Nonlinear Gravitational Waves and Solitons

  • Francisco R. VillatoroEmail author
Part of the Understanding Complex Systems book series (UCS)


Gravitational wave astronomy is born in 2016. The laser interferometers of Advanced LIGO have detected two gravitational waves, each one generated by two black hole pairs. The observed wave profiles result from the fusion of two stellar-mass black holes into a single rotating black hole, with the emission of gravitational radiation with energy in the solar-mass scale. Indeed, they are the most violent astrophysical events recorded to date. Since gravitational waves solve the weak-field approximation of the Einstein equations in vacuum, in this limit, they evolve as linear waves. However, gravitational waves are intrinsically nonlinear waves; in fact, the chirp of the signal, the change in frequency observed by Advanced LIGO detectors, is due to the nonlinearity at the sources, even being negligible far away from them. Both cylindrical and planar nonlinear gravitational waves can be interpreted as soliton solutions of Einstein’s equations outside the sources. Actually, even black holes, the main sources of gravitational radiation, are two-soliton solutions of Einstein’s equations in vacuum. Gravitational solitons differ from standard nonlinear solitons in several aspects, including new phenomena such as multi-soliton coalescence, a phenomenon that emits low-amplitude radiation. Indeed, the pair-of-pants solution for the fusion of two black holes can be interpreted in such a way. In conclusion, although gravitational waves propagate in spacetime like linear waves, at their sources they are nonlinear gravitational waves and, even, gravitational solitons.


Gravitational waves Black holes Gravitational solitons Inverse scattering method 



The author acknowledges financial support from project TIN2014-56494-C4-1-P from Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia del Ministerio de Ciencia e Innovación (MICINN) of Spain.


