Abstract
We study the waveforms of time signals produced by scalar perturbations in static hairy black holes, in which the perturbations can be governed by a double-peak effective potential. The inner potential peak would give rise to echoes, which provide a powerful tool to test the Kerr hypothesis. The waveforms are constructed in the time and frequency domains, and we find that the late-time waveforms are determined by the long-lived and sub-long-lived quasinormal modes, which are trapped in the potential valley and near the smaller peak, respectively. When the distance between the peaks is significantly larger than the width of the peaks, a train of decaying echo pulses is produced by the superposition of the long-lived and sub-long-lived modes. In certain cases, the echoes can vanish and then reappear. When the peaks are close enough, one detects far fewer echo signals and a following sinusoid tail, which is controlled by the long-lived or sub-long-lived mode and hence decays very slowly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett. 875 (2019) L1 [arXiv:1906.11238] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. II. Array and Instrumentation, Astrophys. J. Lett. 875 (2019) L2 [arXiv:1906.11239] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. III. Data Processing and Calibration, Astrophys. J. Lett. 875 (2019) L3 [arXiv:1906.11240] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole, Astrophys. J. Lett. 875 (2019) L4 [arXiv:1906.11241] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring, Astrophys. J. Lett. 875 (2019) L5 [arXiv:1906.11242] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole, Astrophys. J. Lett. 875 (2019) L6 [arXiv:1906.11243] [INSPIRE].
H.-P. Nollert, TOPICAL REVIEW: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quant. Grav. 16 (1999) R159 [INSPIRE].
E. Berti, V. Cardoso, J.A. Gonzalez and U. Sperhake, Mining information from binary black hole mergers: A comparison of estimation methods for complex exponentials in noise, Phys. Rev. D 75 (2007) 124017 [gr-qc/0701086] [INSPIRE].
V. Cardoso, E. Franzin and P. Pani, Is the gravitational-wave ringdown a probe of the event horizon?, Phys. Rev. Lett. 116 (2016) 171101 [Erratum ibid. 117 (2016) 089902] [arXiv:1602.07309] [INSPIRE].
R.H. Price and G. Khanna, Gravitational wave sources: reflections and echoes, Class. Quant. Grav. 34 (2017) 225005 [arXiv:1702.04833] [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].
J.P.S. Lemos and O.B. Zaslavskii, Black hole mimickers: Regular versus singular behavior, Phys. Rev. D 78 (2008) 024040 [arXiv:0806.0845] [INSPIRE].
P.V.P. Cunha, J.A. Font, C. Herdeiro, E. Radu, N. Sanchis-Gual and M. Zilhão, Lensing and dynamics of ultracompact bosonic stars, Phys. Rev. D 96 (2017) 104040 [arXiv:1709.06118] [INSPIRE].
P.V.P. Cunha and C.A.R. Herdeiro, Shadows and strong gravitational lensing: a brief review, Gen. Rel. Grav. 50 (2018) 42 [arXiv:1801.00860] [INSPIRE].
R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, A novel gravitational lensing feature by wormholes, Phys. Lett. B 789 (2019) 270 [Erratum ibid. 791 (2019) 422] [arXiv:1811.08245] [INSPIRE].
D.-C. Dai and D. Stojkovic, Observing a Wormhole, Phys. Rev. D 100 (2019) 083513 [arXiv:1910.00429] [INSPIRE].
H. Huang and J. Yang, Charged Ellis Wormhole and Black Bounce, Phys. Rev. D 100 (2019) 124063 [arXiv:1909.04603] [INSPIRE].
J.H. Simonetti, M.J. Kavic, D. Minic, D. Stojkovic and D.-C. Dai, Sensitive searches for wormholes, Phys. Rev. D 104 (2021) L081502 [arXiv:2007.12184] [INSPIRE].
M. Wielgus, J. Horak, F. Vincent and M. Abramowicz, Reflection-asymmetric wormholes and their double shadows, Phys. Rev. D 102 (2020) 084044 [arXiv:2008.10130] [INSPIRE].
