Abstract
Quasinormal modes and frequencies are the eigenvectors and eigenvalues of a non-Hermitian differential operator. They hold crucial significance in the physics of black holes. The analysis of quasinormal modes of black holes in asymptotically Anti-de Sitter geometries plays also a key role in the study of strongly coupled quantum many-body systems via gauge/gravity duality. In contrast to normal Sturm-Liouville operators, the spectrum of non-Hermitian (and non-normal) operators generally is unstable under small perturbations. This research focuses on the stability analysis of the spectrum of quasinormal frequencies pertaining to asymptotically planar AdS black holes, employing pseudospectrum analysis. Specifically, we concentrate on the pseudospectra of scalar and transverse gauge fields, shedding light on their relevance within the framework of gauge/gravity duality.
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References
H.-P. Nollert, Topical Review: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quant. Grav. 16 (1999) R159 [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].
N. Franchini and S.H. Völkel, Testing General Relativity with Black Hole Quasi-Normal Modes, arXiv:2305.01696 [INSPIRE].
J.L. Jaramillo, Pseudospectrum and binary black hole merger transients, Class. Quant. Grav. 39 (2022) 217002 [arXiv:2206.08025] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
O. Aharony et al., Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/gravity duality: Foundations and applications, Cambridge University Press, Cambridge (2015) [https://doi.org/10.1017/cbo9780511846373].
J. Zaanen, Y. Liu, Y.W. Sun and K. Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press (2015) [https://doi.org/10.1017/cbo9781139942492].
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic Quantum Matter, MIT Press (2018) [ISBN: 9780262038430].
G. Policastro, D.T. Son and A.O. Starinets, The shear viscosity of strongly coupled 𝑁 = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].
R. Baier et al., Relativistic viscous hydrodynamics, conformal invariance, and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].
S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
C.P. Herzog, P. Kovtun, S. Sachdev and D.T. Son, Quantum critical transport, duality, and M-theory, Phys. Rev. D 75 (2007) 085020 [hep-th/0701036] [INSPIRE].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].
D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [INSPIRE].
P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].
C.P. Herzog and S.S. Pufu, The Second Sound of SU(2), JHEP 04 (2009) 126 [arXiv:0902.0409] [INSPIRE].
I. Amado, M. Kaminski and K. Landsteiner, Hydrodynamics of Holographic Superconductors, JHEP 05 (2009) 021 [arXiv:0903.2209] [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
I. Amado, C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Hydrodynamics and beyond in the strongly coupled 𝑁 = 4 plasma, JHEP 07 (2008) 133 [arXiv:0805.2570] [INSPIRE].
B. Withers, Short-lived modes from hydrodynamic dispersion relations, JHEP 06 (2018) 059 [arXiv:1803.08058] [INSPIRE].
S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, The complex life of hydrodynamic modes, JHEP 11 (2019) 097 [arXiv:1904.12862] [INSPIRE].
S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, Convergence of the Gradient Expansion in Hydrodynamics, Phys. Rev. Lett. 122 (2019) 251601 [arXiv:1904.01018] [INSPIRE].
H.-P. Nollert, About the significance of quasinormal modes of black holes, Phys. Rev. D 53 (1996) 4397 [gr-qc/9602032] [INSPIRE].
H.-P. Nollert and R.H. Price, Quantifying excitations of quasinormal mode systems, J. Math. Phys. 40 (1999) 980 [gr-qc/9810074] [INSPIRE].
J.M. Aguirregabiria and C.V. Vishveshwara, Scattering by black holes: A simulated potential approach, Phys. Lett. A 210 (1996) 251 [INSPIRE].
J.L. Jaramillo, R. Panosso Macedo and L. Al Sheikh, Pseudospectrum and Black Hole Quasinormal Mode Instability, Phys. Rev. X 11 (2021) 031003 [arXiv:2004.06434] [INSPIRE].
K. Destounis et al., Pseudospectrum of Reissner-Nordström black holes: Quasinormal mode instability and universality, Phys. Rev. D 104 (2021) 084091 [arXiv:2107.09673] [INSPIRE].
