Sturm–Liouville and Carroll: at the heart of the memory effect

  • P.-M. Zhang
  • M. ElbistanEmail author
  • G. W. Gibbons
  • P. A. Horvathy
Research Article


For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a \(2\times 2\) symmetric traceless matrix K(U), the matrix Sturm–Liouville equation \(\ddot{P}=KP\) plays a multiple and central rôle: (i) it determines the isometries; (ii) it appears as the key tool for switching from Brinkmann to BJR coordinates and vice versa; (iii) it determines the trajectories of particles initially at rest. All trajectories can be obtained from trivial “Carrollian” ones by a suitable action of the (broken) Carrollian isometry group.


Gravitational waves Sturm–Liouville equation Carroll group 



We are grateful to Christian Duval for his contribution at the early stages of this project, and to an anonymous referee for drawing our attention to [27] of which were were previously unaware. ME and PH thank the Institute of Modern Physics of the Chinese Academy of Sciences in Lanzhou for hospitality. This work was supported by the Chinese Academy of Sciences President’s International Fellowship Initiative (No. 2017PM0045), and by the National Natural Science Foundation of China (Grant No. 11575254). PH would like to acknowledge also the organizers of the “Workshop on Applied Newton–Cartan Geometry” and the Mainz Institute for Theoretical Physics (MITP), where part of this work was completed. We are grateful to our colleagues to inform us about their work in progress [33].