J. Yang and H. Huang, Trapping horizons of the evolving charged wormhole and black bounce, Phys. Rev. D 104 (2021) 084005 [arXiv:2104.11134] [INSPIRE].
C. Bambi and D. Stojkovic, Astrophysical Wormholes, Universe 7 (2021) 136 [arXiv:2105.00881] [INSPIRE].
J. Peng, M. Guo and X.-H. Feng, Observational signature and additional photon rings of an asymmetric thin-shell wormhole, Phys. Rev. D 104 (2021) 124010 [arXiv:2102.05488] [INSPIRE].
P. Bueno, P.A. Cano, F. Goelen, T. Hertog and B. Vercnocke, Echoes of Kerr-like wormholes, Phys. Rev. D 97 (2018) 024040 [arXiv:1711.00391] [INSPIRE].
R.A. Konoplya, Z. Stuchlík and A. Zhidenko, Echoes of compact objects: new physics near the surface and matter at a distance, Phys. Rev. D 99 (2019) 024007 [arXiv:1810.01295] [INSPIRE].
Y.-T. Wang, J. Zhang and Y.-S. Piao, Primordial gravastar from inflation, Phys. Lett. B 795 (2019) 314 [arXiv:1810.04885] [INSPIRE].
Y.-T. Wang, Z.-P. Li, J. Zhang, S.-Y. Zhou and Y.-S. Piao, Are gravitational wave ringdown echoes always equal-interval?, Eur. Phys. J. C 78 (2018) 482 [arXiv:1802.02003] [INSPIRE].
V. Cardoso and P. Pani, Testing the nature of dark compact objects: a status report, Living Rev. Rel. 22 (2019) 4 [arXiv:1904.05363] [INSPIRE].
J.T. Gálvez Ghersi, A.V. Frolov and D.A. Dobre, Echoes from the scattering of wavepackets on wormholes, Class. Quant. Grav. 36 (2019) 135006 [arXiv:1901.06625] [INSPIRE].
H. Liu, P. Liu, Y. Liu, B. Wang and J.-P. Wu, Echoes from phantom wormholes, Phys. Rev. D 103 (2021) 024006 [arXiv:2007.09078] [INSPIRE].
M.-Y. Ou, M.-Y. Lai and H. Huang, Echoes from asymmetric wormholes and black bounce, Eur. Phys. J. C 82 (2022) 452 [arXiv:2111.13890] [INSPIRE].
J. Abedi, H. Dykaar and N. Afshordi, Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons, Phys. Rev. D 96 (2017) 082004 [arXiv:1612.00266] [INSPIRE].
J. Abedi, H. Dykaar and N. Afshordi, Echoes from the Abyss: The Holiday Edition!, arXiv:1701.03485 [INSPIRE].
R. Dey, S. Biswas and S. Chakraborty, Ergoregion instability and echoes for braneworld black holes: Scalar, electromagnetic, and gravitational perturbations, Phys. Rev. D 103 (2021) 084019 [arXiv:2010.07966] [INSPIRE].
Y. Yang, D. Liu, Z. Xu, Y. Xing, S. Wu and Z.-W. Long, Echoes of novel black-bounce spacetimes, Phys. Rev. D 104 (2021) 104021 [arXiv:2107.06554] [INSPIRE].
S.H. Völkel and K.D. Kokkotas, Ultra Compact Stars: Reconstructing the Perturbation Potential, Class. Quant. Grav. 34 (2017) 175015 [arXiv:1704.07517] [INSPIRE].
Z. Mark, A. Zimmerman, S.M. Du and Y. Chen, A recipe for echoes from exotic compact objects, Phys. Rev. D 96 (2017) 084002 [arXiv:1706.06155] [INSPIRE].
Q. Wang, N. Oshita and N. Afshordi, Echoes from Quantum Black Holes, Phys. Rev. D 101 (2020) 024031 [arXiv:1905.00446] [INSPIRE].
