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Foundations of Physics

, Volume 42, Issue 4, pp 488–511 | Cite as

Vectorial Form of the Successive Lorentz Transformations. Application: Thomas Rotation

  • Riad ChamseddineEmail author
Article

Abstract

A complete treatment of the Thomas rotation involves algebraic manipulations of overwhelming complexity. In this paper, we show that a choice of convenient vectorial forms for the relativistic addition law of velocities and the successive Lorentz transformations allows us to obtain straightforwardly the Thomas rotation angle by three new methods: (a) direct computation as the angle between the composite vectors of the non-collinear velocities, (b) vectorial approach, and (c) matrix approach. The new expression of the Thomas rotation angle permits us to simply obtain the Thomas precession. Original diagrams are given for the first time.

Keywords

Successive Lorentz transformations Relativistic addition law of velocities Thomas rotation Thomas precession 

Notes

Acknowledgements

I would like to thank the referees for their very helpful comments which have led to improvement of this paper.

References

  1. 1.
    Thomas, L.H.: The motion of the spinning electron. Nature 117, 514 (1926) ADSCrossRefGoogle Scholar
  2. 2.
    Thomas, L.H.: The kinematics of an electron with an axis. Philos. Mag. 3, 1–22 (1927) Google Scholar
  3. 3.
    Silberstein, L.: The Theory of Relativity. Macmillan, London (1924) zbMATHGoogle Scholar
  4. 4.
    Bacry, H.: Lectures on Group Theory and Particle Theory. Gordon and Breach, New York (1977) zbMATHGoogle Scholar
  5. 5.
    Kennedy, W.L.: Thomas rotation: a Lorentz matrix approach. Eur. J. Phys. 23, 235–247 (2002) zbMATHCrossRefGoogle Scholar
  6. 6.
    Ungar, A.A.: Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1, 57–89 (1988) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ungar, A.A.: The relativistic velocity composition paradox and the Thomas rotation. Found. Phys. 19, 1385–1396 (1989) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Ungar, A.A.: Thomas precession and its associated group—like structure. Am. J. Phys. 54, 824–834 (1991) MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Sexl, R., Urbantke, H.K.: Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer, Wien (2001) zbMATHGoogle Scholar
  10. 10.
    Ben-Menahem, A.: Wigner’s rotation revisited. Am. J. Phys. 53, 62–66 (1985) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Mocanu, C.I.: On the relativistic velocity composition paradox and the Thomas rotation. Found. Phys. Lett. 5, 443–456 (1992) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Vigoureux, J.M.: Calculations of the Wigner angle. Eur. J. Phys. 22, 149–155 (2001) zbMATHCrossRefGoogle Scholar
  13. 13.
    Macfarlane, A.J.: On the restricted Lorentz group and groups homomorphically related to it. J. Math. Phys. 3, 1116–1129 (1962) MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Urbantke, H.: Physical holonomy, Thomas precession, and Clifford algebra. Am. J. Phys. 58, 747–750 (1990) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Salingaros, N.: The Lorentz group and the Thomas precession: Exact results for the product of two boosts. J. Math. Phys. 27, 157–162 (1986) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    van Wyk, C.B.: Rotation associated with the product of two Lorentz transformations. Am. J. Phys. 52, 853–854 (1984) ADSCrossRefGoogle Scholar
  17. 17.
    Hestenes, D.: Space–Time Algebra. Gordon and Breach, New York (1966) zbMATHGoogle Scholar
  18. 18.
    Rivas, M., Valle, M.A., Aguirregabiria, J.M.: Composition law and contractions of the Poincare group. Eur. J. Phys. 7, 1–5 (1986) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Farach, H.A., Aharonov, Y., Poole, C.P., Zanette, S.I.: Application of the nonlinear vector product to Lorenz transformations. Am. J. Phys. 47, 247–249 (1979) ADSCrossRefGoogle Scholar
  20. 20.
    Hirshfeld, A.C., Metzger, F.: A simple formula for combining rotations and Lorentz boosts. Am. J. Phys. 54, 550–552 (1986) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Belloni, L., Reina, C.: Sommerfeld’s way to the Thomas precession. Eur. J. Phys. 7, 55–61 (1986) CrossRefGoogle Scholar
  22. 22.
    Fahnline, D.E.: A covariant four-dimensional expression for Lorentz transformations. Am. J. Phys. 50, 818–821 (1982) ADSCrossRefGoogle Scholar
  23. 23.
    Møller, C.: The Theory of Relativity. Oxford University Press, New York (1952) Google Scholar
  24. 24.
    Taylor, E.F., Wheeler, J.A.: Spacetime Physics, Introduction to Special Relativity. Freeman, New York (1992) Google Scholar
  25. 25.
    Sears, F.W., Zemansky, M.W., Young, H.D.: University Physics. Addison-Wesley, Reading (1979) Google Scholar
  26. 26.
    Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1980) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Sciences-ILebanese UniversityBeirutLebanon

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