World-Line Perturbation Theory

  • Jan-Willem van HoltenEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 196)


The motion of a compact body in space and time is commonly described by the world line of a point representing the instantaneous position of the body. In General Relativity such a world-line formalism is not quite straightforward because of the strict impossibility to accommodate point masses and rigid bodies. In many situations of practical interest it can still be made to work using an effective hamiltonian or energy-momentum tensor for a finite number of collective degrees of freedom of the compact object. Even so exact solutions of the equations of motion are often not available. In such cases families of world lines of compact bodies in curved space-times can be constructed by a perturbative procedure based on generalized geodesic deviation equations. Examples for simple test masses and for spinning test bodies are presented.



I am indebted to Richard Kerner, Roberto Collistete jr., Gideon Koekoek, Giuseppe d’Ambrosi, S. Satish Kumar and Jorinde van de Vis for pleasant and informative discussions and collaboration on various aspects of the topics discussed. This work is supported by the Foundation for Fundamental Research of Matter (FOM) in the Netherlands.

Supplementary material


  1. 1.
    C. Møller, Sur la dynamique des systèmes ayant un moment angulaire interne. Ann. Inst. Henri Poincaré 11, 251 (1949)zbMATHGoogle Scholar
  2. 2.
    C. Møller, The Theory of Relativity (Clarendon Press, Oxford, 1952)zbMATHGoogle Scholar
  3. 3.
    L.F. Costa, J. Natário, Center of mass, spin supplementary conditions, and the momentum of spinning particles. Fund. Theor. Phys. 179, 215 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)Google Scholar
  5. 5.
    E. Hackmann, Geodesic equations and algebro-geometric methods (2015), arXiv:1506.00804v1 [gr–qc]
  6. 6.
    R. Kerner, J.W. van Holten, R. Collistete jr., Relativistic epicycles: another approach to geodesic deviations. Class. Quantum Gravity 18, 4725 (2001)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Colistete Jr., C. Leygnac, R. Kerner, Higher-order geodesic deviations applied to the Kerr metric. Class. Quantum Gravity 19, 4573 (2002)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Ehlers, F.A.E. Pirani, A. Schild, The geometry of free fall and light propagation, in General Relativity: Papers in Honnor of J.L. Synge, ed. by L. O’Raifeartaigh (Oxford University Press, Oxford, 1972)Google Scholar
  9. 9.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (Freeman and Co., San Francisco, 1970)Google Scholar
  10. 10.
    P. Szekeres, The gravitational compass. J. Math. Phys. 6, 1387 (1965)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Puetzfeld, Y.N. Obukhov, Generalized deviation equation and determination of the curvature in general relativity. Phys. Rev. D 93, 044073 (2016)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    D. Philipp, D. Puetzfeld, C. Lämmerzahl, On the applicability of the geodesic deviation equation in general relativity (2016), arXiv:1604.07173 [gr–qc]
  13. 13.
    G. Koekoek, J.W. van Holten, Epicycles and Poincaré resonances in general relativity. Phys. Rev. D 83, 064041 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    G. Koekoek, J.W. van Holten, Geodesic deviations: modeling extreme mass-ratio systems and their gravitational waves. Class. Quantum Gravity 28, 225022 (2011)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Mathison, Neue Mechanik materieller Systeme. Acta Phys. Pol. 6, 163 (1937)zbMATHGoogle Scholar
  16. 16.
    W. Tulczyjew, Motion of multipole particles in general relativity theory. Acta Phys. Pol. 18, 393 (1959)MathSciNetzbMATHGoogle Scholar
  17. 17.
    J. Steinhoff, Canonical formulation of spin in general relativity, Ph.D. thesis (Jena University) (2011), arXiv:1106.4203v1 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    G. d’Ambrosi, S.S. Kumar, J.W. van Holten, Covariant Hamiltonian spin dynamics in curved space-time. Phys. Lett. B 743, 478 (2015)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    J.W. van Holten, Spinning bodies in general relativity. Int. J. Geom. Methods Mod. Phys. 13, 1640002 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    G. d’Ambrosi, S.S. Kumar, J.W. van Holten, J. van de Vis, Spinning bodies in curved space-time. Phys. Rev. D 93, 04451 (2016)CrossRefGoogle Scholar
  21. 21.
    J. Ehlers, E. Rudolph, Dynamics of extended bodies in general relativity: center-of-mass description and quasi-rigidity. Gen. Relativ. Gravit. 8, 197 (1977)ADSCrossRefGoogle Scholar
  22. 22.
    R. Ruediger, Conserved quantities of spinning test particles in general relativity. Proc. R. Soc. Lond. A375, 185 (1981)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    I. Khriplovich, A. Pomeransky, Equations of motion for spinning relativistic particle in external fields. Surv. High Energy Phys. 14, 145 (1999)ADSCrossRefGoogle Scholar
  24. 24.
    J.W. van Holten, On the electrodynamics of spinning particles. Nucl. Phys. B 356, 3–26 (1991)ADSCrossRefGoogle Scholar
  25. 25.
    J.W. van Holten, Relativistic time dilation in an external field. Phys. A 182, 279 (1992)CrossRefGoogle Scholar
  26. 26.
    G. d’Ambrosi, J.W. van Holten, Ballistic orbits in Schwarzschild space-time and gravitational waves from EMR binary mergers. Class. Quantum Gravity 32, 015012 (2015)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.NikhefAmsterdamThe Netherlands

Personalised recommendations