Czechoslovak Journal of Physics

, Volume 55, Issue 2, pp 105–118 | Cite as

On the uniqueness of harmonic coordinates

  • Jiří Bičák
  • Joseph Katz


Harmonic coordinate conditions in stationary asymptotically flat spacetimes with matter sources have more than one solution. The solutions depend on the degree of smoothness of the metric and its first derivatives, which we wish to impose across the material boundary, and on the conditions at infinity and at a suitable point inside the matter. This is illustrated in detail by simple fully solvable examples of static spherically symmetric spacetimes in global harmonic coordinates. Examples of stationary electrovacuum spacetimes described simply in harmonic coordinates are also given. They can represent the exterior fields of material discs.

The use of an appropriate background metric considerably simplifies the calculations.

Key words

harmonic coordinates stationary metrics and their sources 


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  1. [1]
    V. Fock: The Theory of Space, Time and Gravitation, 2nd revised edition, Pergamon Press, Oxford, 1964.Google Scholar
  2. [2]
    L. Blanchet: Living Rev. Rel. 5 (2002) 3.Google Scholar
  3. [3]
    L. Blanchet, T. Damour, and G. Esposito-Farèse: Phys. Rev. D 69 (2004) 124007.Google Scholar
  4. [4]
    M. Babiuc, B. Szilágyi, and J. Winicour: Some mathematical problems in numerical relativity, 21 April 2004, gr-qc/0404092.Google Scholar
  5. [5]
    C. Bona, T. Ledvinka, C. Palenzuela, and M. Žáček: Phys. Rev. D 67 (2003) 104005.Google Scholar
  6. [6]
    H. Friedrich and A. D. Rendall: in Einstein’s field equations and their physical implications — Selected essays in honour of Jürgen Ehlers (Ed. B. G. Schmidt), Lec. Notes Phys. Vol. 540, Springer-Verlag, Berlin-Heidelberg, 2000, p. 127.Google Scholar
  7. [7]
    H. Lindblad and I. Rodnianski: Global existence for the Einstein vacuum equations in wave coordinates, 29 Dec. 2003, math. AP/0312479; submitted to Commun. Math. Phys.Google Scholar
  8. [8]
    Q.-H. Liu: J. Math. Phys. 39 (1998) 6086.Google Scholar
  9. [9]
    S. Weinberg: Gravitation and Cosmology: Principles and Applications of the General heory of Relativity, Wiley, New York, 1972.Google Scholar
  10. [10]
    L. Á. Gergely: J. Math. Phys. 40 (1999) 4177.Google Scholar
  11. [11]
    Q.-H. Liu: J. Math. Phys. 40 (1999) 4179.Google Scholar
  12. [12]
    J.M. Aguirregabiria, Ll. Bel, J. Martín, A. Molina, and E. Ruiz: Gen. Rel. Grav. 33 (2001) 1809.Google Scholar
  13. [13]
    J.L. Hernández-Pastora, J. Martín, and E. Ruiz: Admissible Lichnerowicz coordinates for the Schwarzschild metric, 21 Jan 2002, gr-qc/0109031; in Advances in General Relativity and Cosmology (in memory of A. Lichnerowicz), (Ed. G. Ferrarese), Pitagora Editrice, Bologna, 2003.Google Scholar
  14. [14]
    J. Martín-Martín, A. Molina, and E. Ruiz: in Proc. 27th Spanish Relativity Meeting, Alicante 2003, Univ. Alicante Press, 2004.Google Scholar
  15. [15]
    A. Lichnerowicz: Théories relativistes de la Gravitation et de l’Électromagnétisme, Masson, Paris, 1955.Google Scholar
  16. [16]
    K. Heun: Math. Ann. 33 (1889) 161. E. L. Ince: Ordinary Differential Equations, Dover, New York, 1956, p. 394.Google Scholar
  17. [17]
    H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers, and E. Herl: Exact Solutions to Einstein’s Field Equations, Second edition, Cambridge University Press, Cambridge, 2003.Google Scholar
  18. [18]
    B.V. Ivanov: Phys. Rev. D 65 (2002) 104001.Google Scholar
  19. [19]
    W.B. Bonnor: Mon. Not. Roy. Astron. Soc. 129 (1965) 443.Google Scholar
  20. [20]
    J. Katz, J. Bičák, and D. Lynden-Bell: Class. Quantum Grav. 16 (1999) 4023.Google Scholar
  21. [21]
    E. Ruiz: Gen. Rel. Grav. 18 (1986) 805.Google Scholar
  22. [22]
    M. Abe, S. Ichinose, and N. Nakanishi: Prog. Theor. Phys. 78 (1987) 1186.Google Scholar
  23. [23]
    J. Bičák and T. Ledvinka: Phys. Rev. Let. 71 (1993) 1669.Google Scholar
  24. [24]
    C. Pichon and D. Lynden-Bell: Mon. Not. Roy. Astron. Soc. 280 (1996) 1007.Google Scholar
  25. [25]
    J. Katz, J. Bičák, and D. Lynden-Bell: Phys. Rev. D 55 (1997) 5957.Google Scholar
  26. [26]
    D. Kramer and G. Neugebauer: Annalen Physik 27 (1971) 129.Google Scholar
  27. [27]
    W.B. Bonnor: Class. Quantum Grav. 16 (1999) 4125.Google Scholar
  28. [28]
    W.B. Bonnor: Class. Quantum Grav. 15 (1998) 351.Google Scholar
  29. [29]
    D. Lynden-Bell, J. Bičák, and J. Katz: Ann. Phys. (N. Y.) 271 (1999) 1.Google Scholar
  30. [30]
    H. Pfister and M. King: Class. Quantum Grav. 20 (2003) 205.Google Scholar
  31. [31]
    J. Novak and E. Marcq: Class. Quantum Grav. 20 (2003) 3051.Google Scholar
  32. [32]
    T. Ledvinka, M. Žofka, and J. Bičák: in Proc. 8th M. Grossmann Meeting on Gen. Rel., (Ed. T. Piran), World Scientific, Singapore, 1999, p. 339.Google Scholar
  33. [33]
    G.B. Cook: Living Rev. Rel. 3 (2000) 5.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Jiří Bičák
    • 1
    • 2
  • Joseph Katz
    • 1
    • 3
  1. 1.Institute of AstronomyThe ObservatoriesCambridgeUK
  2. 2.Institute of Theoretical Physics, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  3. 3.Racah Institute of PhysicsThe Hebrew UniversityJerusalemIsrael

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