K. Saraswat and N. Afshordi, Quantum Nature of Black Holes: Fast Scrambling versus Echoes, JHEP 04 (2020) 136 [arXiv:1906.02653] [INSPIRE].
R. Dey, S. Chakraborty and N. Afshordi, Echoes from braneworld black holes, Phys. Rev. D 101 (2020) 104014 [arXiv:2001.01301] [INSPIRE].
N. Oshita, D. Tsuna and N. Afshordi, Quantum Black Hole Seismology I: Echoes, Ergospheres, and Spectra, Phys. Rev. D 102 (2020) 024045 [arXiv:2001.11642] [INSPIRE].
S. Chakraborty, E. Maggio, A. Mazumdar and P. Pani, Implications of the quantum nature of the black hole horizon on the gravitational-wave ringdown, arXiv:2202.09111 [INSPIRE].
H. Liu, W.-L. Qian, Y. Liu, J.-P. Wu, B. Wang and R.-H. Yue, Alternative mechanism for black hole echoes, Phys. Rev. D 104 (2021) 044012 [arXiv:2104.11912] [INSPIRE].
K. Chakravarti, R. Ghosh and S. Sarkar, Signature of nonuniform area quantization on black hole echoes, Phys. Rev. D 105 (2022) 044046 [arXiv:2112.10109] [INSPIRE].
G. D’Amico and N. Kaloper, Black hole echoes, Phys. Rev. D 102 (2020) 044001 [arXiv:1912.05584] [INSPIRE].
H.-S. Liu, Z.-F. Mai, Y.-Z. Li and H. Lü, Quasi-topological Electromagnetism: Dark Energy, Dyonic Black Holes, Stable Photon Spheres and Hidden Electromagnetic Duality, Sci. China Phys. Mech. Astron. 63 (2020) 240411 [arXiv:1907.10876] [INSPIRE].
H. Huang, M.-Y. Ou, M.-Y. Lai and H. Lü, Echoes from classical black holes, Phys. Rev. D 105 (2022) 104049 [arXiv:2112.14780] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
J. Zhang and S.-Y. Zhou, Can the graviton have a large mass near black holes?, Phys. Rev. D 97 (2018) 081501 [arXiv:1709.07503] [INSPIRE].
R. Dong and D. Stojkovic, Gravitational wave echoes from black holes in massive gravity, Phys. Rev. D 103 (2021) 024058 [arXiv:2011.04032] [INSPIRE].
C.A.R. Herdeiro, E. Radu, N. Sanchis-Gual and J.A. Font, Spontaneous Scalarization of Charged Black Holes, Phys. Rev. Lett. 121 (2018) 101102 [arXiv:1806.05190] [INSPIRE].
R.A. Konoplya and A. Zhidenko, Analytical representation for metrics of scalarized Einstein-Maxwell black holes and their shadows, Phys. Rev. D 100 (2019) 044015 [arXiv:1907.05551] [INSPIRE].
P. Wang, H. Wu and H. Yang, Scalarized Einstein-Born-Infeld black holes, Phys. Rev. D 103 (2021) 104012 [arXiv:2012.01066] [INSPIRE].
G. Guo, P. Wang, H. Wu and H. Yang, Scalarized Einstein-Maxwell-scalar black holes in anti-de Sitter spacetime, Eur. Phys. J. C 81 (2021) 864 [arXiv:2102.04015] [INSPIRE].
G. Guo, P. Wang, H. Wu and H. Yang, Thermodynamics and phase structure of an Einstein-Maxwell-scalar model in extended phase space, Phys. Rev. D 105 (2022) 064069 [arXiv:2107.04467] [INSPIRE].
W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776 [INSPIRE].
