvalue solver. Results for black hole spins ranging from a/M = 0 (black dots) to a/M = 0.99 (red squares). This method enables the simultaneous extraction of multiple QNMs—including higher harmonics and overtones—without the need for fine￾tuned initial seeds in root-finding algorithms or prior assumptions about angular stru… view at source ↗
Figure 3
Figure 3. Representation of the QNM modes with notations [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Relative error ϵq,ℓ,m as a function of discretized operador Lˆ’s total size, cf. (48). Convergence tests performed in the m = 2-sector for both the prograde and retrograde modes, and different individual eigenvalues. Both choices of hyperboloidal foliation — radial fix…
Figure 5
Figure 5. Figure 5: Eigenfunctions associated with the mode (q, ℓ, m) = (0, 2, 2) computed using the radial fixing gauge. On the left panel the real part of the function is represented while on the right panel the imaginary part is. The hyperboloidal framework allows for a regular represe…
Figure 6
Figure 6. Figure 6: Real part of the eigenfunctions associated with the mode [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Angular structure of QNM eigenfunctions sΦ¯ q,ℓ,m for (q, ℓ, m) = (0, 2, 2) evaluated at future null infinity (σ = 0). Color code used for the legend is given on figure Fig.5. Left panel: The mild variation with respect to the angular coordinate x ∈ [−1, 1] is a conseq…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

56 extracted references · 54 canonical work pages

  1. [1]

    Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations

    Teukolsky Saul A. Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett., 29:1114–1118, 1972

    work page 1972

  2. [2]

    Perturbations of a rotating Black hole

    Teukolsky Saul A. Perturbations of a rotating Black hole. I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations. The American Astronomical Society , 185:635:647, 1973

    work page 1973

  3. [3]

    Press and Saul A

    William H. Press and Saul A. Teukolsky. Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric. Astrophys. J., 185:649–674, 1973

    work page 1973

  4. [4]

    S. A. Teukolsky and W. H. Press. Perturbations of a rotating black hole. III - Interaction of the hole with gravitational and electromagnet ic radiation. Astrophys. J., 193:443–461, 1974

    work page 1974

  5. [5]

    The mathematical theory of black holes

    Subrahmanyan Chandrasekhar. The mathematical theory of black holes . Oxford University Press, 1985

    work page 1985

  6. [6]

    Bernard F. Whiting. Mode Stability of the Kerr Black Hole. J. Math. Phys., 30:1301, 1989

    work page 1989

  7. [7]

    Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case |a| ≪M

    Mihalis Dafermos, Gustav Holzegel, and Igor Rodnianski. Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case |a| ≪M. 11 2017

    work page 2017

  8. [8]

    Stability for linearized gravity on the Kerr spacetime

    Lars Andersson, Thomas B ¨ackdahl, Pieter Blue, and Siyuan Ma. Stability for linearized gravity on the Kerr spacetime. 3 2019

    work page 2019

Show all 56 references
  1. [9]

    Black Hole Perturbation Theory and Gravitational Self-F orce , pages 1–119

    Adam Pound and Barry Wardell. Black Hole Perturbation Theory and Gravitational Self-F orce , pages 1–119. Springer Singapore, Singapore, 2020

    2020

  2. [10]

    Waveform Modelling for the Laser Interferometer Space Antenna

    Niayesh Afshordi et al. Waveform Modelling for the Laser Interferometer Space Antenna. 11 2023

    2023

  3. [11]

    Black hole spectroscopy: from theory to experiment

    Emanuele Berti et al. Black hole spectroscopy: from theory to experiment. 5 2025

    2025

  4. [12]

    Quasinormal modes of stars and black holes

    Schmidt B.G Kokkotas, K.D. Quasinormal modes of stars and black holes. Living Rev. Rel., 1999

    1999

  5. [13]

    H.P. Nollert. Topical review: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars. Class. Quant. Grav., 1999

    1999

  6. [14]

    Starinets E

    A.O. Starinets E. Berti, V . Cardoso. Quasinormal modes of black holes and black branes. Class. Quant. Grav., 26:163001, 2009

    2009

  7. [15]

    Zhidenko R.A

    A. Zhidenko R.A. Konoplya. Quasinormal modes of black holes : From astrophysics to string theory. Reviews of Modern Physics, 2011

    2011

  8. [16]

    Kelly, Badri Krishnan, Lee Samuel Finn, David Garrison, and Ramon Lopez- Aleman

    Olaf Dreyer, Bernard J. Kelly, Badri Krishnan, Lee Samuel Finn, David Garrison, and Ramon Lopez- Aleman. Black hole spectroscopy: Testing general relativity through gravitational wave observations. Class. Quant. Grav., 21:787–804, 2004

    2004

  9. [17]

    Clifford M. Will E. Berti, V . Cardoso. On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA. arXiv preprint arXiv:gr-qc/0512160, 2005

