- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Origin of Quasinormal Modes in Semi-Open Systems

Leonardo Solidoro1,2,*, Sam Patrick3, Silke Weinfurtner1,2,4, and Ruth Gregory3,5

  • *Contact author: leonardo.solidoro@nottingham.ac.uk
PDF

Phys. Rev. Lett. 135, 051401 – Published 28 July, 2025

DOI: https://doi.org/10.1103/7kjp-vrml

Abstract

Astrophysical black holes are open systems which, when perturbed, radiate quasinormal modes (QNMs) to infinity. By contrast, laboratory analogs are necessarily finite sized, presenting a potential obstacle to exciting QNMs in experiments. We explore how the QNM spectrum of a toy-model black hole changes when enclosed by a partially reflecting wall with adjustable reflectivity. Our results reveal a continuous connection between the QNM spectra of open and finite-sized systems. Additionally, we demonstrate that QNMs in this setup are easily excited by incoherent background noise. This Letter opens new avenues for studying QNMs of black holes and compact objects in laboratory settings, where finite-size effects and noise are unavoidable.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (66)

  1. B. P. Abbott et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  2. V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole, Phys. Rev. D 30, 295 (1984).
  3. F. Echeverria, Gravitational-wave measurements of the mass and angular momentum of a black hole, Phys. Rev. D 40, 3194 (1989).
  4. K. D. Kokkotas and B. G. Schmidt, Quasi-normal modes of stars and black holes, Living Rev. Relativity 2, 1 (1999).
  5. V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Geodesic stability, Lyapunov exponents, and quasinormal modes, Phys. Rev. D 79, 064016 (2009).
  6. E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  7. R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83, 793 (2011).
  8. P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender, E. Berti, P. Binetruy, M. Born, D. Bortoluzzi et al., Laser interferometer space antenna, arXiv:1702.00786.
  9. M. Maggiore, C. Van Den Broeck, N. Bartolo, E. Belgacem, D. Bertacca, M. A. Bizouard, M. Branchesi, S. Clesse, S. Foffa, J. García-Bellido et al., Science case for the Einstein telescope, J. Cosmol. Astropart. Phys. 03 (2020) 050.
  10. E. Cannizzaro, T. F. Spieksma, V. Cardoso, and T. Ikeda, The impact of plasma on the relaxation of black holes, Phys. Rev. D 110, L021302 (2024).
  11. M. H.-Y. Cheung, V. Baibhav, E. Berti, V. Cardoso, G. Carullo, R. Cotesta, W. Del Pozzo, F. Duque, T. Helfer, E. Shukla, and K. W. K. Wong, Nonlinear effects in black hole ringdown, Phys. Rev. Lett. 130, 081401 (2023).
  12. H.-P. Nollert, About the significance of quasinormal modes of black holes, Phys. Rev. D 53, 4397 (1996).
  13. J. Aguirregabiria and C. Vishveshwara, Scattering by black holes: A simulated potential approach, Phys. Lett. A 210, 251 (1996).
  14. H.-P. Nollert and R. H. Price, Quantifying excitations of quasinormal mode systems, J. Math. Phys. (N.Y.) 40, 980 (1999).
  15. Mark Ho-Yeuk Cheung, K. Destounis, R. P. Macedo, E. Berti, and V. Cardoso, Destabilizing the fundamental mode of black holes: The elephant and the flea, Phys. Rev. Lett. 128, 111103 (2022).
  16. H. Witek, V. Cardoso, A. Ishibashi, and U. Sperhake, Superradiant instabilities in astrophysical systems, Phys. Rev. D 87, 043513 (2013).
  17. Y. Huang and H. Zhang, Quasibound states of charged dilatonic black holes, Phys. Rev. D 103, 044062 (2021).
  18. J. L. Jaramillo, R. P. Macedo, and L. A. Sheikh, Pseudospectrum and black hole quasinormal mode instability, Phys. Rev. X 11, 031003 (2021).
  19. J. L. Jaramillo, R. P. Macedo, and L. A. Sheikh, Gravitational wave signatures of black hole quasinormal mode instability, Phys. Rev. Lett. 128, 211102 (2022).
  20. E. Maggio, P. Pani, and V. Ferrari, Exotic compact objects and how to quench their ergoregion instability, Phys. Rev. D 96, 104047 (2017).