  1. 1.
    Zel’dovich, Y.B., Polnarev, A.G.: Radiation of gravitational waves by a cluster of superdense stars, Astron. Zh. 51, 30 (1974) [Sov. Astron. 18, 17 (1974)]Google Scholar
  2. 2.
    Braginsky, V.B., Grishchuk, L.P:. Kinematic resonance and the memory effect in free mass gravitational antennas. Zh. Eksp. Teor. Fiz. 89, 744–750 (1985) [Sov. Phys. JETP 62, 427 (1985)]Google Scholar
  3. 3.
    Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Witten, L. (ed.) Gravitation, an Introduction to Current Research. Wiley, New York (1962)Google Scholar
  4. 4.
    Souriau, J-M.: Le milieu élastique soumis aux ondes gravitationnelles, Ondes et radiations gravitationnelles. Colloques Internationaux du CNRS No 220, p. 243. Paris (1973)Google Scholar
  5. 5.
    Braginsky, V.B., Thorne, K.S.: Gravitational-wave burst with memory and experimental prospects. Nature (London) 327, 123 (1987)ADSCrossRefGoogle Scholar
  6. 6.
    Bondi, H., Pirani, F.A.E.: Gravitational waves in general relativity. 13: caustic property of plane waves. Proc. R. Soc. Lond. A 421, 395 (1989)ADSCrossRefGoogle Scholar
  7. 7.
    Grishchuk, L.P., Polnarev, A.G.: Gravitational wave pulses with ‘velocity coded memory’. Sov. Phys. JETP 69, (1989) 653 [Zh. Eksp. Teor. Fiz. 96, (1989) 1153]Google Scholar
  8. 8.
    Zhang, P.-M., Duval, C., Gibbons, G.W., Horvathy, P.A.: The memory effect for plane gravitational waves. Phys. Lett. B 772, 743 (2017). arXiv:1704.05997 [gr-qc]
  9. 9.
    Zhang, P.-M., Duval, C., Gibbons, G.W., Horvathy, P.A.: Soft gravitons and the memory effect for plane gravitational waves, Phys. Rev. D 96(6), 064013 (2017). arXiv:1705.01378 [gr-qc]
  10. 10.
    Lasenby, A.: Black Holes and Gravitational Waves, Talks Given at the Royal Society Workshop on ‘Black Holes’. Chichley Hall, UK and KIAA, Beijing (2017)Google Scholar
  11. 11.
    Zhang, P.M., Duval, C., Gibbons, G.W., Horvathy, P.A.: Velocity memory effect for polarized gravitational waves. J. Cosmol. Astropart. Phys. 2018(05), 030 (2018). arXiv:1802.09061 [gr-qc]
  12. 12.
    Gibbons, G.W.: Quantized fields propagating in plane wave space-times. Commun. Math. Phys. 45, 191 (1975)ADSCrossRefGoogle Scholar
  13. 13.
    Lévy-Leblond, J.M.: Une nouvelle limite non-relativiste du group de Poincaré. Ann. Inst. H Poincaré 3, 1 (1965)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gupta, V.D.S.: On an analogue of the Galileo group. Il Nuovo Cimento 54, 512 (1966)CrossRefGoogle Scholar
  15. 15.
    Bondi, H., Pirani, F.A.E., Robinson, I.: Gravitational waves in general relativity. 3. Exact plane waves. Proc. R. Soc. Lond. A 251, 519 (1959). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kramer, D., Stephani, H., McCallum, M., Herlt, E.: Exact Solutions of Einstein’s field Equations, 2nd edn, p. 385. Cambridge University Press, Cambridge (2003). (sec 24.5 Table 24.2)zbMATHGoogle Scholar
  17. 17.
    Duval, C., Gibbons, G.W., Horvathy, P.A., Zhang, P.-M.: Carroll symmetry of plane gravitational waves. Class. Quant. Grav. 34 (2017). arXiv:1702.08284 [gr-qc]
  18. 18.
    Ngendakumana, A., Nzotungicimpaye, J., Todjihounde, L.: Group theoretical construction of planar noncommutative phase spaces. J. Math. Phys. 55, 013508 (2014). arXiv:1308.3065 [math-ph]
  19. 19.
    Duval, C., Gibbons, G.W., Horvathy, P.A., Zhang, P.M.: Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time. Class. Quant. Grav. 31, 085016 (2014). arXiv:1402.5894 [gr-qc]
  20. 20.
    Bergshoeff, E., Gomis, J., Longhi, G.: Dynamics of Carroll Particles, Class. Quant. Grav. 31(20), 205009 (2014) arXiv:1405.2264 [hep-th]
  21. 21.
    Brinkmann, M.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94, 119–145 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, P.-M., Duval, C., Horvathy, P. A.: Memory effect for impulsive gravitational waves. Class. Quant. Grav. 35(6), 065011 (2018). arXiv:1709.02299 [gr-qc]
  23. 23.
    Podolský, J., Sämann, C., Steinbauer, R., Svarc, R.: The global existence, uniqueness and \(C^1\)-regularity of geodesics in nonexpanding impulsive gravitational waves. Class. Quant. Grav. 32(2), 025003 (2015). arXiv:1409.1782 [gr-qc]
  24. 24.
    Torre, C.G.: Gravitational waves: just plane symmetry. Gen. Relativ. Gravit. 38, 653 (2006). arXiv:gr-qc/9907089
  25. 25.
    Duval, C., Burdet, G., Künzle, H.P., Perrin, M.: Bargmann structures and Newton–Cartan theory. Phys. Rev. D 31, 1841 (1985)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Duval, C., Gibbons, G.W., Horvathy, P.: Celestial mechanics, conformal structures and gravitational waves. Phys. Rev. D 43, 3907 (1991). arXiv:hep-th/0512188
  27. 27.
    Ehrlich, P.E., Emch, G.G.: Gravitational waves and causality. Rev. Math. Phys. 4 (1992) 163. (Erratum: [Rev. Math. Phys. 4, (1992) 501])
  28. 28.
    Baldwin, O.R., Jeffery, G.B.: The relativity theory of plane waves. Proc. R. Soc. Lond. A 111, 95 (1926)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Rosen, N.: Plane polarized waves in the general theory of relativity. Phys. Z. Sowjetunion 12, 366 (1937)zbMATHGoogle Scholar
  30. 30.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Volume 2 of A Course of Theoretical Physics. Pergamon Press, Oxford (1971)zbMATHGoogle Scholar
  31. 31.
    Gibbons, G.W., Pope, C.N.: Kohn’s theorem, Larmor’s equivalence principle and the Newton–Hooke group. Ann. Phys. 326, 1760 (2011). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhang, P.M., Horvathy, P.A., Andrzejewski, K., Gonera, J., Kosinski, P.: Newton–Hooke type symmetry of anisotropic oscillators. Ann. Phys. 333, 335 (2013)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Andrzejewski, K., Prencel, S.: Memory effect, conformal symmetry and gravitational plane waves. Phys. Lett. B 782, 421 (2018). ADSCrossRefGoogle Scholar

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Authors and Affiliations

  • P.-M. Zhang
    • 1
  • M. Elbistan
    • 1
    Email author
  • G. W. Gibbons
    • 2
  • P. A. Horvathy
    • 1
    • 3
  1. 1.Institute of Modern PhysicsChinese Academy of SciencesLanzhouChina
  2. 2.D.A.M.T.P.Cambridge UniversityCambridgeUK
  3. 3.Laboratoire de Mathématiques et de Physique ThéoriqueUniversité de ToursToursFrance

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