Skip to main content
SpringerLink
Log in

Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves

  • Reference work entry
  • First Online:
Handbook of Gravitational Wave Astronomy

  • 1350 Accesses

Abstract

The largest part of any gravitational-wave inspiral of a compact binary can be understood as a slow, adiabatic drift between the trajectories of a certain referential conservative system. In many contexts, the phase space of this conservative system is smooth, and there are no “topological transitions” in the phase space, meaning that there are no sudden qualitative changes in the character of the orbital motion during the inspiral. However, in this chapter, we discuss the cases where this assumption fails and nonlinear and/or non-smooth transitions come into play. In integrable conservative systems under perturbation, topological transitions suddenly appear at resonances, and we sketch how to implement the passage through such regions in an inspiral model. Even though many of the developments of this chapter apply to general inspirals, we focus on a particular scenario known as the extreme mass ratio inspiral (EMRI). An EMRI consists of a compact stellar-mass object inspiraling into a supermassive black hole. At leading order, the referential conservative system is simply geodesic motion in the field of the supermassive black hole, and the rate of the drift is given by radiation reaction. In Einstein gravity, the supermassive black hole field is the Kerr spacetime in which the geodesic motion is integrable. However, the equations of motion can be perturbed in various ways so that prolonged resonances and chaos appear in phase space as well as the inspiral, which we demonstrate in simple physically motivated examples.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 699.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 849.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Amaro-Seoane P, Audley H, Babak S et al (2017) Laser interferometer space antenna. arXiv:1702.00786

    Google Scholar 

  2. Arnold V, Kozlov V, Neishtadt A (2006) Mathematical aspects of classical and celestial mechanics, 3rd edn. Springer International Publishing

    Book  MATH  Google Scholar 

  3. Arnold VI (1963) Proof of a Theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ Math Surv 18(5):9–36

    Google Scholar 

  4. Bambi C (2011) Testing the Kerr black hole hypothesis. Mod Phys Lett A 26(33):2453–2468

    Article  ADS  MathSciNet  Google Scholar 

  5. Banks J, Brooks J, Cairns G et al (1992) On Devaney’s definition of chaos. Am Math Mon 99(4):332–334

    Article  MathSciNet  MATH  Google Scholar 

  6. Barack L, Pound A (2019) Self-force and radiation reaction in general relativity. Rep Prog Phys 82(1):016904

    Article  ADS  Google Scholar 

  7. Barausse E, Berti E, Hertog T et al (2020) Prospects for fundamental physics with LISA. Gen Relativ Gravit 52(8):81

    Article  ADS  Google Scholar 

  8. Basovník M, Semerák O (2016) Geometry of deformed black holes. II. Schwarzschild hole surrounded by a Bach-Weyl ring. Phys Rev D 94(4):044007

    Google Scholar 

  9. Berry CPL, Cole RH, Cañizares P, Gair JR (2016) Importance of transient resonances in extreme-mass-ratio inspirals. Phys Rev D 94:124042

    Article  ADS  Google Scholar 

  10. Bičák J, Ledvinka T (1993) Relativistic disks as sources of the Kerr metric. Phys Rev Lett 71(11):1669–1672

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Birkhoff GD (1913) Proof of Poincaré’s geometric theorem. Trans Am Math Soc 14(1):14–22

    MATH  Google Scholar 

  12. Brink J, Geyer M, Hinderer T (2015) Astrophysics of resonant orbits in the Kerr metric. Phys Rev D 91:083001

    Article  ADS  MathSciNet  Google Scholar 

  13. Brink J, Geyer M, Hinderer T (2015) Orbital resonances around black holes. Phys Rev Lett 114(8):081102

    Article  ADS  Google Scholar 

  14. Carter B (1968) Global structure of the Kerr family of gravitational fields. Phys Rev 174(5):1559–1571

    Article  ADS  MATH  Google Scholar 

  15. Carter B (1968). Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s Equations. Commun Math Phys 10(4):280–310

    Article  ADS  MATH  Google Scholar 

  16. Contopoulos G (2004) Order and chaos in dynamical astronomy. Springer Science & Business Media

    MATH  Google Scholar 

  17. Destounis K, Suvorov AG, Kokkotas KD (2020) Testing spacetime symmetry through gravitational waves from extreme-mass-ratio inspirals. Phys Rev D 102:064041

