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Extension of trigonometric and hyperbolic functions to vectorial arguments and its application to the representation of rotations and Lorentz transformations

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Abstract

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations bring a Lorentz-invariant fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics.

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Authors and Affiliations

  1. Department of Physics, Kyushu Dental College, Kitakyushu, Japan

    H. Yamasaki

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  1. H. Yamasaki
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Yamasaki, H. Extension of trigonometric and hyperbolic functions to vectorial arguments and its application to the representation of rotations and Lorentz transformations. Found Phys 13, 1139–1154 (1983). https://doi.org/10.1007/BF00728141

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  • DOI: https://doi.org/10.1007/BF00728141

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