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.
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References
Thomas, L.H.: The motion of the spinning electron. Nature 117, 514 (1926)
Thomas, L.H.: The kinematics of an electron with an axis. Philos. Mag. 3, 1–22 (1927)
Silberstein, L.: The Theory of Relativity. Macmillan, London (1924)
Bacry, H.: Lectures on Group Theory and Particle Theory. Gordon and Breach, New York (1977)
Kennedy, W.L.: Thomas rotation: a Lorentz matrix approach. Eur. J. Phys. 23, 235–247 (2002)
Ungar, A.A.: Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1, 57–89 (1988)
Ungar, A.A.: The relativistic velocity composition paradox and the Thomas rotation. Found. Phys. 19, 1385–1396 (1989)
Ungar, A.A.: Thomas precession and its associated group—like structure. Am. J. Phys. 54, 824–834 (1991)
Sexl, R., Urbantke, H.K.: Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer, Wien (2001)
Ben-Menahem, A.: Wigner’s rotation revisited. Am. J. Phys. 53, 62–66 (1985)
Mocanu, C.I.: On the relativistic velocity composition paradox and the Thomas rotation. Found. Phys. Lett. 5, 443–456 (1992)
Vigoureux, J.M.: Calculations of the Wigner angle. Eur. J. Phys. 22, 149–155 (2001)
Macfarlane, A.J.: On the restricted Lorentz group and groups homomorphically related to it. J. Math. Phys. 3, 1116–1129 (1962)
Urbantke, H.: Physical holonomy, Thomas precession, and Clifford algebra. Am. J. Phys. 58, 747–750 (1990)
Salingaros, N.: The Lorentz group and the Thomas precession: Exact results for the product of two boosts. J. Math. Phys. 27, 157–162 (1986)
van Wyk, C.B.: Rotation associated with the product of two Lorentz transformations. Am. J. Phys. 52, 853–854 (1984)
Hestenes, D.: Space–Time Algebra. Gordon and Breach, New York (1966)
Rivas, M., Valle, M.A., Aguirregabiria, J.M.: Composition law and contractions of the Poincare group. Eur. J. Phys. 7, 1–5 (1986)
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)
Hirshfeld, A.C., Metzger, F.: A simple formula for combining rotations and Lorentz boosts. Am. J. Phys. 54, 550–552 (1986)
Belloni, L., Reina, C.: Sommerfeld’s way to the Thomas precession. Eur. J. Phys. 7, 55–61 (1986)
Fahnline, D.E.: A covariant four-dimensional expression for Lorentz transformations. Am. J. Phys. 50, 818–821 (1982)
Møller, C.: The Theory of Relativity. Oxford University Press, New York (1952)
Taylor, E.F., Wheeler, J.A.: Spacetime Physics, Introduction to Special Relativity. Freeman, New York (1992)
Sears, F.W., Zemansky, M.W., Young, H.D.: University Physics. Addison-Wesley, Reading (1979)
Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1980)
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Appendix
Appendix
Demonstration of
The left side of (75) can be written as
When substituting tanφ 1 (31) into (76), this equation can be rewritten in the form
where
Substituting tanχ 1 from the first of (22a) into these quantities, and taking into account the expressions of sinχ 2 and cosχ 2 given by the second and the third of (22b), respectively, and cosχ 1 given by the third of (22a), we get
Substituting these quantities A, D, B, and C into (77), we obtain (75).
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Chamseddine, R. Vectorial Form of the Successive Lorentz Transformations. Application: Thomas Rotation. Found Phys 42, 488–511 (2012). https://doi.org/10.1007/s10701-011-9617-5
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DOI: https://doi.org/10.1007/s10701-011-9617-5