  21. A. Testa and P. Pani, Analytical template for gravitational-wave echoes: Signal characterization and prospects of detection with current and future interferometers, Phys. Rev. D 98, 044018 (2018).
  22. E. Maggio, V. Cardoso, S. R. Dolan, and P. Pani, Ergoregion instability of exotic compact objects: Electromagnetic and gravitational perturbations and the role of absorption, Phys. Rev. D 99, 064007 (2019).
  23. E. Maggio, L. Buoninfante, A. Mazumdar, and P. Pani, How does a dark compact object ringdown?, Phys. Rev. D 102, 064053 (2020).
  24. S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Measurement of stimulated Hawking emission in an analogue system, Phys. Rev. Lett. 106, 021302 (2011).
  25. L.-P. Euvé, F. Michel, R. Parentani, T. G. Philbin, and G. Rousseaux, Observation of noise correlated by the Hawking effect in a water tank, Phys. Rev. Lett. 117, 121301 (2016).
  26. J. R. Muñoz de Nova, K. Golubkov, V. I. Kolobov, and J. Steinhauer, Observation of thermal hawking radiation and its temperature in an analogue black hole, Nature (London) 569, 688 (2019).
  27. M. Visser and S. Weinfurtner, Vortex analogue for the equatorial geometry of the Kerr black hole, Classical Quantum Gravity 22, 2493 (2005).
  28. T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford, and S. Weinfurtner, Rotational superradiant scattering in a vortex flow, Nat. Phys. 13, 833 (2017).
  29. D. D. Solnyshkov, C. Leblanc, S. V. Koniakhin, O. Bleu, and G. Malpuech, Quantum analogue of a Kerr black hole and the Penrose effect in a Bose-Einstein condensate, Phys. Rev. B 99, 214511 (2019).
  30. M. C. Braidotti, R. Prizia, C. Maitland, F. Marino, A. Prain, I. Starshynov, N. Westerberg, E. M. Wright, and D. Faccio, Measurement of Penrose superradiance in a photon superfluid, Phys. Rev. Lett. 128, 013901 (2022).
  31. M. Cromb, G. M. Gibson, E. Toninelli, M. J. Padgett, E. M. Wright, and D. Faccio, Amplification of waves from a rotating body, Nat. Phys. 16, 1069 (2020).
  32. M. J. Jacquet, L. Giacomelli, Q. Valnais, M. Joly, F. Claude, E. Giacobino, Q. Glorieux, I. Carusotto, and A. Bramati, Quantum vacuum excitation of a quasinormal mode in an analog model of black hole spacetime, Phys. Rev. Lett. 130, 111501 (2023).
  33. C. Burgess, S. Patrick, T. Torres, R. Gregory, and F. König, Quasinormal modes of optical solitons, Phys. Rev. Lett. 132, 053802 (2024).
  34. S. Patrick, A. Coutant, M. Richartz, and S. Weinfurtner, Black hole quasibound states from a draining bathtub vortex flow, Phys. Rev. Lett. 121, 061101 (2018).
  35. T. Torres, S. Patrick, M. Richartz, and S. Weinfurtner, Quasinormal mode oscillations in an analogue black hole experiment, Phys. Rev. Lett. 125, 011301 (2020).
  36. P. Švančara, P. Smaniotto, L. Solidoro, J. F. MacDonald, S. Patrick, R. Gregory, C. F. Barenghi, and S. Weinfurtner, Rotating curved spacetime signatures from a giant quantum vortex, Nature (London) 628, 66 (2024).
  37. V. I. Kolobov, K. Golubkov, J. R. Muñoz de Nova, and J. Steinhauer, Observation of stationary spontaneous Hawking radiation and the time evolution of an analogue black hole, Nat. Phys. 17, 362 (2021).
  38. A. Delhom, K. Guerrero, P. Calizaya Cabrera, K. Falque, A. Bramati, A. J. Brady, M. J. Jacquet, and I. Agullo, Entanglement from superradiance and rotating quantum fluids of light, Phys. Rev. D 109, 105024 (2024).
  39. T. Torres, S. Patrick, and R. Gregory, Imperfect draining vortex as analog extreme compact object, Phys. Rev. D 106, 045026 (2022).

  40. Note that our choice of ϵ as real and positive corresponds to a Neumann boundary condition in the case of a complete reflection, whereas choosing ϵ as negative (complex) would lead to a Dirichlet (Robin) boundary condition. In general, ϵ could depend on ω, but for the sake of simplicity, we model it here as a constant.

