## 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.

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## References and Notes

- 1.
L. H. Thomas,

*Nature*,**117**, 514 (1926);*Phil. Mag.***3**, 1 (1927). - 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). - 3.
N. Salingaros, J. Math. Phys.

**27**, 157 (1986), and references therein. - 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. - 5.
M. C. Møller,

*The Theory of Relativity*, pp. 53–56 (Clarendon Press, Oxford, 1952). - 6.
H. Goldstein,

*Classical Mechanics*, pp. 285–288, 2nd edn. (Addison-Wesley, Menlo-Park, California, 1980). - 7.
D. Hestenes,

*Space-Time Algebra*(Gordon & Breach, New York, 1966). - 8.
C. B. van Wyk,

*Am. J. Phys.***52**, 853 (1984). - 9.
A. Ben-Menahem,

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M. Rivas, M. A. Valle and J. M. Aguirregabiria,

*Eur. J. Phys.***7**, 1 (1986). - 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). - 12.
V. S. Varadarajan,

*Lie Groups, Lie Algebras and their Applications*, (Prentice-Hall, Engelwood Cliffs, 1974). - 13.
W. E. Baylis and G. Jones,

*J. Mat. Phys.***29**, 57 (1988). - 14.
D. Han, Y. S. Kim and D. Son,

*J. Mat. Phys.***27**, 2228 (1986). - 15.
K. Chen and C. Pei,

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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). - 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. - 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. - 19.
J. Krause,

*J. Mat. Phys.***18**, 889 (1977). - 20.
D. M. Fradkin,

*J. Mat. Phys.***23**, 2520 (1982). - 21.
C. B. van Wyk,

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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). - 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. - 24.
Citation from G. E. Uhlenbeck,

*Phys. Today***29**, 43 (June 1976). - 25.
See n. 4 in ref. 14.

- 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). - 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. - 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. - 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. - 30.
See, for instance, R. H. Rand,

*Computer Algebra in Applied Mathematics: An Introduction to MACSYMA*, (Pitman, Boston, 1984). - 31.
A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, to appear.

- 32.
See, for instance, Ben Menahem

^{(9)}and G. P. Fisher,*Am. J. Phys.***40**, 1772 (1972). - 33.
C. I. Mocanu,

*Rev. Roum. Techn. - Electrotechn. Energ.***30**, 119 and 367 (1985) and references therein. - 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. - 35.
J. Wittenburg,

*Dynamics of Systems of Rigid Bodies*, (Teubner, Stuttgart, 1977), pp. 23–25. - 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). - 37.
J. L. Synge,

*Relativity: The Special Theory*, (North-Holland, Amsterdam, 1967), 2nd ed., p. 79. - 38.
A. C. Hirshfeld and F. Metzger,

*Am. J. Phys.***54**, 550 (1986). - 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). - 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. - 41.
Y. S. Kim and M. E. Noz,

*Theory and Applications of the Poincare Group*(Reidel, Boston, 1986), p. 215. - 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). - 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. - 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.

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Ungar, A.A. Thomas rotation and the parametrization of the Lorentz transformation group.
*Found Phys Lett* **1, **57–89 (1988). https://doi.org/10.1007/BF00661317

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### Key words

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