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Effective field theory of black hole echoes


Gravitational wave ‘echoes’ during black-hole merging events have been advocated as possible signals of modifications to gravity in the strong-field (but semiclassical) regime. In these proposals the observable effect comes entirely from the appearance of nonzero reflection probability at the horizon, which vanishes for a standard black hole. We show how to apply EFT reasoning to these arguments, using and extending earlier work for localized systems that relates choices of boundary condition to the action for the physics responsible for these boundary conditions. EFT reasoning applied to this action argues that linear ‘Robin’ boundary conditions dominate at low energies, and we determine the relationship between the corresponding effective coupling (whose value is the one relevant low-energy prediction of particular modifications to General Relativity for these systems) and the phenomenologically measurable near-horizon reflection coefficient. Because this connection involves only near-horizon physics it is comparatively simple to establish, and we do so for perturbations in both the Schwarzschild geometry (which is the one most often studied theoretically) and the Kerr geometry (which is the one of observational interest for post-merger ring down). In passing we identify the renormalization-group evolution of the effective couplings as a function of a regularization distance from the horizon, that enforces how physics does not depend on the precise position where the boundary conditions are imposed. We show that the perfect-absorber/perfect-emitter boundary conditions of General Relativity correspond to the only fixed points of this evolution. Nontrivial running of all other RG evolution reflects how modifications to gravity necessarily introduce new physics near the horizon.

A preprint version of the article is available at ArXiv.


  1. [1]

    Virgo and LIGO Scientific collaborations, B.P. Abbott et al., GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116 (2016) 241103 [arXiv:1606.04855] [INSPIRE].

  2. [2]

    Virgo and LIGO Scientific collaborations, B.P. Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].

  3. [3]

    VIRGO and LIGO Scientific collaborations, B.P. Abbott et al., GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett. 118 (2017) 221101 [arXiv:1706.01812] [INSPIRE].

  4. [4]

    Virgo and LIGO Scientific collaborations, B.P. Abbott et al., GW170608: Observation of a 19-solar-mass Binary Black Hole Coalescence, Astrophys. J. 851 (2017) L35 [arXiv:1711.05578] [INSPIRE].

  5. [5]

    T. Damour and J.H. Taylor, Strong field tests of relativistic gravity and binary pulsars, Phys. Rev. D 45 (1992) 1840 [INSPIRE].

  6. [6]

    I.H. Stairs, Testing general relativity with pulsar timing, Living Rev. Rel. 6 (2003) 5 [astro-ph/0307536] [INSPIRE].

  7. [7]

    M. Krämer et al., Tests of general relativity from timing the double pulsar, Science 314 (2006) 97 [astro-ph/0609417] [INSPIRE].

  8. [8]

    M. Krämer and N. Wex, The double pulsar system: A unique laboratory for gravity, Class. Quant. Grav. 26 (2009) 073001 [INSPIRE].

  9. [9]

    C.M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Rel. 17 (2014) 4 [arXiv:1403.7377] [INSPIRE].

  10. [10]

    E. Berti et al., Testing General Relativity with Present and Future Astrophysical Observations, Class. Quant. Grav. 32 (2015) 243001 [arXiv:1501.07274] [INSPIRE].

  11. [11]

    K. Yagi and L.C. Stein, Black Hole Based Tests of General Relativity, Class. Quant. Grav. 33 (2016) 054001 [arXiv:1602.02413] [INSPIRE].

  12. [12]

    J.T. Jebsen, Über die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum (On the General Spherically Symmetric Solutions of Einstein’s Gravitational Equations in Vacuo), Ark. Mat. Astron. Fys. 15 (1921) 19.

  13. [13]

    G.D. Birkhoff, Relativity and Modern Physics, Harvard University Press, Cambridge, U.S.A, (1923).

  14. [14]

    W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776 [INSPIRE].

  15. [15]

    W. Israel, Event horizons in static electrovac space-times, Commun. Math. Phys. 8 (1968) 245 [INSPIRE].

  16. [16]

    B. Carter, Axisymmetric Black Hole Has Only Two Degrees of Freedom, Phys. Rev. Lett. 26 (1971) 331 [INSPIRE].

  17. [17]

    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].

  18. [18]

    L. Susskind, The Transfer of Entanglement: The Case for Firewalls, arXiv:1210.2098 [INSPIRE].

  19. [19]

    O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].

  20. [20]

    O. Lunin and S.D. Mathur, Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, Phys. Rev. Lett. 88 (2002) 211303 [hep-th/0202072] [INSPIRE].

  21. [21]

    S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].

  22. [22]

    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].

  23. [23]

    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].

  24. [24]

    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].

  25. [25]

    A. Maselli, S.H. Völkel and K.D. Kokkotas, Parameter estimation of gravitational wave echoes from exotic compact objects, Phys. Rev. D 96 (2017) 064045 [arXiv:1708.02217] [INSPIRE].

  26. [26]

    R.S. Conklin, B. Holdom and J. Ren, Gravitational wave echoes through new windows, Phys. Rev. D 98 (2018) 044021 [arXiv:1712.06517] [INSPIRE].

  27. [27]

    Q. Wang and N. Afshordi, Black hole echology: The observer’s manual, Phys. Rev. D 97 (2018) 124044 [arXiv:1803.02845] [INSPIRE].

  28. [28]

    J. Abedi and N. Afshordi, Echoes from the Abyss: A highly spinning black hole remnant for the binary neutron star merger GW170817, arXiv:1803.10454 [INSPIRE].

  29. [29]

    K.W. Tsang et al., A morphology-independent data analysis method for detecting and characterizing gravitational wave echoes, Phys. Rev. D 98 (2018) 024023 [arXiv:1804.04877] [INSPIRE].

  30. [30]

    N. Oshita and N. Afshordi, Probing microstructure of black hole spacetimes with gravitational wave echoes, arXiv:1807.10287 [INSPIRE].

  31. [31]

    H. Nakano, N. Sago, H. Tagoshi and T. Tanaka, Black hole ringdown echoes and howls, PTEP 2017 (2017) 071E01 [arXiv:1704.07175] [INSPIRE].

  32. [32]

    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].

  33. [33]

    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].

  34. [34]

    M.R. Correia and V. Cardoso, Characterization of echoes: A Dyson-series representation of individual pulses, Phys. Rev. D 97 (2018) 084030 [arXiv:1802.07735] [INSPIRE].

  35. [35]

    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 (2018) 044018 [arXiv:1806.04253] [INSPIRE].

  36. [36]

    J. Abedi, H. Dykaar and N. Afshordi, Echoes from the Abyss: The Holiday Edition!, arXiv:1701.03485 [INSPIRE].

  37. [37]

    G. Ashton et al., Comments on: “Echoes from the abyss: Evidence for Planck-scale structure at black hole horizons”, arXiv:1612.05625 [INSPIRE].

  38. [38]

    J. Westerweck et al., Low significance of evidence for black hole echoes in gravitational wave data, Phys. Rev. D 97 (2018) 124037 [arXiv:1712.09966] [INSPIRE].

  39. [39]

    J. Abedi, H. Dykaar and N. Afshordi, Comment on: “Low significance of evidence for black hole echoes in gravitational wave data”, arXiv:1803.08565 [INSPIRE].

  40. [40]

    E. Maggio, P. Pani and V. Ferrari, Exotic Compact Objects and How to Quench their Ergoregion Instability, Phys. Rev. D 96 (2017) 104047 [arXiv:1703.03696] [INSPIRE].

  41. [41]

    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, arXiv:1807.08840 [INSPIRE].

  42. [42]

    G. Allwright and D.M. Jacobs, Robin boundary conditions are generic in quantum mechanics, arXiv:1610.09581.

  43. [43]

    C.P. Burgess, P. Hayman, M. Williams and L. Zalavari, Point-Particle Effective Field Theory I: Classical Renormalization and the Inverse-Square Potential, JHEP 04 (2017) 106 [arXiv:1612.07313] [INSPIRE].

  44. [44]

    S.A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185 (1973) 635 [INSPIRE].

  45. [45]

    C.P. Burgess, P. Hayman, M. Rummel, M. Williams and L. Zalavari, Point-Particle Effective Field Theory II: Relativistic Effects and Coulomb/Inverse-Square Competition, JHEP 07 (2017) 072 [arXiv:1612.07334] [INSPIRE].

  46. [46]

    C.P. Burgess, P. Hayman, M. Rummel and L. Zalavari, Point-Particle Effective Field Theory III: Relativistic Fermions and the Dirac Equation, JHEP 09 (2017) 007 [arXiv:1706.01063] [INSPIRE].

  47. [47]

    H.E. Camblong and C.R. Ordóñez, Anomaly in conformal quantum mechanics: From molecular physics to black holes, Phys. Rev. D 68 (2003) 125013 [hep-th/0303166] [INSPIRE].

  48. [48]

    H.E. Camblong and C.R. Ordóñez, Black hole thermodynamics from near-horizon conformal quantum mechanics, Phys. Rev. D 71 (2005) 104029 [hep-th/0411008] [INSPIRE].

  49. [49]

    H.E. Camblong and C.R. Ordóñez, Semiclassical methods in curved spacetime and black hole thermodynamics, Phys. Rev. D 71 (2005) 124040 [hep-th/0412309] [INSPIRE].

  50. [50]

    H.E. Camblong and C.R. Ordóñez, Conformal Enhancement of Holographic Scaling in Black Hole Thermodynamics: A Near-Horizon Heat-Kernel Framework, JHEP 12 (2007) 099 [arXiv:0709.2942] [INSPIRE].

  51. [51]

    H.E. Camblong and C.R. Ordóñez, Conformal Tightness of Holographic Scaling in Black Hole Thermodynamics, Class. Quant. Grav. 30 (2013) 175007 [arXiv:0903.1960] [INSPIRE].

  52. [52]

    S. Carlip, Effective Conformal Descriptions of Black Hole Entropy: A Review, AIP Conf. Proc. 1483 (2012) 54 [arXiv:1207.1488] [INSPIRE].

  53. [53]

    S. Moroz and R. Schmidt, Nonrelativistic inverse square potential, scale anomaly and complex extension, Annals Phys. 325 (2010) 491 [arXiv:0909.3477] [INSPIRE].

  54. [54]

    R. Plestid, C.P. Burgess and D.H.J. O’Dell, Fall to the Centre in Atom Traps and Point-Particle EFT for Absorptive Systems, JHEP 08 (2018) 059 [arXiv:1804.10324] [INSPIRE].

  55. [55]

    W. Kinnersley, Field of an Arbitrarily Accelerating Point Mass, Phys. Rev. 186 (1969) 1335 [INSPIRE].

  56. [56]

    E. Berti, V. Cardoso and M. Casals, Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions, Phys. Rev. D 73 (2006) 024013 [Erratum ibid. D 73 (2006) 109902] [gr-qc/0511111] [INSPIRE].

  57. [57]

    R. Brito, V. Cardoso and P. Pani, Superradiance: Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics, Lect. Notes Phys. 906 (2015) 1 [arXiv:1501.06570].

  58. [58]

    R.H. Price and K.S. Thorne, Membrane Viewpoint on Black Holes: Properties and Evolution of the Stretched Horizon, Phys. Rev. D 33 (1986) 915 [INSPIRE].

  59. [59]

    V. Efimov, Energy levels arising form the resonant two-body forces in a three-body system, Phys. Lett. B 33 (1970) 563 [INSPIRE].

  60. [60]

    A.C. Fonseca, E.F. Redish and P.E. Shanley, Efimov effect in an analytically solvable model, Nucl. Phys. A 320 (1979) 273 [INSPIRE].

  61. [61]

    T. Kraemer et al., Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature 440 (2006) 315.

  62. [62]

    E. Braaten and H.W. Hammer, Efimov Physics in Cold Atoms, Annals Phys. 322 (2007) 120 [cond-mat/0612123] [INSPIRE].

  63. [63]

    L. Platter, Few-Body Systems and the Pionless Effective Field Theory, PoS(CD09)104 (2009) [arXiv:0910.0031] [INSPIRE].

  64. [64]

    H.W. Hammer and L. Platter, Efimov physics from a renormalization group perspective, Philos. Trans. A. Math. Phys. Eng. Sci. 369 (2011) 2679.

  65. [65]

    D.J. MacNeill and F. Zhou, Pauli Blocking Effect on Efimov States near a Feshbach Resonance, Phys. Rev. Lett. 106 (2011) 145301.

  66. [66]

    A.C. Ottewill and P. Taylor, Static Kerr Green’s Function in Closed Form and an Analytic Derivation of the Self-Force for a Static Scalar Charge in Kerr Space-Time, Phys. Rev. D 86 (2012) 024036 [arXiv:1205.5587] [INSPIRE].

  67. [67]

    V.P. Frolov and I.D. Novikov, Black hole physics: Basic concepts and new developments, Fundam. Theor. Phys. 96 (1998).

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Correspondence to Markus Rummel.

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ArXiv ePrint: 1808.00847

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Burgess, C.P., Plestid, R. & Rummel, M. Effective field theory of black hole echoes. J. High Energ. Phys. 2018, 113 (2018).

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  • Black Holes
  • Classical Theories of Gravity
  • Models of Quantum Gravity