    Article  ADS  MathSciNet  Google Scholar 

  18. Destounis K, Suvorov AG, Kokkotas KD (2021) Gravitational-wave glitches in chaotic extreme-mass-ratio inspirals. arXiv e-prints, page arXiv:2103.05643

    Google Scholar 

  19. Dixon WG (1974) Dynamics of extended bodies in general relativity. III. Equations of motion. Philos Trans R Soc Lond Ser A 277(1264):59–119

    ADS  Google Scholar 

  20. Doroshkevich A, Zel’Dovich YB, Novikov I (1966) Gravitational collapse of nonsymmetric and rotating masses. Sov Phys JETP 22:122–30

    ADS  Google Scholar 

  21. Efthymiopoulos C, Contopoulos G, Voglis N, Dvorak R (1997) Stickiness and cantori. J Phys A Math Gen 30(23):8167–8186

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Eleni A, Apostolatos TA (2020) Newtonian analogue of a Kerr black hole. Phys Rev D 101(4):044056

    Article  ADS  MathSciNet  Google Scholar 

  23. Flanagan ÉÉ, Hinderer T (2012) Transient resonances in the inspirals of point particles into black holes. Phys Rev Lett 109(7):071102

    Article  ADS  Google Scholar 

  24. Flanagan EE, Hughes SA, Ruangsri U (2014) Resonantly enhanced and diminished strong-field gravitational-wave fluxes. Phys Rev D 89:084028

    Article  ADS  Google Scholar 

  25. Frolov VP, Krtouš P, Kubizňák D (2017) Black holes, hidden symmetries, and complete integrability. Living Rev Relativ 20(1):6

    Article  ADS  MATH  Google Scholar 

  26. Grobman DM (1959) Homeomorphisms of systems of differential equations. Dokl Akad Nauk SSSR 128:880–881

    MathSciNet  MATH  Google Scholar 

  27. Hansen RO (1974) Multipole moments of stationary space-times. J Math Phys 15(1):46–52

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Hartman P (1960) A lemma in the theory of structural stability of differential equations. Proc Am Math Soc 11(4):610–620

    Article  MathSciNet  MATH  Google Scholar 

  29. Hu W-R, Wu Y-L (2017) The Taiji Program in Space for gravitational wave physics and the nature of gravity. Natl Sci Rev 4(5):685–686

    Article  Google Scholar 

  30. Isoyama S, Fujita R, Nakano H et al (2013) Evolution of the carter constant for resonant inspirals into a Kerr black hole: I. The scalar case. Progress Theoretical Exp Phys 2013(6):063E01

    Google Scholar 

  31. Isoyama S, Fujita R, Nakano H et al (2019) “flux-balance formulae” for extreme mass-ratio inspirals. Progress Theoretical Exp Phys 2019(1):013E01

    Google Scholar 

  32. Johannsen T (2013) Regular black hole metric with three constants of motion. Phys Rev D 88(4):044002

    Article  ADS  Google Scholar 

  33. Kerr RP (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys Rev Lett 11(5):237–238

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Kevorkian JK, Cole JD (2012) Multiple scale and singular perturbation methods, volume 114. Springer Science & Business Media

    Google Scholar 

  35. Kolmogorov AN (1954) On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl Akad Nauk SSSR 98:527–530

    MathSciNet  MATH  Google Scholar 

  36. Lemos JPS, Letelier PS (1994) Exact general relativistic thin disks around black holes. Phys Rev D 49(10):5135–5143

    Article  ADS  Google Scholar 

  37. Lukes-Gerakopoulos G, Apostolatos TA, Contopoulos G (2010) Observable signature of a background deviating from the Kerr metric. Phys Rev D 81:124005