    2005 arXiv

  10. [18]

    Rico K. L. Lo, Leart Sabani, and Vitor Cardoso. Quasinormal modes and excitation factors of Kerr black holes. 3 2025

    2025

  11. [19]

    Marcilhacy

    G. Marcilhacy. On the Teukolsky Equation. Lettere al Nuovo Cimento, 37(8):300, 1983

    1983

  12. [20]

    Blandin, R

    J. Blandin, R. Pons, and G. Marcilhacy. General Solution of Teukolsky’s Equation. Lettere al Nuovo Cimento, 38(17):561, 1983

    1983

  13. [21]

    Hortacsu

    M. Hortacsu. Heun Functions and Some of Their Applications in Physics. pages 23–39, 2012

    2012

  14. [22]

    The confluent Heun functions in black hole perturbation theory: a spacetime interpretation

    Marica Minucci and Rodrigo Panosso Macedo. The confluent Heun functions in black hole perturbation theory: a spacetime interpretation. Gen. Rel. Grav., 57(2):33, 2025

    2025

  15. [23]

    E.W. Leaver. An analytic representation for the quasi-normal modes of kerr black holes. proc. R. Soc. , 1985

    1985

  16. [24]

    Quasinormal modes of Schwarzschild black holes: The determination of quasinormal frequencies with very large imaginary parts

    Hiemensans-Peter Nollert. Quasinormal modes of Schwarzschild black holes: The determination of quasinormal frequencies with very large imaginary parts. Phys. Rev. D, 47:5253–5258, 1993. Quasinormal modes in Kerr spacetime as a 2D Eigenvalue problem 24

    1993

  17. [25]

    Cook and Maxim Zalutskiy

    Gregory B. Cook and Maxim Zalutskiy. Gravitational perturbations of the Kerr geometry: High-accuracy study. Phys. Rev. D, 90(12):124021, 2014

    2014

  18. [26]

    Hyperboloidal foliations and scri-fixing

    Anil Zenginoglu. Hyperboloidal foliations and scri-fixing. Class. Quant. Grav., 25:145002, 2008

    2008

  19. [27]

    A Geometric framework for black hole perturbations

    Anil Zenginoglu. A Geometric framework for black hole perturbations. Phys. Rev., D83:127502, 2011

    2011

  20. [28]

    Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics

    Rodrigo Panosso Macedo. Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics. Phil. Trans. Roy. Soc. Lond. A , 382:20230046, 2024

    2024

  21. [29]

    Hyperboloidal Approach to Quasinormal Modes

    Rodrigo Panosso Macedo and Anil Zenginoglu. Hyperboloidal Approach to Quasinormal Modes. Front. Phys., 12:1497601, 2025

    2025

  22. [30]

    Pseudospectrum and black hole quasinormal mode instability

    Jos ´e Luis Jaramillo, Rodrigo Panosso Macedo, and Lamis Al Sheikh. Pseudospectrum and black hole quasinormal mode instability. Physical Review X, 11(3):031003, 2021

    2021

  23. [31]

    Pseudospectrum and binary black hole merger transients

    Jos ´e Luis Jaramillo. Pseudospectrum and binary black hole merger transients. Class. Quant. Grav. , 39(21):217002, 2022

    2022

  24. [32]

    Numerical investigation of the late-time Kerr tails

    Istvan Racz and Gabor Zsolt Toth. Numerical investigation of the late-time Kerr tails. Class. Quant. Grav., 28:195003, 2011

    2011

  25. [33]

    Hyperboloidal slices for the wave equation of Kerr-Schild metrics and numerical applications

    Michael Jasiulek. Hyperboloidal slices for the wave equation of Kerr-Schild metrics and numerical applications. Class. Quant. Grav., 29:015008, 2012

    2012

  26. [34]

    A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime

    Enno Harms, Sebastiano Bernuzzi, Alessandro Nagar, and An Zenginoglu. A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime. Class. Quant. Grav. , 31(24):245004, 2014

    2014

  27. [35]

    Axisymmetric fully spectral code for hyperbolic equations

    Rodrigo Panosso Macedo and Marcus Ansorg. Axisymmetric fully spectral code for hyperbolic equations . Computational Physics, 276:357–379, 2014

    2014

  28. [36]

    Numerical investigation of the dynamics of linear spin s fields on a Kerr background: Late-time tails of spin s = ±1, ±2 fields

    K ´aroly Csuk´as, Istv´an R´acz, and G´abor Zsolt T´oth. Numerical investigation of the dynamics of linear spin s fields on a Kerr background: Late-time tails of spin s = ±1, ±2 fields. Phys. Rev. D, 100(10):104025, 2019

    2019

  29. [37]

    Numerical investigation of the dynamics of linear spin s fields on a Kerr background II: Superradiant scattering

