Lorentz invariants in particle-wave mechanical systems
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2024
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Hill, J.M.
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Zeitschrift fur Angewandte Mathematik und Physik, 2024; 75(6):1-13
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In these notes we detail a number of new results involving the Lorentz invariants associated with the special relativistic extension of Newton’s second law proposed in and not included in that text. We first summarise existing results for the two Lorentz invariants ξ and η and the angle θ which is the angle in which Lorentz invariance appears as a translational invariance. We then determine new integral formulae involving these quantities and a new relationship which connects the applied forces with the same invariances. This latter relation turns out to be equivalent to the partial differential equation arising from the invariance of the velocity equation dx/dt=u(x, t) under a Lorentz transformation. We then provide some specific examples which confirm the validity of the new relation connecting the applied forces with the invariances.
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Copyright 2024 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. (https://creativecommons.org/licenses/by/4.0/)
Access Condition Notes: Open Access funding enabled and organized by CAUL and its Member Institutions. The author received no funding for this paper.