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Vectorial Form of the Successive Lorentz Transformations. Application: Thomas Rotation

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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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Acknowledgements

I would like to thank the referees for their very helpful comments which have led to improvement of this paper.

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Correspondence to Riad Chamseddine.

Appendix

Appendix

Demonstration of

$$ \tan(\varphi_1-\chi_1)=\gamma\tan(\chi_2+\varphi).$$
(75)

The left side of (75) can be written as

$$ \tan(\varphi_1-\chi_1)=\frac{\tan\varphi_1-\tan\chi_1}{1+\tan\varphi_1\tan\chi_1}.$$
(76)

When substituting tanφ 1 (31) into (76), this equation can be rewritten in the form

$$ \tan(\varphi_1-\chi_1)=\gamma\frac{A\cos\varphi+B\sin\varphi}{C\cos\varphi-D\sin\varphi},$$
(77)

where

(78)

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

(79)
(80)

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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