Application of the double numbers in the representation of the Lorentz transformations
Keywords:Lorentz transformations, double numbers, Wigner angle, Thomas precession
We show that the orthochronous proper Lorentz transformations that leave one of the Cartesian coordinates fixed can be represented by $2 \times 2$ unitary matrices with determinant equal to 1, whose entries are double numbers. This representation is employed in the calculation of the Wigner angle, which arises in the composition of two boosts in arbitrary directions.
G.F. Torres del Castillo, 3-D Spinors, Spin-Weighted Functions and their Applications (Springer, New York, 2003), Sec. 1.1. https://doi.org/10.1007/978-0-8176-8146-3
G.F. Torres del Castillo, Some applications in classical mechanics of the double and the dual numbers, Rev. Mex. F´ıs. E 65 (2019) 152. https://rmf.smf.mx/ojs/rmf-e/author/submission/550
G.F. Torres del Castillo, Applications of the double and the dual numbers. The Bianchi models, Rev. Mex. F´ıs. E 17 (2020) 146. https://doi.org/10.31349/RevMexFisE.17.146
G.F. Torres del Castillo and K.C. Gutiérrez-Herrera, Double and dual numbers. SU(2) groups, two-component spinors and
generating functions, Rev. Mex. F´ıs. 66 (2020) 418. https://doi.org/10.31349/RevMexFis.66.418
W. Rindler, Relativity, Special, General, and Cosmological, 2nd ed. (Oxford University Press, Oxford, 2006).
J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998), Sec. 11.8.
T. Padmanabhan, Sleeping Beauties in Theoretical Physics (Springer, Cham, 2015), Chap. 13. https://doi.org/10.1007/978-3-319-13443-7
C.B. Van Wyk, Rotation associated with the product of two Lorentz transformations, Am. J. Phys. 52 (1984) 853. https://doi.org/10.1119/1.13538
A. Ben-Menachem, Wigner’s rotation revisited, Am. J. Phys. 53 (1985) 62. https://doi.org/10.1119/1.13953
M.W.P. Strandberg, Abstract group theoretical reduction of products of Lorentz-group representations, Phys. Rev. A 34 (1986) 2458. https://doi.org/10.1103/PhysRevA.34.2458
P.K. Aravind, The Wigner angle as an anholonomy in rapidity space, Am. J. Phys. 65 (1997) 634. https://doi.org/10.1119/1.18620
J.A. Rhodes, M.D. Semon, Relativistic velocity space, Wigner rotation and Thomas precession, Am. J. Phys. 72 (2004) 943. https://doi.org/10.1119/1.1652040
G.P. Fisher, The Thomas precession, Am. J. Phys. 40 (1972) 1772. https://doi.org/10.1119/1.1987061
A.A. Ungar, Thomas precession and its associated grouplike structure, Am. J. Phys. 59 (1991) 824. https://doi.org/10.1119/1.16730
How to Cite
Copyright (c) 2023 Gerardo Francisco Torres del Castillo
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.