Abstract
We give a fresh look into the age-old rotational kinematic formula which was originally devised by L. Euler in the eighteenth century. Instead of some verbose explanations for its logical validity, an argument of a covariant differentiation with respect to coordinate time with a tetrad-based relativistic account in an anholonomic frame will be given as a viewpoint that is mathematically sound and self-explanatory. The familiar “ω(t) × q(t)” term is replaced by a linear combination of space-frame fields of a tetrad with ‘Ricci’s connection coefficients’ of infinitesimal generators, so as to be expressed by \(\sum\nolimits_{i,j}^3 { = 1} e(i)c{\gamma ^i}j0\mathop q\limits^{oj} (t)\) with one subscript index set to zero, ‘o’, for the time coordinate. This recognition gives a new interpretation of the “time derivative for the space set of rotated axes” as the formal covariant time derivative in this tetrad-based coordinate transformation of the four-dimensional space-time that is curved due to an implicit Galilean transformation.
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Acknowledgments
I would like to thank my colleagues, especially Prof. B. S. Mun for encouragement for publication. For technical discussions, special thanks go to Profs. D.-H. Kim and K.-Y. Kim. I would appreciate one of the reviewers greatly, who found an error in my hand simplification of Eq. (25) for the Galilean transformation. The work was supported in part by the GIST Research Institute Fund.
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Song, G.H. The Rotational Kinematic Formula Viewed from a Tetrad-based Anholonomic Frame. J. Korean Phys. Soc. 76, 357–367 (2020). https://doi.org/10.3938/jkps.76.357
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DOI: https://doi.org/10.3938/jkps.76.357