Il Nuovo Cimento (1955-1965)

, Volume 34, Issue 2, pp 317–339 | Cite as

A covariant multipole formalism for extended test bodies in general relativity

  • W. G. Dixon
Article

Summary

A discussion and criticism is given of various forms that have been put forward for the multipole theory of an extended test body in curved space-time, and a new treatment is proposed, in which the effect of an electromagnetic field is included and in which the covariance is retained throughout by the use of bitensors. Included in this is a covariant definition of the world-line of the mass centre. Equations of motion are derived in the pole-dipole approximation and a comparison is made with the previous theories. It is seen that, although the new theory avoids the defects mentioned in the former theories, solutions of our equations satisfy those of the former theories if terms quadratic in the spin are neglected.

Riassunto

Si discutono e si criticano le varie forme, che sono state esposte sinora, della teoria multipolare di un corpo di prova esteso nello spazio-tempo curvo, e si propone un nuovo trattamento, in cui si tien conto dell’effetto di un campo elettromagnetico nel quale si mantiene dappertutto l’evidente covarianza con l’uso di bitensori. In questo si comprende una definiziore covariante della linea universale del centro di massa. si deducono le equazioni del moto nell’approssimazione polo-dipolo e si fa un confronto con le teorie precedenti. Si vede che, sebbene la nuova teoria eviti i difetti menzionati nelle precedenti teorie, le soluzioni delle nostre equazioni soddisfano quelle delle teorie precedenti se si trascurano termini quadratici nello spin.

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Literatur

  1. (1).
    M. Mathisson:Acta Phys. Polon.,6, 163 (1937).Google Scholar
  2. (2).
    We have adopted a slightly different notation from that of the papers quoted. For details of the conventions and notation used here, see Sect.2.Google Scholar
  3. (3).
    J. Weyssenhoff andA. Raabe:Acta Phys. Polon.,9, 7 (1947).MathSciNetGoogle Scholar
  4. (4).
    A. Papapetrou:Proc. Roy. Soc.,A 209, 248 (1951).MathSciNetADSCrossRefGoogle Scholar
  5. (5).
    E. Corinaldesi andA. Papapetrou:Proc. Roy. Soc.,A 209, 259 (1951).MathSciNetADSCrossRefGoogle Scholar
  6. (6).
    W. Tulczyjew:Acta Phys. Polon.,18, 393 (1959).MathSciNetMATHGoogle Scholar
  7. (7).
    B. Tulczyjew andW. Tulczyjew: article inRecent Developments in General Relativity (London, 1962), p. 465.Google Scholar
  8. (8).
    C. Møller:Comm. Dublin Inst. Adv. Studies, A 5, (1949), has shown that an extended body of positive definite energy density and of given rotational angular momentum about its mass centre must be larger than a certain minimum size.Google Scholar
  9. (9).
    Loc. cit. ref. (8).Google Scholar
  10. (10).
    D. Bohm andJ.-P. Vigier:Phys. Rev.,109, 1882 (1958).MathSciNetADSCrossRefMATHGoogle Scholar
  11. (11).
    J. A. Schouten:Ricci-Calculus. An Introduction to Tensor Analysis and its Geometrical Applications, 2nd. ed. (Berlin, 1954).Google Scholar
  12. (12).
    B. S. DeWitt andR. W. Brehme:Ann. Phys. (N. Y.),9, 220 (1960). See also Chapter II ofJ. L. Synge:Relativity: The General Theory (Amsterdam, 1960).MathSciNetADSCrossRefGoogle Scholar
  13. (13).
  14. (15).
    For a thorough study of various possibilities, seeM. H. L. Pryce:Proc. Roy. Soc.,A 195, 62 (1948), and alsoC. Møller: loc. cit. ref. (8).MathSciNetADSCrossRefGoogle Scholar
  15. (18).
    See Sect.2 and footnotes (12, 14).Google Scholar
  16. (20).
    Loc. cit. ref. (15).Google Scholar
  17. (21).
    We here follow the definition of spin given byD. W. Sciama: article inRecent Developments in General Relativity (London, 1962), p. 415. See alsoT. W. B. Kibble:Journ. Math. Phys.,2, 212 (1961).MathSciNetADSCrossRefMATHGoogle Scholar
  18. (22).
    L. Rosenfeld:Mem. Acad. Roy. Belg.,18, 6 (1940). See also ref. (21).MathSciNetGoogle Scholar
  19. (23).
    R. Utiyama:Phys. Rev.,101, 1597 (1956).MathSciNetADSCrossRefMATHGoogle Scholar
  20. (24).
    SeeL. Rosenfeld: loc. cit. ref. (22).MathSciNetCrossRefMATHGoogle Scholar
  21. (26).
    SeeJ. A. Schouten: loc. cit. ref. (11), Chapter II, Sect.10.MathSciNetCrossRefGoogle Scholar
  22. (27).
    For the technique of obtaining these covariant expansions, seeB. S. DeWitt andR. W. Brehme: loc. cit. ref. (12).MathSciNetCrossRefGoogle Scholar
  23. (31).
    L. I. Schiff:Phys. Rev. Lett.,4, 215 (1962).ADSCrossRefGoogle Scholar
  24. (32).
    A. Peres:Nuovo Cimento,28, 1091 (1963).CrossRefMATHGoogle Scholar
  25. (33).
    See,e.g.,J. L. Synge: loc. cit. ref. (12).CrossRefGoogle Scholar
  26. (34).
    SeeB. S. DeWitt andR. W. Brehme: loc. cit. ref. (12).CrossRefMATHGoogle Scholar

Copyright information

© Proprietà Letteraria Riservata 1964

Authors and Affiliations

  • W. G. Dixon
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridge

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