Thomas rotation and the parametrization of the Lorentz transformation group


Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical observables: the (3-dimensional) relative velocity and orientation between inertial frames. The resulting parametrization of the Lorentz group, in turn, enables the composition of successive Lorentz transformations to be given by parameter composition. This composition is continuously deformed into a corresponding, well-known Galilean transformation composition by letting the speed of light approach infinity. Finally, as an application the Lorentz transformation with given orientation parameter is uniquely expressed in terms of an initial and a final time-like 4-vector.

This is a preview of subscription content, log in to check access.

References and Notes

  1. 1.

    L. H. Thomas,Nature,117, 514 (1926);Phil. Mag. 3, 1 (1927).

    Google Scholar 

  2. 2.

    Discussed by many authors, see for instance H. Goldstein(6); W. H. Weihofen,Am. J. Phys. 43, 39 (1975); S. Margulies,Am. J. Phys. 50, 434 (1980); D. E. Fahnline,Am. J. Phys. 50, 818 (1982); C. B. van Wyk,Am. J. Phys. 52, 853 (1984); and A. C. Hirshfeld and F. Metzger,Am. J. Phys. 54, 550 (1986).

    Google Scholar 

  3. 3.

    N. Salingaros, J. Math. Phys.27, 157 (1986), and references therein.

    Google Scholar 

  4. 4.

    E. P. Wigner,Ann. Math. 40, 149 (1939). For some more refs. on the Wigner rotation, see for instance E. C. G. Sudarshan and N. Mukunda,Classical Dynamics: A Modern Perspective, (Wiley, New York, 1974), S. Gasiorowicz,Elementary Particle Physics (Wiley, New York, 1967), and refs. therein; and refs. 8,9,14,15. It seems that the termWigner rotation, used by several authors to describe the rotation that we callThomas rotation, was introduced into the English literature from German literature by Gasiorowicz. An objection to the use of this term to describe the Thomas rotation is expressed in n. 4 of ref. 14.

    Google Scholar 

  5. 5.

    M. C. Møller,The Theory of Relativity, pp. 53–56 (Clarendon Press, Oxford, 1952).

    Google Scholar 

  6. 6.

    H. Goldstein,Classical Mechanics, pp. 285–288, 2nd edn. (Addison-Wesley, Menlo-Park, California, 1980).

    Google Scholar 

  7. 7.

    D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966).

    Google Scholar 

  8. 8.

    C. B. van Wyk,Am. J. Phys. 52, 853 (1984).

    Google Scholar 

  9. 9.

    A. Ben-Menahem,Am. J. Phys. 53, 62 (1985).

    Google Scholar 

  10. 10.

    M. Rivas, M. A. Valle and J. M. Aguirregabiria,Eur. J. Phys. 7, 1 (1986).

    Google Scholar 

  11. 11.

    A. Chakrabarti,J. Mat. Phys. 5, 1747 (1964); V. I. Ritus,Soviet Phys. JETP 13, 240 (1961); H. P. Stapp,Phys. Rev. 103, 425 (1956); and V. Lalan,C. R. Acad. Sci. (Paris) 236, 2297 (1953).

    Google Scholar 

  12. 12.

    V. S. Varadarajan,Lie Groups, Lie Algebras and their Applications, (Prentice-Hall, Engelwood Cliffs, 1974).

    Google Scholar 

  13. 13.

    W. E. Baylis and G. Jones,J. Mat. Phys. 29, 57 (1988).

    Google Scholar 

  14. 14.

    D. Han, Y. S. Kim and D. Son,J. Mat. Phys. 27, 2228 (1986).

    Google Scholar 

  15. 15.

    K. Chen and C. Pei,Chem. Phys. Lett. 124, 365 (1985).

    Google Scholar 

  16. 16.

    See for instance, in addition to Goldstein,(6) a remark in the paragraph following eq. (39) in J. T. Cushing,Am. J. Phys. 35, 858 (1967).

    Google Scholar 

  17. 17.

