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.
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References and Notes
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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).
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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.
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See n. 4 in ref. 14.
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).
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.
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.
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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.
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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).
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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
- Special theory of relativity
- Lorentz transformation
- Thomas rotation