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

  • P.-M. Zhang
  • M. Elbistan
  • G. W. Gibbons
  • P. A. Horvathy
Research Article
  • 36 Downloads

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.

Keywords

Gravitational waves Sturm–Liouville equation Carroll group 

Notes

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].

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Authors and Affiliations

  • P.-M. Zhang
    • 1
  • M. Elbistan
    • 1
  • 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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