  41. C. F. B. Macedo, T. Stratton, S. Dolan, and L. C. B. Crispino, Spectral lines of extreme compact objects, Phys. Rev. D 98, 104034 (2018).
  42. S. Dolan and T. Torres, Spectral (in)stability of black holes under potential perturbations (to be published).
  43. J. J. Florentin, M. Abramowitz, and I. A. Stegun, Handbook of mathematical functions, Am. Math. Mon. 73, 1143 (1966).
  44. See Supplemental Material at http://link.aps.org/supplemental/10.1103/7kjp-vrml for details on the Pöschl-Teller reflection coefficient (Appendix A) and the numerical simulations (Appendix B).
  45. E. W. Leaver, An analytic representation for the quasi-normal modes of Kerr black holes, Proc. R. Soc. A 402, 285 (1985).
  46. C. V. Vishveshwara, Scattering of gravitational radiation by a Schwarzschild black-hole, Nature (London) 227, 936 (1970).
  47. P. Smaniotto, L. Solidoro, P. Švančara, S. Patrick, M. Richartz, C. F. Barenghi, R. Gregory, and S. Weinfurtner, Black-hole spectroscopy from a giant quantum vortex, arXiv:2502.11209.
  48. P. T. Leung, S. Y. Liu, and K. Young, Completeness and orthogonality of quasinormal modes in leaky optical cavities, Phys. Rev. A 49, 3057 (1994).
  49. G. T. Horowitz and V. E. Hubeny, Quasinormal modes of ads black holes and the approach to thermal equilibrium, Phys. Rev. D 62, 024027 (2000).
  50. D. T. Son and A. O. Starinets, Viscosity, black holes, and quantum field theory, Annu. Rev. Nucl. Part. Sci. 57, 95 (2007).
  51. B. J. Dalton, S. M. Barnett, and B. M. Garraway, Theory of pseudomodes in quantum optical processes, Phys. Rev. A 64, 053813 (2001).
  52. P. T. Kristensen, R. C. Ge, and S. Hughes, Normalization of quasinormal modes in leaky optical cavities and plasmonic resonators, Phys. Rev. A 92, 053810 (2015).
  53. S. Hod, Bohr’s correspondence principle and the area spectrum of quantum black holes, Phys. Rev. Lett. 81, 4293 (1998).
  54. M. Maggiore, Physical interpretation of the spectrum of black hole quasinormal modes, Phys. Rev. Lett. 100, 141301 (2008).
  55. S. Patrick, A. Gupta, R. Gregory, and C. F. Barenghi, Stability of quantized vortices in two-component condensates, Phys. Rev. Res. 5, 033201 (2023).
  56. S. Lan, X. Li, J. Mo, Y. Tian, Y. Yan, P. Yang, and H. Zhang, Splitting of doubly quantized vortices in holographic superfluid of finite temperature, J. High Energy Phys. 05 (2023) 223.
  57. K. D. Kokkotas, Pulsating relativistic stars, arXiv:gr-qc/9603024.
  58. E. Barausse, V. Cardoso, and P. Pani, Can environmental effects spoil precision gravitational-wave astrophysics?, Phys. Rev. D 89, 104059 (2014).
  59. P. S. Cole, G. Bertone, A. Coogan, D. Gaggero, T. Karydas, B. J. Kavanagh, T. F. M. Spieksma, and G. M. Tomaselli, Distinguishing environmental effects on binary black hole gravitational waveforms, Nat. Astron. 7, 943 (2023).
  60. O. A. Hannuksela, K. W. K. Wong, R. Brito, E. Berti, and T. G. F. Li, Probing the existence of ultralight bosons with a single gravitational-wave measurement, Nat. Astron. 3, 447 (2019).
  61. D. Baumann, G. Bertone, J. Stout, and G. M. Tomaselli, Sharp signals of boson clouds in black hole binary inspirals, Phys. Rev. Lett. 128, 221102 (2022).
  62. L. London, D. Shoemaker, and J. Healy, Modeling ringdown: Beyond the fundamental quasinormal modes, Phys. Rev. D 90, 124032 (2014).
  63. K. Mitman, M. Lagos, L. C. Stein, S. Ma, L. Hui, Y. Chen, N. Deppe, F. Hébert, L. E. Kidder, J. Moxon et al., Nonlinearities in black hole ringdowns, Phys. Rev. Lett. 130, 081402 (2023).
  64. H. Yoshino and H. Kodama, Bosenova collapse of axion cloud around a rotating black hole, Prog. Theor. Phys. 128, 153 (2012).
  65. E. Cannizzaro, F. Corelli, and P. Pani, Nonlinear photon-plasma interaction and the black hole superradiant instability, Phys. Rev. D 109, 023007 (2024).
  66. M. Baryakhtar, M. Galanis, R. Lasenby, and O. Simon, Black hole superradiance of self-interacting scalar fields, Phys. Rev. D 103, 095019 (2021).

Outline

Information

Phys. Rev. Lett. 135, 051401– Published 28 July, 2025
  • Received 6 October 2024
  • Revised 5 June 2025
  • Accepted 7 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation