Abstract
Quasinormal modes describe the ringdown of compact objects deformed by small perturbations. In generic theories of gravity that extend General Relativity, the linearized dynamics of these perturbations is described by a system of coupled linear differential equations of second order. We first show, under general assumptions, that such a system can be brought to a Schrödinger-like form. We then devise an analytic approximation scheme to compute the spectrum of quasinormal modes. We validate our approach using a toy model with a controllable mixing parameter ε and showing that the analytic approximation for the fundamental mode agrees with the numerical computation when the approximation is justified. The accuracy of the analytic approximation is at the (sub-) percent level for the real part and at the level of a few percent for the imaginary part, even when ε is of order one. Our approximation scheme can be seen as an extension of the approach of Schutz and Will [1] to the case of coupled systems of equations, although our approach is not phrased in terms of a WKB analysis, and offers a new viewpoint even in the case of a single equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B.F. Schutz and C.M. Will, Black hole normal modes: a semianalytic approach, Astrophys. J. Lett. 291 (1985) L33 [INSPIRE].
E.S.C. Ching et al., Quasinormal-mode expansion for waves in open systems, Rev. Mod. Phys. 70 (1998) 1545 [gr-qc/9904017] [INSPIRE].
K.D. Kokkotas and B.G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Rel. 2 (1999) 2 [gr-qc/9909058] [INSPIRE].
E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 (2009) 163001 [arXiv:0905.2975] [INSPIRE].
C.V. Vishveshwara, Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936 [INSPIRE].
W.H. Press, Long Wave Trains of Gravitational Waves from a Vibrating Black Hole, Astrophys. J. Lett. 170 (1971) L105 [INSPIRE].
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 (1974) 443 [INSPIRE].
S. Chandrasekhar and S.L. Detweiler, The quasi-normal modes of the Schwarzschild black hole, Proc. Roy. Soc. Lond. A 344 (1975) 441 [arXiv:1975.0112] [INSPIRE].
S. Chandrasekhar, The mathematical theory of black holes, Clarendon Press, Oxford, UK (1992).
E.W. Leaver, An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A 402 (1985) 285 [arXiv:1985.0119] [INSPIRE].
R.A. Konoplya, A. Zhidenko and A.F. Zinhailo, Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations, Class. Quant. Grav. 36 (2019) 155002 [arXiv:1904.10333] [INSPIRE].
Y. Hatsuda, Quasinormal modes of black holes and Borel summation, Phys. Rev. D 101 (2020) 024008 [arXiv:1906.07232] [INSPIRE].
LIGO Scientific, Virgo collaborations, Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog, Phys. Rev. D 103 (2021) 122002 [arXiv:2010.14529] [INSPIRE].
J.L. Blázquez-Salcedo et al., Perturbed black holes in Einstein-dilaton-Gauss-Bonnet gravity: Stability, ringdown, and gravitational-wave emission, Phys. Rev. D 94 (2016) 104024 [arXiv:1609.01286] [INSPIRE].
J.L. Blázquez-Salcedo, F.S. Khoo and J. Kunz, Quasinormal modes of Einstein-Gauss-Bonnet-dilaton black holes, Phys. Rev. D 96 (2017) 064008 [arXiv:1706.03262] [INSPIRE].
L. Pierini and L. Gualtieri, Quasi-normal modes of rotating black holes in Einstein-dilaton Gauss-Bonnet gravity: the first order in rotation, Phys. Rev. D 103 (2021) 124017 [arXiv:2103.09870] [INSPIRE].
P. Wagle, N. Yunes and H.O. Silva, Quasinormal modes of slowly-rotating black holes in dynamical Chern-Simons gravity, Phys. Rev. D 105 (2022) 124003 [arXiv:2103.09913] [INSPIRE].
M. Srivastava, Y. Chen and S. Shankaranarayanan, Analytical computation of quasinormal modes of slowly rotating black holes in dynamical Chern-Simons gravity, Phys. Rev. D 104 (2021) 064034 [arXiv:2106.06209] [INSPIRE].