B. Carter, Axisymmetric Black Hole Has Only Two Degrees of Freedom, Phys. Rev. Lett. 26 (1971) 331 [INSPIRE].
R. Ruffini and J.A. Wheeler, Introducing the black hole, Phys. Today 24 (1971) 30 [INSPIRE].
P.G.S. Fernandes, C.A.R. Herdeiro, A.M. Pombo, E. Radu and N. Sanchis-Gual, Spontaneous Scalarisation of Charged Black Holes: Coupling Dependence and Dynamical Features, Class. Quant. Grav. 36 (2019) 134002 [Erratum ibid. 37 (2020) 049501] [arXiv:1902.05079] [INSPIRE].
P.G.S. Fernandes, C.A.R. Herdeiro, A.M. Pombo, E. Radu and N. Sanchis-Gual, Charged black holes with axionic-type couplings: Classes of solutions and dynamical scalarization, Phys. Rev. D 100 (2019) 084045 [arXiv:1908.00037] [INSPIRE].
J.L. Blázquez-Salcedo, C.A.R. Herdeiro, J. Kunz, A.M. Pombo and E. Radu, Einstein-Maxwell-scalar black holes: the hot, the cold and the bald, Phys. Lett. B 806 (2020) 135493 [arXiv:2002.00963] [INSPIRE].
D.-C. Zou and Y.S. Myung, Scalarized charged black holes with scalar mass term, Phys. Rev. D 100 (2019) 124055 [arXiv:1909.11859] [INSPIRE].
P.G.S. Fernandes, Einstein-Maxwell-scalar black holes with massive and self-interacting scalar hair, Phys. Dark Univ. 30 (2020) 100716 [arXiv:2003.01045] [INSPIRE].
Y. Peng, Scalarization of horizonless reflecting stars: neutral scalar fields non-minimally coupled to Maxwell fields, Phys. Lett. B 804 (2020) 135372 [arXiv:1912.11989] [INSPIRE].
Y.S. Myung and D.-C. Zou, Instability of Reissner-Nordström black hole in Einstein-Maxwell-scalar theory, Eur. Phys. J. C 79 (2019) 273 [arXiv:1808.02609] [INSPIRE].
Y.S. Myung and D.-C. Zou, Stability of scalarized charged black holes in the Einstein-Maxwell-Scalar theory, Eur. Phys. J. C 79 (2019) 641 [arXiv:1904.09864] [INSPIRE].
D.-C. Zou and Y.S. Myung, Radial perturbations of the scalarized black holes in Einstein-Maxwell-conformally coupled scalar theory, Phys. Rev. D 102 (2020) 064011 [arXiv:2005.06677] [INSPIRE].
Y.S. Myung and D.-C. Zou, Onset of rotating scalarized black holes in Einstein-Chern-Simons-Scalar theory, Phys. Lett. B 814 (2021) 136081 [arXiv:2012.02375] [INSPIRE].
Z.-F. Mai and R.-Q. Yang, Stability analysis of a charged black hole with a nonlinear complex scalar field, Phys. Rev. D 104 (2021) 044008 [arXiv:2101.00026] [INSPIRE].
D. Astefanesei, C. Herdeiro, J. Oliveira and E. Radu, Higher dimensional black hole scalarization, JHEP 09 (2020) 186 [arXiv:2007.04153] [INSPIRE].
Y.S. Myung and D.-C. Zou, Quasinormal modes of scalarized black holes in the Einstein-Maxwell-Scalar theory, Phys. Lett. B 790 (2019) 400 [arXiv:1812.03604] [INSPIRE].
J. Luis Blázquez-Salcedo, C.A.R. Herdeiro, S. Kahlen, J. Kunz, A.M. Pombo and E. Radu, Quasinormal modes of hot, cold and bald Einstein-Maxwell-scalar black holes, Eur. Phys. J. C 81 (2021) 155 [arXiv:2008.11744] [INSPIRE].
Y.S. Myung and D.-C. Zou, Scalarized charged black holes in the Einstein-Maxwell-Scalar theory with two U(1) fields, Phys. Lett. B 811 (2020) 135905 [arXiv:2009.05193] [INSPIRE].
Y.S. Myung and D.-C. Zou, Scalarized black holes in the Einstein-Maxwell-scalar theory with a quasitopological term, Phys. Rev. D 103 (2021) 024010 [arXiv:2011.09665] [INSPIRE].
H. Guo, X.-M. Kuang, E. Papantonopoulos and B. Wang, Horizon curvature and spacetime structure influences on black hole scalarization, Eur. Phys. J. C 81 (2021) 842 [arXiv:2012.11844] [INSPIRE].
Y. Brihaye, B. Hartmann, N.P. Aprile and J. Urrestilla, Scalarization of asymptotically anti-de Sitter black holes with applications to holographic phase transitions, Phys. Rev. D 101 (2020) 124016 [arXiv:1911.01950] [INSPIRE].
Y. Brihaye, C. Herdeiro and E. Radu, Black Hole Spontaneous Scalarisation with a Positive Cosmological Constant, Phys. Lett. B 802 (2020) 135269 [arXiv:1910.05286] [INSPIRE].
C.-Y. Zhang, P. Liu, Y. Liu, C. Niu and B. Wang, Dynamical charged black hole spontaneous scalarization in anti-de Sitter spacetimes, Phys. Rev. D 104 (2021) 084089 [arXiv:2103.13599] [INSPIRE].
C. de Rham and J. Zhang, Perturbations of stealth black holes in degenerate higher-order scalar-tensor theories, Phys. Rev. D 100 (2019) 124023 [arXiv:1907.00699] [INSPIRE].
Y. Xie, J. Zhang, H.O. Silva, C. de Rham, H. Witek and N. Yunes, Square Peg in a Circular Hole: Choosing the Right Ansatz for Isolated Black Holes in Generic Gravitational Theories, Phys. Rev. Lett. 126 (2021) 241104 [arXiv:2103.03925] [INSPIRE].
Q. Gan, P. Wang, H. Wu and H. Yang, Photon spheres and spherical accretion image of a hairy black hole, Phys. Rev. D 104 (2021) 024003 [arXiv:2104.08703] [INSPIRE].
Q. Gan, P. Wang, H. Wu and H. Yang, Photon ring and observational appearance of a hairy black hole, Phys. Rev. D 104 (2021) 044049 [arXiv:2105.11770] [INSPIRE].
G. Guo, P. Wang, H. Wu and H. Yang, Quasinormal Modes of Black Holes with Multiple Photon Spheres, arXiv:2112.14133 [INSPIRE].
N. Andersson, Evolving test fields in a black hole geometry, Phys. Rev. D 55 (1997) 468 [gr-qc/9607064] [INSPIRE].
V. Cardoso, S. Hopper, C.F.B. Macedo, C. Palenzuela and P. Pani, Gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale, Phys. Rev. D 94 (2016) 084031 [arXiv:1608.08637] [INSPIRE].
V. Cardoso and P. Pani, The observational evidence for horizons: from echoes to precision gravitational-wave physics, arXiv:1707.03021 [INSPIRE].
N. Sago, S. Isoyama and H. Nakano, Fundamental Tone and Overtones of Quasinormal Modes in Ringdown Gravitational Waves: A Detailed Study in Black Hole Perturbation, Universe 7 (2021) 357 [arXiv:2108.13017] [INSPIRE].
R.H. Price, Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields, Phys. Rev. D 5 (1972) 2439 [INSPIRE].
E.S.C. Ching, P.T. Leung, W.M. Suen and K. Young, Late time tail of wave propagation on curved space-time, Phys. Rev. Lett. 74 (1995) 2414 [gr-qc/9410044] [INSPIRE].
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: 2204.00982
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
Guo, G., Wang, P., Wu, H. et al. Echoes from hairy black holes. J. High Energ. Phys. 2022, 73 (2022). https://doi.org/10.1007/JHEP06(2022)073
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2022)073