Skip to main content
SpringerLink
Log in

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

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. The Carroll group [13, 14] is the subgroup of the Bargmann group with no time translations; the latter is itself the subgroup of the Poincaré group which leaves \({\partial }_V\) invariant. The Bargmann group is a 1-parameter central extension of the Galilei group upon which it projects when translations along V are factored out. For circularly polarised periodic waves the symmetry can be extend to a 6-parameter group [11, 16].

  2. A sandwich wave is one which vanishes outside some finite interval \([U_i,U_f]\) [6, 15].

  3. The BJR coordinates are valid only in finite intervals before becoming singular [6], see Sect. 3.

  4. The conserved quantity associated with the “vertical” Killing vector \({\partial }_V\) can be used to show that proper time and u are proportional.

  5. Remember that the 4D gravitational wave spacetime is in fact the “Bargmann space” for both a non-relativistic and for a Carroll particle in the transverse plane [19, 25]; geodesics in 4D project to motions in \(2+1\) dimensions.

  6. Comparison with the trajectories obtained by solving directly the equations of motion numerically shows a perfect overlapping. This is a third appearance of the solution P of the SL eqn (3.2). In Souriau’s approach it is the determinant of the metric (2.1) which satisfies a Sturm–Liouville equation.

References

  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)]

  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)]

  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. Souriau, J-M.: Le milieu élastique soumis aux ondes gravitationnelles, Ondes et radiations gravitationnelles. Colloques Internationaux du CNRS No 220, p. 243. Paris (1973)

  5. Braginsky, V.B., Thorne, K.S.: Gravitational-wave burst with memory and experimental prospects. Nature (London) 327, 123 (1987)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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]

  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). https://doi.org/10.1016/j.physletb.2017.07.050. arXiv:1704.05997 [gr-qc]

  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). https://doi.org/10.1103/PhysRevD.96.064013. arXiv:1705.01378 [gr-qc]

  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)

  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). https://doi.org/10.1088/1475-7516/2018/05/030. arXiv:1802.09061 [gr-qc]

  12. Gibbons, G.W.: Quantized fields propagating in plane wave space-times. Commun. Math. Phys. 45, 191 (1975)

    Article  ADS  Google Scholar 

  13. Lévy-Leblond, J.M.: Une nouvelle limite non-relativiste du group de Poincaré. Ann. Inst. H Poincaré 3, 1 (1965)

    MathSciNet  MATH  Google Scholar 

  14. Gupta, V.D.S.: On an analogue of the Galileo group. Il Nuovo Cimento 54, 512 (1966)

    Article  Google Scholar 

  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). https://doi.org/10.1098/rspa.1959.0124

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  17. Duval, C., Gibbons, G.W., Horvathy, P.A., Zhang, P.-M.: Carroll symmetry of plane gravitational waves. Class. Quant. Grav. 34 (2017). doi.org/10.1088/1361-6382/aa7f62. arXiv:1702.08284 [gr-qc]

  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. 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. Bergshoeff, E., Gomis, J., Longhi, G.: Dynamics of Carroll Particles, Class. Quant. Grav. 31(20), 205009 (2014) https://doi.org/10.1088/0264-9381/31/20/205009. arXiv:1405.2264 [hep-th]

  21. Brinkmann, M.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94, 119–145 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, P.-M., Duval, C., Horvathy, P. A.: Memory effect for impulsive gravitational waves. Class. Quant. Grav. 35(6), 065011 (2018). https://doi.org/10.1088/1361-6382/aaa987. arXiv:1709.02299 [gr-qc]

  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). https://doi.org/10.1088/0264-9381/32/2/025003. arXiv:1409.1782 [gr-qc]

  24. Torre, C.G.: Gravitational waves: just plane symmetry. Gen. Relativ. Gravit. 38, 653 (2006). arXiv:gr-qc/9907089

  25. Duval, C., Burdet, G., Künzle, H.P., Perrin, M.: Bargmann structures and Newton–Cartan theory. Phys. Rev. D 31, 1841 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  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. Ehrlich, P.E., Emch, G.G.: Gravitational waves and causality. Rev. Math. Phys. 4 (1992) 163. https://doi.org/10.1142/S0129055X92000066 (Erratum: [Rev. Math. Phys. 4, (1992) 501])

  28. Baldwin, O.R., Jeffery, G.B.: The relativity theory of plane waves. Proc. R. Soc. Lond. A 111, 95 (1926)

    Article  ADS  MATH  Google Scholar 

  29. Rosen, N.: Plane polarized waves in the general theory of relativity. Phys. Z. Sowjetunion 12, 366 (1937)

    MATH  Google Scholar 

  30. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Volume 2 of A Course of Theoretical Physics. Pergamon Press, Oxford (1971)

    MATH  Google Scholar 

  31. Gibbons, G.W., Pope, C.N.: Kohn’s theorem, Larmor’s equivalence principle and the Newton–Hooke group. Ann. Phys. 326, 1760 (2011). https://doi.org/10.1016/j.aop.2011.03.003

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  33. Andrzejewski, K., Prencel, S.: Memory effect, conformal symmetry and gravitational plane waves. Phys. Lett. B 782, 421 (2018). https://doi.org/10.1016/j.physletb.2018.05.072

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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].

Author information

Authors and Affiliations

  1. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China

    P.-M. Zhang, M. Elbistan & P. A. Horvathy

  2. D.A.M.T.P., Cambridge University, Cambridge, UK

    G. W. Gibbons

  3. Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Tours, France

    P. A. Horvathy

Authors
  1. P.-M. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Elbistan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. W. Gibbons
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. A. Horvathy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Elbistan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, PM., Elbistan, M., Gibbons, G.W. et al. Sturm–Liouville and Carroll: at the heart of the memory effect. Gen Relativ Gravit 50, 107 (2018). https://doi.org/10.1007/s10714-018-2430-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-018-2430-0

Keywords

Search

Navigation