D. Langlois, K. Noui and H. Roussille, Linear perturbations of Einstein-Gauss-Bonnet black holes, JCAP 09 (2022) 019 [arXiv:2204.04107] [INSPIRE].
S. Endlich, V. Gorbenko, J. Huang and L. Senatore, An effective formalism for testing extensions to General Relativity with gravitational waves, JHEP 09 (2017) 122 [arXiv:1704.01590] [INSPIRE].
O.J. Tattersall, P.G. Ferreira and M. Lagos, General theories of linear gravitational perturbations to a Schwarzschild Black Hole, Phys. Rev. D 97 (2018) 044021 [arXiv:1711.01992] [INSPIRE].
G. Franciolini et al., Effective Field Theory of Black Hole Quasinormal Modes in Scalar-Tensor Theories, JHEP 02 (2019) 127 [arXiv:1810.07706] [INSPIRE].
L. Hui, A. Podo, L. Santoni and E. Trincherini, Effective Field Theory for the perturbations of a slowly rotating black hole, JHEP 12 (2021) 183 [arXiv:2111.02072] [INSPIRE].
P.A. Cano, K. Fransen and T. Hertog, Ringing of rotating black holes in higher-derivative gravity, Phys. Rev. D 102 (2020) 044047 [arXiv:2005.03671] [INSPIRE].
P.A. Cano, K. Fransen, T. Hertog and S. Maenaut, Gravitational ringing of rotating black holes in higher-derivative gravity, Phys. Rev. D 105 (2022) 024064 [arXiv:2110.11378] [INSPIRE].
S. Mukohyama and V. Yingcharoenrat, Effective field theory of black hole perturbations with timelike scalar profile: formulation, JCAP 09 (2022) 010 [arXiv:2204.00228] [INSPIRE].
S. Mukohyama, K. Takahashi and V. Yingcharoenrat, Generalized Regge-Wheeler equation from Effective Field Theory of black hole perturbations with a timelike scalar profile, JCAP 10 (2022) 050 [arXiv:2208.02943] [INSPIRE].
J. Khoury, T. Noumi, M. Trodden and S.S.C. Wong, Stability of Hairy Black Holes in Shift-Symmetric Scalar-Tensor Theories via the Effective Field Theory Approach, arXiv:2208.02823 [INSPIRE].
G. Aminov, A. Grassi and Y. Hatsuda, Black Hole Quasinormal Modes and Seiberg–Witten Theory, Annales Henri Poincare 23 (2022) 1951 [arXiv:2006.06111] [INSPIRE].
M. Bianchi, D. Consoli, A. Grillo and J.F. Morales, QNMs of branes, BHs and fuzzballs from quantum SW geometries, Phys. Lett. B 824 (2022) 136837 [arXiv:2105.04245] [INSPIRE].
G. Bonelli, C. Iossa, D.P. Lichtig and A. Tanzini, Exact solution of Kerr black hole perturbations via CFT2 and instanton counting: Greybody factor, quasinormal modes, and Love numbers, Phys. Rev. D 105 (2022) 044047 [arXiv:2105.04483] [INSPIRE].
M. Bianchi, D. Consoli, A. Grillo and J.F. Morales, More on the SW-QNM correspondence, JHEP 01 (2022) 024 [arXiv:2109.09804] [INSPIRE].
M. Giesler, M. Isi, M.A. Scheel and S. Teukolsky, Black Hole Ringdown: The Importance of Overtones, Phys. Rev. X 9 (2019) 041060 [arXiv:1903.08284] [INSPIRE].
M. Isi et al., Testing the no-hair theorem with GW150914, Phys. Rev. Lett. 123 (2019) 111102 [arXiv:1905.00869] [INSPIRE].
K. Mitman et al., Nonlinearities in Black Hole Ringdowns, Phys. Rev. Lett. 130 (2023) 081402 [arXiv:2208.07380] [INSPIRE].
M. Lagos and L. Hui, Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole, Phys. Rev. D 107 (2023) 044040 [arXiv:2208.07379] [INSPIRE].
