# Thomas rotation and the parametrization of the Lorentz transformation group

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## Abstract

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-known*Thomas 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.

## Key words

Special theory of relativity Lorentz transformation parametrization Thomas rotation## Preview

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## References and Notes

- 1.
- 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.N. Salingaros, J. Math. Phys.
**27**, 157 (1986), and references therein.Google Scholar - 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 term*Wigner rotation,*used by several authors to describe the rotation that we call*Thomas 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.M. C. Møller,
*The Theory of Relativity*, pp. 53–56 (Clarendon Press, Oxford, 1952).Google Scholar - 6.H. Goldstein,
*Classical Mechanics*, pp. 285–288, 2nd edn. (Addison-Wesley, Menlo-Park, California, 1980).Google Scholar - 7.D. Hestenes,
*Space-Time Algebra*(Gordon & Breach, New York, 1966).Google Scholar - 8.
- 9.
- 10.
- 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.V. S. Varadarajan,
*Lie Groups, Lie Algebras and their Applications*, (Prentice-Hall, Engelwood Cliffs, 1974).Google Scholar - 13.
- 14.
- 15.
- 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.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 approximation*v ≪ 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 when*v/c*is negligible, due to the high accelerations that may be involved in orbital motions.Google Scholar - 18.J. M. Lévy-Leblond, in
*Group 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.
- 20.
- 21.
- 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 matrix*B*(v) can be found: Equation (9) on p. 497 in G. Herglotz,*Ann. Phys. (Leipzig)***36**, 393 (1911).Google Scholar - 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.
- 25.See n. 4 in ref. 14.Google Scholar
- 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.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 Rivas*et al.*^{(10)}Objection for this use of the term*Wigner 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 boost*followed,*rather than*preceded,*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.Google Scholar - 28.Elegant derivations of the
*rhs*of eq. (19) corresponding to ω_{θ}≠ 0 can be found in J. Mathew,*Am. J. Phys.***44**, 1210 (1976), and in J. P. Fillmore, IEEE*Comp. 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.For the theory of Cartesian tensors see, for instance, G. Temple,
*Cartesian Tensors*(Wiley, New York, 1960); H. Jeffreys*Cartesian Tensors*, (Cambridge Univ. Press, Cambridge, 1965); and E. C. Young,*Vector and Tensor Analysis*(Dekker, New York, 1978), Chap. 5.Google Scholar - 30.See, for instance, R. H. Rand,
*Computer Algebra in Applied Mathematics: An Introduction to MACSYMA*, (Pitman, Boston, 1984).Google Scholar - 31.A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, to appear.Google Scholar
- 32.
- 33.C. I. Mocanu,
*Rev. Roum. Techn. - Electrotechn. Energ.***30**, 119 and 367 (1985) and references therein.Google Scholar - 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.J. Wittenburg,
*Dynamics of Systems of Rigid Bodies*, (Teubner, Stuttgart, 1977), pp. 23–25.Google Scholar - 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.J. L. Synge,
*Relativity: The Special Theory*, (North-Holland, Amsterdam, 1967), 2nd ed., p. 79.Google Scholar - 38.
- 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.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.Y. S. Kim and M. E. Noz,
*Theory and Applications of the Poincare Group*(Reidel, Boston, 1986), p. 215.Google Scholar - 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.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.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