Relative binary and ternary 4D velocities in the Special Relativity in terms of manifestly covariant Lorentz transformation

TL;DR

This paper reviews and generalizes Oziewicz's 4D covariant relative velocities in special relativity, revealing their ternary nature and the role of Lorentz transformations as inherently ternary operations.

Contribution

It provides a comprehensive review and new generalizations of Oziewicz's binary and ternary 4D relative velocities, emphasizing their ternary structure and covariant Lorentz transformations.

Findings

Lorentz transformation of velocity is inherently ternary.

Oziewicz's relative velocities are mathematically precise and physically logical.

The work unifies various velocity concepts under a covariant framework.

Abstract

Zbigniew Oziewicz was a pioneer of the 4D space-time approach to covariant relative velocities. In the 1980s (according to private correspondence) he discovered two types of 4D relative velocities: binary and ternary, along with the rules for adding them. They were first published in conference materials in 2004, and the second time in a peer-reviewed journal in 2007. These physically logical and mathematically precise concepts are so subtle that Oziewicz's numerous preprints have yet to receive the recognition they deserve. This work was planned to be a more review, but a thorough review of the little-known results was made in an original synthetic manner with numerous generalizations. The Part I presents the Oziewicz-\'Swierk-Bol\'os (and Matolsci or Bini-Carini-Jantzen) binary relative velocity and the Oziewicz-Ungar-Dragan (also Celakoska-Chakmakov-Petrushevski on the basis of Urbantke, as well as Wyk) ternary relative velocity. The Einstein-Oziewicz and Einstein-Minkowski velocities, which are a four-dimensional generalization of Einstein velocities addition, also have a ternary character. This also applies to Oziewicz-Minkowski relative velocity, which is a time-like equivalent (generalization) of the canonical ternary velocity. It turns out that the Lorentz transformation of velocity itself is already ternary anchored. This fact is explicitly revealed by the manifestly covariant Lorentz transformation of velocity, which is the main tool of the work.