M.H.-Y. Cheung et al., Nonlinear Effects in Black Hole Ringdown, Phys. Rev. Lett. 130 (2023) 081401 [arXiv:2208.07374] [INSPIRE].
V. Cardoso et al., Parametrized black hole quasinormal ringdown: Decoupled equations for nonrotating black holes, Phys. Rev. D 99 (2019) 104077 [arXiv:1901.01265] [INSPIRE].
R. McManus et al., Parametrized black hole quasinormal ringdown. II. Coupled equations and quadratic corrections for nonrotating black holes, Phys. Rev. D 100 (2019) 044061 [arXiv:1906.05155] [INSPIRE].
S.H. Völkel, N. Franchini and E. Barausse, Theory-agnostic reconstruction of potential and couplings from quasinormal modes, Phys. Rev. D 105 (2022) 084046 [arXiv:2202.08655] [INSPIRE].
K. Glampedakis and H.O. Silva, Eikonal quasinormal modes of black holes beyond General Relativity, Phys. Rev. D 100 (2019) 044040 [arXiv:1906.05455] [INSPIRE].
H.O. Silva and K. Glampedakis, Eikonal quasinormal modes of black holes beyond general relativity. II. Generalized scalar-tensor perturbations, Phys. Rev. D 101 (2020) 044051 [arXiv:1912.09286] [INSPIRE].
A. Bryant, H.O. Silva, K. Yagi and K. Glampedakis, Eikonal quasinormal modes of black holes beyond general relativity. III. Scalar Gauss-Bonnet gravity, Phys. Rev. D 104 (2021) 044051 [arXiv:2106.09657] [INSPIRE].
D. Langlois, K. Noui and H. Roussille, Asymptotics of linear differential systems and application to quasinormal modes of nonrotating black holes, Phys. Rev. D 104 (2021) 124043 [arXiv:2103.14744] [INSPIRE].
D. Langlois, K. Noui and H. Roussille, Black hole perturbations in modified gravity, Phys. Rev. D 104 (2021) 124044 [arXiv:2103.14750] [INSPIRE].
D. Langlois, K. Noui and H. Roussille, On the effective metric of axial black hole perturbations in DHOST gravity, JCAP 08 (2022) 040 [arXiv:2205.07746] [INSPIRE].
M.H.-Y. Cheung et al., Destabilizing the Fundamental Mode of Black Holes: The Elephant and the Flea, Phys. Rev. Lett. 128 (2022) 111103 [arXiv:2111.05415] [INSPIRE].
M. Mirbabayi, The Quasinormal Modes of Quasinormal Modes, JCAP 01 (2020) 052 [arXiv:1807.04843] [INSPIRE].
L. Hui, D. Kabat and S.S.C. Wong, Quasinormal modes, echoes and the causal structure of the Green’s function, JCAP 12 (2019) 020 [arXiv:1909.10382] [INSPIRE].
F. Strocchi, Complex Coordinates and Quantum Mechanics, Rev. Mod. Phys. 38 (1966) 36.
S. Iyer and C.M. Will, Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering, Phys. Rev. D 35 (1987) 3621 [INSPIRE].
J. Matyjasek and M. Opala, Quasinormal modes of black holes. The improved semianalytic approach, Phys. Rev. D 96 (2017) 024011 [arXiv:1704.00361] [INSPIRE].
H.-J. Blome and B. Mashhoon, Quasi-normal oscillations of a schwarzschild black hole, Phys. Lett. A 100 (1984) 231.
V. Ferrari and B. Mashhoon, Oscillations of a Black Hole, Phys. Rev. Lett. 52 (1984) 1361 [INSPIRE].
V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole, Phys. Rev. D 30 (1984) 295 [INSPIRE].
O.B. Zaslavsky, Black hole normal modes and quantum anharmonic oscillator, Phys. Rev. D 43 (1991) 605 [INSPIRE].
S. Weinberg, Eikonal Method in Magnetohydrodynamics, Physical Review 126 (1962) 1899.
J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Cambridge University Press (2020) [INSPIRE].