S. Sarkar, M. Rahman and S. Chakraborty, Perturbing the perturbed: Stability of quasinormal modes in presence of a positive cosmological constant, Phys. Rev. D 108 (2023) 104002 [arXiv:2304.06829] [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].
E. Berti et al., Stability of the fundamental quasinormal mode in time-domain observations against small perturbations, Phys. Rev. D 106 (2022) 084011 [arXiv:2205.08547] [INSPIRE].
R.A. Konoplya and A. Zhidenko, First few overtones probe the event horizon geometry, arXiv:2209.00679 [INSPIRE].
A. Courty, K. Destounis and P. Pani, Spectral instability of quasinormal modes and strong cosmic censorship, Phys. Rev. D 108 (2023) 104027 [arXiv:2307.11155] [INSPIRE].
R.G. Daghigh, M.D. Green and J.C. Morey, Significance of Black Hole Quasinormal Modes: A Closer Look, Phys. Rev. D 101 (2020) 104009 [arXiv:2002.07251] [INSPIRE].
W.-L. Qian et al., Asymptotical quasinormal mode spectrum for piecewise approximate effective potential, Phys. Rev. D 103 (2021) 024019 [arXiv:2009.11627] [INSPIRE].
L. Al Sheikh, Scattering resonances and Pseudospectrum: stability and completeness aspects in optical and gravitational systems, Ph.D. thesis, Université Bourgogne Franche-Comté, CEDEX, France (2022).
K. Destounis and F. Duque, Black-hole spectroscopy: quasinormal modes, ringdown stability and the pseudospectrum, arXiv:2308.16227 [INSPIRE].
T. Torres, From Black Hole Spectral Instability to Stable Observables, Phys. Rev. Lett. 131 (2023) 111401 [arXiv:2304.10252] [INSPIRE].
V. Boyanov et al., Pseudospectrum of horizonless compact objects: A bootstrap instability mechanism, Phys. Rev. D 107 (2023) 064012 [arXiv:2209.12950] [INSPIRE].
L.N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton (2005) [https://doi.org/10.1515/9780691213101].
J. Sjöstrand, Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Birkhäuser Cham (2019) [https://doi.org/10.1007/978-3-030-10819-9].
E.B. Davies, Linear Operators and their Spectra, Cambridge Studies in Advanced Mathematics, Cambridge University Press (2007) [https://doi.org/10.1017/CBO9780511618864].
M. Isi et al., Testing the no-hair theorem with GW150914, Phys. Rev. Lett. 123 (2019) 111102 [arXiv:1905.00869] [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].
C.D. Capano et al., Multimode Quasinormal Spectrum from a Perturbed Black Hole, Phys. Rev. Lett. 131 (2023) 221402 [arXiv:2105.05238] [INSPIRE].
C.D. Capano et al., Statistical validation of the detection of a sub-dominant quasi-normal mode in GW190521, arXiv:2209.00640 [INSPIRE].
C.M. Warnick, On quasinormal modes of asymptotically anti-de Sitter black holes, Commun. Math. Phys. 333 (2015) 959 [arXiv:1306.5760] [INSPIRE].
L.N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, U.S.A. (2000) [https://doi.org/10.1137/1.9780898719598].
J.P. Boyd, Chebyshev and Fourier spectral methods, Dover Publications Inc. (2000) [ISBN: 9780486411835].
T. Kato, Perturbation theory for linear operators, Springer Science & Business Media (2013) [https://doi.org/10.1007/978-3-662-12678-3].
D.H. Richard Courant, Methods of Mathematical Physics. Volume 1, Wiley-VCH (1989) [https://doi.org/10.1002/9783527617210].
R.A. Horn and C.R. Johnson, Matrix analysis, Cambridge University Press (2013) [https://doi.org/10.1017/cbo9780511810817].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
F. Ficek and C. Warnick, Quasinormal modes of Reissner-Nordström-AdS: the approach to extremality, arXiv:2308.16035 [INSPIRE].
B.G. Schmidt, On relativistic stellar oscillations, [INSPIRE].
A. Zenginoglu, A geometric framework for black hole perturbations, Phys. Rev. D 83 (2011) 127502 [arXiv:1102.2451] [INSPIRE].