    Article  ADS  Google Scholar 

  38. Lukes-Gerakopoulos G, Katsanikas M, Patsis PA, Seyrich J (2016) Dynamics of a spinning particle in a linear in spin Hamiltonian approximation. Phys Rev D 94(2):024024

    Article  ADS  MathSciNet  Google Scholar 

  39. Lynden-Bell D (2000) Carter separable electromagnetic fields. Mon Not R Astron Soc 312(2):301–315

    Article  ADS  Google Scholar 

  40. Manko VS, Novikov ID (1992) Generalizations of the Kerr and Kerr-Newman metrics possessing an arbitrary set of mass-multipole moments. Class Quan Grav 9(11):2477–2487

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Manko VS, Sanabria-Gómez JD, Manko OV (2000) Nine-parameter electrovac metric involving rational functions. Phys Rev D 62(4):044048

    Article  ADS  MathSciNet  Google Scholar 

  42. Markakis C (2014) Constants of motion in stationary axisymmetric gravitational fields. Mon Not R Astron Soc 441(4):2974–2985

    Article  ADS  Google Scholar 

  43. Mathisson M (1937) Neue mechanik materieller systemes. Acta Phys Polon 6:163–2900

    MATH  Google Scholar 

  44. Miller J, Pound A (2020) Two-timescale evolution of extreme-mass-ratio inspirals: waveform generation scheme for quasicircular orbits in Schwarzschild spacetime. arXiv preprint arXiv:2006.11263

    Google Scholar 

  45. Mino Y (2003) Perturbative approach to an orbital evolution around a supermassive black hole. Phys Rev D 67(8):084027

    Article  ADS  Google Scholar 

  46. Morbidelli A (2002) Modern celestial mechanics: aspects of solar system dynamics, 1st edn. CRC Press

    MATH  Google Scholar 

  47. Moser J (1962) On invariant curves of area-preserving mappings of an annulus. Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse, pp 1–20

    Google Scholar 

  48. Neugebauer G, Meinel R (1993) The Einsteinian gravitational field of the rigidly rotating disk of dust. Astrophys J Lett 414:L97

    Article  ADS  Google Scholar 

  49. Papapetrou A (1951) Spinning test particles in general relativity. 1. Proc R Soc Lond A209:248–258

    ADS  MathSciNet  MATH  Google Scholar 

  50. Pesin YB (1977) Characteristic Lyapunov exponents and smooth ergodic theory. Russ Math Surv 32(4):55–114

    Article  MATH  Google Scholar 

  51. Poincaré H (1912) Sur un théorème de géométrie. Rendiconti del Circolo Matematico di Palermo 33:375–407

    Article  MATH  Google Scholar 

  52. Poincaré H (1993) New methods of celestial mechanics. American Institute of Physics, Woodbury

    MATH  Google Scholar 

  53. Poisson E, Pound A, Vega I (2011) The motion of point particles in curved spacetime. Living Rev Relativ 14(1):7

    Article  ADS  MATH  Google Scholar 

  54. Polcar L, Semerák O (2019) Free motion around black holes with discs or rings: Between integrability and chaos. VI. the Melnikov method. Phys Rev D 100(10):103013

    Google Scholar 

  55. Rüdiger R (1981) Conserved quantities of spinning test particles in general relativity. I. Proc R Soc Lond A 375(1761):185–193

    Article  ADS  MathSciNet  Google Scholar 

  56. Rüdiger R (1983) Conserved quantities of spinning test particles in general relativity. II. Proc R Soc Lond A 385(1788):229–239

    Article  ADS  MathSciNet  Google Scholar 

  57. Sano Y, Tagoshi H (2014) Gravitational perturbation induced by a rotating ring around a Kerr black hole. arXiv preprint arXiv:1412.8607

    Google Scholar 

  58. Semerák O (2003) Gravitating discs around a Schwarzschild black hole: III. Class Quan Grav 20(9):1613–1634

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. Semerák O, Suková P (2010) Free motion around black holes with discs or rings: between integrability and chaos–I. Mon Not R Astron Soc 404(2):545–574