    See, for instance, statement no. (3), 2nd paragraph, in P. S. Farago,Am. J. Phys. 35, 246 (1967), according to which “The resultant of two Lorentz transformations in succession is different from the resultant of two Galilean transformations even in the approximationv ≪ c.” Farago needed this statement to explain why the angular velocity of the Thomas rotation is not negligible even when it is associated with nonrelativistic velocities. The correct explanation follows from the fact that the angular velocity,ω T, of the Thomas rotation need not be negligible even whenv/c is negligible, due to the high accelerations that may be involved in orbital motions.

    Google Scholar 

  18. 18.

    J. M. Lévy-Leblond, inGroup Theory and its Applications, Vol. 2, E. M. Loebl ed. (Academic Press, New York, 1971), pp. 221–299, where additional references may be found.

    Google Scholar 

  19. 19.

    J. Krause,J. Mat. Phys. 18, 889 (1977).

    Google Scholar 

  20. 20.

    D. M. Fradkin,J. Mat. Phys. 23, 2520 (1982).

    Google Scholar 

  21. 21.

    C. B. van Wyk,J. Mat. Phys. 27, 1311 (1986).

    Google Scholar 

  22. 22.

    M. C. Møller,The Theory of Relativity, p. 42, (Clarendon Press, Oxford, 1952). A simple derivation of the pure Lorentz transformation, in a vector form, may be found in W. Pauli,Theory of Relativity, p. 10, (Pergamon Press, New York, 1958). He mentions an earlier writer in whom the boost matrixB(v) can be found: Equation (9) on p. 497 in G. Herglotz,Ann. Phys. (Leipzig) 36, 393 (1911).

    Google Scholar 

  23. 23.

    Calculations of the decomposition in eq. (11) can be found, for instance, in F. R. Halpern,Special Relativity and Quantum Mechanics (Prentice-Hall, Englewod Cliffs, NJ, 1968), Appendix 3; and in ref. 2, D. E. Fahnline. See also ref. 6, H. Goldstein, Prob. 13, p. 336 and refs. 8–11,13,14.

    Google Scholar 

  24. 24.

    Citation from G. E. Uhlenbeck,Phys. Today 29, 43 (June 1976).

    Google Scholar 

  25. 25.

    See n. 4 in ref. 14.

  26. 26.

    Equations equivalent to eq. (13) for the Thomas rotation are common in the literature, see for instance, eq. (60) in M. C. Møller,The Theory of Relativity, p. 55, (Clarendon Press, Oxford, 1952). For further understanding of composite Lorentz transformations one must study properties of the Thomas rotation, tom[u; v], that are not readily obtainable from eq. (13).

    Google Scholar 

  27. 27.

    The definition of the Thomas rotation in eq. (11) is identical with the definition of the Wigner rotation made by several authors;(3,7,9,10) see for instance eq. (11) in Rivaset al. (10) Objection for this use of the termWigner rotation is expressed by Han, Kim, and Son.(25) Some authors define the Wigner (or Thomas or, simply, space) rotation slightly different, describing a composite boost as a boostfollowed, rather thanpreceded, by a Wigner rotation, as in Fahnline(2) and in Baylis and Jones.(13) This slightly different definitions of the Wigner rotation do not conflict, as seen from eq. (39) or from eq. (xii) of Section 6.

  28. 28.

    Elegant derivations of therhs of eq. (19) corresponding to ωθ ≠ 0 can be found in J. Mathew,Am. J. Phys. 44, 1210 (1976), and in J. P. Fillmore, IEEEComp. Graph. 4, 30 (1984). See also A. E. Fekete,Real Linear Algebra (Dekker, New York, 1985) pp. 293 and 347 for a version attributed to N. E. Steenrod.

    Google Scholar 

  29. 29.

    For the theory of Cartesian tensors see, for instance, G. Temple,Cartesian Tensors (Wiley, New York, 1960); H. JeffreysCartesian Tensors, (Cambridge Univ. Press, Cambridge, 1965); and E. C. Young,Vector and Tensor Analysis (Dekker, New York, 1978), Chap. 5.