M. Ansorg and R. Panosso Macedo, Spectral decomposition of black-hole perturbations on hyperboloidal slices, Phys. Rev. D 93 (2016) 124016 [arXiv:1604.02261] [INSPIRE].
R. Panosso Macedo, J.L. Jaramillo and M. Ansorg, Hyperboloidal slicing approach to quasi-normal mode expansions: the Reissner-Nordström case, Phys. Rev. D 98 (2018) 124005 [arXiv:1809.02837] [INSPIRE].
P. Bizoń, T. Chmaj and P. Mach, A toy model of hyperboloidal approach to quasinormal modes, Acta Phys. Polon. B 51 (2020) 1007 [arXiv:2002.01770] [INSPIRE].
R. Panosso Macedo, Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics, arXiv:2307.15735 [INSPIRE].
E. Gasperin and J.L. Jaramillo, Energy scales and black hole pseudospectra: the structural role of the scalar product, Class. Quant. Grav. 39 (2022) 115010 [arXiv:2107.12865] [INSPIRE].
S.C. Reddy, P.J. Schmid and D.S. Henningson, Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math. 53 (1993) 15.
L.N. Trefethen, A.E. Trefethen, S.C. Reddy and T.A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993) 578.
I. Amado et al., Holographic Superfluids and the Landau Criterion, JHEP 02 (2014) 063 [arXiv:1307.8100] [INSPIRE].
B. Goutéraux, E. Mefford and F. Sottovia, Critical superflows and thermodynamic instabilities in superfluids, Phys. Rev. D 108 (2023) L081903 [arXiv:2212.10410] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
I. Amado, C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Residues of correlators in the strongly coupled 𝑁 = 4 plasma, Phys. Rev. D 77 (2008) 065004 [arXiv:0710.4458] [INSPIRE].
S. Grozdanov, K. Schalm and V. Scopelliti, Black hole scrambling from hydrodynamics, Phys. Rev. Lett. 120 (2018) 231601 [arXiv:1710.00921] [INSPIRE].
M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10 (2018) 127 [arXiv:1801.00010] [INSPIRE].
M. Blake, R.A. Davison, S. Grozdanov and H. Liu, Many-body chaos and energy dynamics in holography, JHEP 10 (2018) 035 [arXiv:1809.01169] [INSPIRE].
A. Biggs and J. Maldacena, Scaling similarities and quasinormal modes of D0 black hole solutions, JHEP 11 (2023) 155 [arXiv:2303.09974] [INSPIRE].
J. Maldacena, A simple quantum system that describes a black hole, arXiv:2303.11534 [INSPIRE].
J.L. Jaramillo, R. Panosso Macedo and L.A. Sheikh, Gravitational Wave Signatures of Black Hole Quasinormal Mode Instability, Phys. Rev. Lett. 128 (2022) 211102 [arXiv:2105.03451] [INSPIRE].
I. Amado, C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Absorption lengths in the holographic plasma, JHEP 09 (2007) 057 [arXiv:0706.2750] [INSPIRE].
K. Landsteiner, The Sound of Strongly Coupled Field Theories: Quasinormal Modes In AdS, AIP Conf. Proc. 1458 (2012) 174 [arXiv:1202.3550] [INSPIRE].
L. Gavassino, M.M. Disconzi and J. Noronha, Dispersion relations alone cannot guarantee causality, arXiv:2307.05987 [INSPIRE].
Acknowledgments
We thank J.L. Jaramillo and V. Boyanov for valuable discussions, and the anonymous referee for the constructive comments which led to the inclusion of section 3.3. The work of D.A and K.L. is supported through the grants CEX2020-001007-S and PID2021-123017NB-100, PID2021-127726NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”. The work of D.G.F. is supported by JAEIntroICU-2022-IFT-02.
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Areán, D., Fariña, D.G. & Landsteiner, K. Pseudospectra of holographic quasinormal modes. J. High Energ. Phys. 2023, 187 (2023). https://doi.org/10.1007/JHEP12(2023)187
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DOI: https://doi.org/10.1007/JHEP12(2023)187