C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev. 184 (1969) 1231 [INSPIRE].
T. Sulejmanpasic and M. Ünsal, Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica® package, Comput. Phys. Commun. 228 (2018) 273 [arXiv:1608.08256] [INSPIRE].
R. Brito, V. Cardoso and P. Pani, Partially massless gravitons do not destroy general relativity black holes, Phys. Rev. D 87 (2013) 124024 [arXiv:1306.0908] [INSPIRE].
R. Brito, V. Cardoso and P. Pani, Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass, Phys. Rev. D 88 (2013) 023514 [arXiv:1304.6725] [INSPIRE].
R.A. Rosen and L. Santoni, Black hole perturbations of massive and partially massless spin-2 fields in (anti) de Sitter spacetime, JHEP 03 (2021) 139 [arXiv:2010.00595] [INSPIRE].
S. Deser and R.I. Nepomechie, Anomalous Propagation of Gauge Fields in Conformally Flat Spaces, Phys. Lett. B 132 (1983) 321 [INSPIRE].
S. Deser and R.I. Nepomechie, Gauge Invariance Versus Masslessness in De Sitter Space, Annals Phys. 154 (1984) 396 [INSPIRE].
A. Higuchi, Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time, Nucl. Phys. B 282 (1987) 397 [INSPIRE].
L. Bernard, C. Deffayet, K. Hinterbichler and M. von Strauss, Partially Massless Graviton on Beyond Einstein Spacetimes, Phys. Rev. D 95 (2017) 124036 [Erratum ibid. 98 (2018) 069902] [arXiv:1703.02538] [INSPIRE].
T. Kobayashi, H. Motohashi and T. Suyama, Black hole perturbation in the most general scalar-tensor theory with second-order field equations II: the even-parity sector, Phys. Rev. D 89 (2014) 084042 [arXiv:1402.6740] [INSPIRE].
G. Franciolini et al., Stable wormholes in scalar-tensor theories, JHEP 01 (2019) 221 [arXiv:1811.05481] [INSPIRE].
C.F.E. Holzhey and F. Wilczek, Black holes as elementary particles, Nucl. Phys. B 380 (1992) 447 [hep-th/9202014] [INSPIRE].
V. Ferrari, M. Pauri and F. Piazza, Quasinormal modes of charged, dilaton black holes, Phys. Rev. D 63 (2001) 064009 [gr-qc/0005125] [INSPIRE].
P. Benincasa, A. Buchel and A.O. Starinets, Sound waves in strongly coupled non-conformal gauge theory plasma, Nucl. Phys. B 733 (2006) 160 [hep-th/0507026] [INSPIRE].
D.T. Son and A.O. Starinets, Hydrodynamics of r-charged black holes, JHEP 03 (2006) 052 [hep-th/0601157] [INSPIRE].
P. Pani, Advanced Methods in Black-Hole Perturbation Theory, Int. J. Mod. Phys. A 28 (2013) 1340018 [arXiv:1305.6759] [INSPIRE].
Q.-Y. Pan and J.-L. Jing, Quasinormal modes of the Schwarzschild black hole with arbitrary spin fields: Numerical analysis, Mod. Phys. Lett. A 21 (2006) 2671 [INSPIRE].
S.H. Völkel, N. Franchini, E. Barausse and E. Berti, Constraining modifications of black hole perturbation potentials near the light ring with quasinormal modes, Phys. Rev. D 106 (2022) 124036 [arXiv:2209.10564] [INSPIRE].
B. Simon, Advanced Complex Analysis. A Comprehensive Course in Analysis, Part 2B. American Mathematical Society (2015), https://bookstore.ams.org/view?ProductCode=SIMON/2.2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2210.10788
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Hui, L., Podo, A., Santoni, L. et al. An analytic approach to quasinormal modes for coupled linear systems. J. High Energ. Phys. 2023, 60 (2023). https://doi.org/10.1007/JHEP03(2023)060
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2023)060