    Article  ADS  Google Scholar 

  60. Semerák O, Čížek P (2020) Rotating disc around a Schwarzschild black hole. Universe 6(2):27

    Article  ADS  Google Scholar 

  61. Semerák O, Suková P (2015) On geodesic dynamics in deformed black-hole fields. Fund Theor Phys 179:561–586

    MathSciNet  MATH  Google Scholar 

  62. Silverman S (1992) On maps with dense orbits and the definition of chaos. Rocky Mt J Math 22(1):353–375

    Article  MathSciNet  MATH  Google Scholar 

  63. Smale S (1965) Diffeomorphisms with many periodic points. In: Cairns SS (ed) Differential and combinatorial topology: a symposium in honor of Marston Morse. Princeton University Press

    Google Scholar 

  64. Speri L, Gair JR (2021) Assessing the impact of transient orbital resonances. arXiv e-prints, page arXiv:2103.06306

    Google Scholar 

  65. Suzuki S, Maeda K-I (1997) Chaos in Schwarzschild spacetime: the motion of a spinning particle. Phys Rev D 55(8):4848–4859

    Article  ADS  Google Scholar 

  66. van de Meent M (2014) Conditions for sustained orbital resonances in extreme mass ratio inspirals. Phys Rev D 89(8):084033

    Article  ADS  Google Scholar 

  67. van de Meent M (2014) Resonantly enhanced kicks from equatorial small mass-ratio inspirals. Phys Rev D 90(4):044027

    Article  ADS  Google Scholar 

  68. Van De Meent M (2018) Gravitational self-force on generic bound geodesics in Kerr spacetime. Phys Rev D 97(10):104033

    Article  ADS  MathSciNet  Google Scholar 

  69. Vigeland S, Yunes N, Stein LC (2011) Bumpy black holes in alternative theories of gravity. Phys Rev D 83(10):104027

    Article  ADS  Google Scholar 

  70. Witzany V (2019) Hamilton-Jacobi equation for spinning particles near black holes. Phys Rev D 100(10):104030

    Article  ADS  MathSciNet  Google Scholar 

  71. Witzany V, Semerák O, Suková P (2015) Free motion around black holes with discs or rings: between integrability and chaos–IV. Mon Not R Astron Soc 451(2):1770–1794

    Article  ADS  Google Scholar 

  72. Witzany V, Steinhoff J, Lukes-Gerakopoulos G (2019) Hamiltonians and canonical coordinates for spinning particles in curved space-time. Class Quan Grav 36(7):075003

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. Zelenka O, Lukes-Gerakopoulos G, Witzany V, Kopáček O (2020) Growth of resonances and chaos for a spinning test particle in the Schwarzschild background. Phys Rev D 101(2):024037

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

GL-G has been supported by the fellowship Lumina Quaeruntur No. LQ100032102 of the Czech Academy of Sciences. VW was supported by European Union’s Horizon 2020 research and innovation program under grant agreement No 894881. The authors would like to thank Lukáš Polcar for allowing them to use the didactic example in section “External Matter Deformation”.

Author information

Authors and Affiliations

  1. Astronomical Institute of the Czech Academy of Sciences, Prague, Czech Republic

    Georgios Lukes-Gerakopoulos

  2. School of Mathematics and Statistics, University College Dublin, Dublin, Ireland

    Vojtěch Witzany

Authors
  1. Georgios Lukes-Gerakopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vojtěch Witzany
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Physics, Fudan University, Shanghai, China

    Cosimo Bambi

  2. Université de Paris and European Gravitational Observatory, Paris, France

    Stavros Katsanevas

  3. Institute for Astronomy and Astrophysics Eberhard Karls, University of Tübingen, Tübingen, Germany

    Konstantinos D. Kokkotas

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Singapore Pte Ltd.

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Lukes-Gerakopoulos, G., Witzany, V. (2022). Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves. In: Bambi, C., Katsanevas, S., Kokkotas, K.D. (eds) Handbook of Gravitational Wave Astronomy. Springer, Singapore. https://doi.org/10.1007/978-981-16-4306-4_42

Download citation

Publish with us

Policies and ethics

Search

Navigation