    K ´aroly Csuk´as and Istv´an R´acz. Numerical investigation of the dynamics of linear spin s fields on a Kerr background II: Superradiant scattering. Phys. Rev. D, 103(8):084035, 2021

    2021

  30. [38]

    Symmetric integration of the 1+1 Teukolsky equation on hyperboloidal foliations of Kerr spacetimes

    Charalampos Markakis, Sean Bray, and Anıl Zengino ˘glu. Symmetric integration of the 1+1 Teukolsky equation on hyperboloidal foliations of Kerr spacetimes. 3 2023

    2023

  31. [39]

    Ansorg D

    M. Ansorg D. Schinkel, R. Panosso Macedo. Initial data for perturbed kerr black holes on hyperboloidal slices. Class. Quantum Grav, 2014

    2014

  32. [40]

    Ansorg D

    M. Ansorg D. Schinkel, R. Panosso Macedo. Axisymmetric constant mean curvature slices in the kerr spacetime. Class. Quantum Grav, 2014

    2014

  33. [41]

    Hyperboloidal framework for the Kerr spacetime

    Rodrigo Panosso Macedo. Hyperboloidal framework for the Kerr spacetime. Class. Quant. Grav. , 37(6):065019, 2020

    2020

  34. [42]

    Justin L. Ripley. Computing the quasinormal modes and eigenfunctions for the Teukolsky equation using horizon penetrating, hyperboloidally compactified coordinates. Class. Quant. Grav. , 39(14):145009, 2022

    2022

  35. [43]

    Ripley, Alejandro C ´ardenas-Avenda˜no, and Frans Pretorius

    Hengrui Zhu, Justin L. Ripley, Alejandro C ´ardenas-Avenda˜no, and Frans Pretorius. Challenges in quasinormal mode extraction: Perspectives from numerical solutions to the Teukolsky equation. Phys. Rev. D, 109(4):044010, 2024

    2024

  36. [44]

    Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds

    Stefanos Aretakis. Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds. J. Funct. Anal., 263:2770–2831, 2012

    2012

  37. [45]

    Horizon Instability of Extremal Black Holes

    Stefanos Aretakis. Horizon Instability of Extremal Black Holes. Adv. Theor . Math. Phys., 19:507–530, 2015

    2015

  38. [46]

    Oscar J. C. Dias, Mahdi Godazgar, and Jorge E. Santos. Linear Mode Stability of the Kerr-Newman Black Hole and Its Quasinormal Modes. Phys. Rev. Lett., 114(15):151101, 2015

    2015

  39. [47]

    Rodrigo Panosso Macedo, Patrick Bourg, Adam Pound, and Samuel D. Upton. Multidomain spectral method for self-force calculations. Phys. Rev. D, 110(8):084008, 2024

    2024

  40. [48]

    The pseudospectrum for the Kerr black hole: spin s = 0case

    Rong-Gen Cai, Li-Ming Cao, Jia-Ning Chen, Zong-Kuan Guo, Liang-Bi Wu, and Yu-Sen Zhou. The pseudospectrum for the Kerr black hole: spin s = 0case. 1 2025. Quasinormal modes in Kerr spacetime as a 2D Eigenvalue problem 25

    2025

  41. [49]

    In preparation

    Edgar Gasperin, Justin Feng, and Rodrigo Panosso Macedo. In preparation. 2025

    2025

  42. [50]

    Hyperboloidal slicing approach to quasi-normal mode expansions: the Reissner-Nordstr¨om case

    Rodrigo Panosso Macedo, Jos ´e Luis Jaramillo, and Marcus Ansorg. Hyperboloidal slicing approach to quasi-normal mode expansions: the Reissner-Nordstr¨om case. Phys. Rev. D, 98(12):124005, 2018

    2018

  43. [51]

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

    J ´er´emy Besson and Jos ´e Luis Jaramillo. Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach. 12 2024

    2024

  44. [52]

    Pseudospectrum of rotating analog black holes

    Lucas Tobias de Paula, Pedro Henrique Croti Siqueira, Rodrigo Panosso Macedo, and Maur ´ıcio Richartz. Pseudospectrum of rotating analog black holes. Phys. Rev. D, 111(10):104064, 2025

    2025

  45. [53]

    (bhptoolkit.org)

    Black Hole Perturbation Toolkit. (bhptoolkit.org)

  46. [54]

    High precision ringdown modeling: Multimode fits and BMS frames

    Lorena Maga ˜na Zertuche et al. High precision ringdown modeling: Multimode fits and BMS frames. Phys. Rev. D, 105(10):104015, 2022

    2022

  47. [55]

    Personal website: Ringdown data, 2025

    Emanuele Berti. Personal website: Ringdown data, 2025. Accessed: 31 March 2025

    2025

  48. [56]

    Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime

    Patrick Bourg, Rodrigo Panosso Macedo, Andrew Spiers, Benjamin Leather, Bonga B ´eatrice, and Adam Pound. Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime. 3 2025

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.