  1. 1.
    Aasi, J., et al.: Advanced LIGO. Class. Quant. Grav. 32, 074001 (2015)ADSCrossRefGoogle Scholar
  2. 2.
    Abbott, B.P., et al.: Binary black hole mergers in the first advanced LIGO observing run. Phys. Rev. X 6(4), 041015 (2016)Google Scholar
  3. 3.
    Abbott, B.P., et al.: Directly comparing GW150914 with numerical solutions of Einsteins equations for binary black hole coalescence. Phys. Rev. D 94(6), 064035 (2016)ADSCrossRefGoogle Scholar
  4. 4.
    Abbott, B.P., et al.: GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116(24), 241103 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Abbott, B.P., et al.: Improved analysis of GW150914 using a fully spin-precessing waveform model. Phys. Rev. X 6(4), 041014 (2016)Google Scholar
  6. 6.
    Abbott, B.P., et al.: Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116(6), 061102 (2016)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Abbott, B.P., et al.: Properties of the binary black hole merger GW150914. Phys. Rev. Lett. 116(24), 241102 (2016)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Abbott, B.P., et al.: Effects of waveform model systematics on the interpretation of GW150914. Class. Quant. Grav. 34(10), 104002 (2017)ADSCrossRefGoogle Scholar
  9. 9.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, USA (1981)CrossRefGoogle Scholar
  10. 10.
    Ackermann, M., et al.: Fermi-LAT observations of the LIGO event GW150914. Astrophys. J. 823(1), L2 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Adrian-Martinez, S., et al.: High-energy Neutrino follow-up search of gravitational wave event GW150914 with ANTARES and IceCube. Phys. Rev. D 93(12), 122010 (2016)ADSCrossRefGoogle Scholar
  12. 12.
    Aldrovandi, R., Pereira, J.G., da Rocha, R., Vu, K.H.: Nonlinear gravitational waves: their form and effects. Int. J. Theor. Phys. 49, 549–563 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Beheshti, S., Tahvildar-Zadeh, S.: Dressing with control: using integrability to generate desired solutions to Einstein’s equations. In: Cuevas-Maraver, J., Kevrekidis, P.G., Williams, F. (eds.) The Sine-Gordon Model and Its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics, pp. 207–231. Springer (2014)Google Scholar
  14. 14.
    Belinski, V., Verdaguer, E.: Gravitational Solitons. Cambridge University Press, Cambridge, UK (2005)zbMATHGoogle Scholar
  15. 15.
    Bohn, A., Kidder, L.E., Teukolsky, S.A.: Toroidal horizons in binary black hole mergers. Phys. Rev. D 94(6), 064009 (2016)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Bondi, H., Pirani, F.A.E., Robinson, I.: Gravitational waves in general relativity iii. exact plane waves. Proc. R. Soc. London A251, 519–533 (1959)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Braginsky, V.B., Bilenko, I.A., Vyatchanin, S.P., Gorodetsky, M.L., Mitrofanov, V.P., Prokhorov, L.G., Strigin, S.E., Khalili, F.: Ya.: The road to the discovery of gravitational waves. Phys. Usp. 59(9), 879–885 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    Cherepashchuk, A.M.: Discovery of gravitational waves: a new chapter in black hole studies. Phys. Usp. 59(9), 910–917 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    Denson Hill, C., Nurowski, P.: How the green light was given for gravitational wave search. arXiv:1608.08673 [physics.hist-ph] (2016)
  20. 20.
    Faraoni, V.: A Common misconception about LIGO detectors of gravitational waves. Gen. Rel. Grav. 39, 677–684 (2007)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Garfinkle, D.: Gauge invariance and the detection of gravitational radiation. Am. J. Phys. 74, 196–199 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, UK (1973)CrossRefGoogle Scholar
  23. 23.
    Israel, W.: Dark stars: the evolution of an idea. In: Hawking, S. W., Israel, W. (eds.) Three Hundred Years of Gravitation, pp. 199–276. Cambridge University Press (1987)Google Scholar
  24. 24.
    Kennefick, D.: Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves. Princeton University Press, Princeton, US (2007)CrossRefGoogle Scholar
  25. 25.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, Course of Theoretical Physics, vol, vol. 2. Pergamon Press, Oxford, UK (1975)CrossRefGoogle Scholar
  26. 26.
    Lipunov, V.M.: Astrophysical meaning of the discovery of gravitational waves. Physics-Uspekhi 59(9), 918–928 (2016)ADSCrossRefGoogle Scholar
  27. 27.
    Maggiore, M.: Gravitational Waves. Vol. 1: Theory and Experiments. Oxford Master Series in Physics. Oxford University Press, New York, USA (2007)CrossRefGoogle Scholar
  28. 28.
    Matzner, R.A., Seidel, H.E., Shapiro, S.L., Smarr, L., Suen, W.M., Teukolsky, S.A., Winicour, J.: Geometry of a black hole collision. Science 270, 941–947 (1995)ADSCrossRefGoogle Scholar
  29. 29.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman, San Francisco, USA (1973)Google Scholar
  30. 30.
    Pretorius, F.: Binary black hole coalescence. In: Colpi, M., Casella, P., Gorini, V., Moschella, U., Possenti, A. (eds.) Physics of Relativistic Objects in Compact Binaries: From Birth to Coalescence, pp. 305–369. Springer (2009)Google Scholar
  31. 31.
    Pustovoit, V.I.: On the direct detection of gravitational waves. Physics-Uspekhi 59(10), 1034–1051 (2016)ADSCrossRefGoogle Scholar
  32. 32.
    Saulson, P.R.: If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves? Am. J. Phys. 65, 501–505 (1997)ADSCrossRefGoogle Scholar
  33. 33.
    Thorne, K.S.: Gravitational radiation. In: Hawking, S.W., Israel, W. (eds.) Three Hundred Years of Gravitation, pp. 330–458. Cambridge University Press (1987)Google Scholar
  34. 34.
    Tomimatsu, A., Sato, H.: New exact solution for the gravitational field of a spinning mass. Phys. Rev. Lett. 29, 1344–1345 (1972)ADSCrossRefGoogle Scholar
  35. 35.
    Tomizawa, S., Mishima, T.: Nonlinear effects for a cylindrical gravitational two-soliton. Phys. Rev. D 91(12), 124058 (2015)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Trautman, A.: Radiation and boundary conditions in the theory of gravitation. Bull. Acad. Polon. Sci. 6, 407–412 (1958)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Weinstein, G.: General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists. Cambridge Scholars Publishing, Cambridge, UK (2015)zbMATHGoogle Scholar
  38. 38.
    Winicour, J.: Characteristic evolution and matching. Living Rev. Rel. 12, lrr-2009-3 (2009)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department Lenguajes y Ciencias de la Computación, Escuela de Ingenierías Industriales, Ampliación del Campus de TeatinosUniversidad de MálagaMálagaSpain

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