    Google Scholar 

  30. 30.

    See, for instance, R. H. Rand,Computer Algebra in Applied Mathematics: An Introduction to MACSYMA, (Pitman, Boston, 1984).

    Google Scholar 

  31. 31.

    A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, to appear.

  32. 32.

    See, for instance, Ben Menahem(9) and G. P. Fisher,Am. J. Phys. 40, 1772 (1972).

    Google Scholar 

  33. 33.

    C. I. Mocanu,Rev. Roum. Techn. - Electrotechn. Energ. 30, 119 and 367 (1985) and references therein.

    Google Scholar 

  34. 34.

    For the use of quaternions to describe rotations see, for instance, L. Brand,Vector and Tensor Analysis (Wiley, New York, 1947), pp. 403–427, and L. A. Pars,A Treatise on Analytical Dynamics, (Wiley, New york, 1965), pp. 90-107.

    Google Scholar 

  35. 35.

    J. Wittenburg,Dynamics of Systems of Rigid Bodies, (Teubner, Stuttgart, 1977), pp. 23–25.

    Google Scholar 

  36. 36.

    For an excellent demonstration of the applicability of the quaternion group in modern physics and extensive relevant bibliography see P. R. Girard,Eur. J. Phys. 5, 25 (1984).

    Google Scholar 

  37. 37.

    J. L. Synge,Relativity: The Special Theory, (North-Holland, Amsterdam, 1967), 2nd ed., p. 79.

  38. 38.

    A. C. Hirshfeld and F. Metzger,Am. J. Phys. 54, 550 (1986).

    Google Scholar 

  39. 39.

    For some other elementary, interesting examples concerning one-parameter matrices see D. Kalman and A. Ungar,Am. Math. Month. 94, 21 (1987), and D. Kalman,Math. Mag. 58, 23 (1982).

    Google Scholar 

  40. 40.

    For such a Galilean transformation in two space dimensions see, for instance, I. M. Yaglom,A simple Non-Euclidean Geometry and its Physical Basis (trans. by A. Shenitzer) (Springer, New York, 1979) p. 20 and ref. 18.

    Google Scholar 

  41. 41.

    Y. S. Kim and M. E. Noz,Theory and Applications of the Poincare Group (Reidel, Boston, 1986), p. 215.

    Google Scholar 

  42. 42.

    The composition law in eq. (58) for the homogeneous Galilean transformation may be found, for instance, in eq. (2.8) of ref. 18; in eq. (I. 3) of J. M. Lévy-Leblond,J. Mat. Phys. 4, 776 (1963); in Vilenkin(43); in Cornwell(43); and in J. Voisin,J. Mat. Phys. 6, 1519 (1965).

    Google Scholar 

  43. 43.

    See, for instance, N. J. Vilenkin,Special Functions and the Theory of Group Representations, (trans. V. N. Singh) (Amer. Math. Soc. Providence, Rhode Island, 1968), p. 197, and J. F. Cornwell,Group Theory in Physics (Academic Press, New York, 1984), Vol. I.

    Google Scholar 

  44. 44.

    The need to consider an orientation parameter in addition to the velocity parameter in the parametrization of the Lorentz transformation in 1+3 dimensions is not well known; see for instance R. Skinner,Relativity for Scientists and Engineers, Dover, New York, 1982. In his eq. (1.194) and Figure 1.109, pp. 109-110, Skinner presents two successive Lorentz transformations parametrized by nonparallel velocities giving rise to an equivalent Lorentz transformation parametrized by velocity, thus ignoring the coordinate rotation involved.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ungar, A.A. Thomas rotation and the parametrization of the Lorentz transformation group. Found Phys Lett 1, 57–89 (1988).

Download citation

Key words

  • Special theory of relativity
  • Lorentz transformation
  • parametrization
  • Thomas rotation