Orthonormal Basis in Minkowski Space

TL;DR

This paper explores the structure of orthonormal bases in Minkowski space within Finsler geometry, analyzing transformations and group actions to understand reference frames in such spaces.

Contribution

It defines orthonormal bases in Minkowski space, examines their transformation properties, and introduces the concepts of motions and quasimotions, contributing to the geometric understanding of Finsler spaces.

Findings

Set of motions forms a not complete group SO(V)

Passive transformations generate a passive representation of SO(V)

Active and passive representations are single transitive

Abstract

Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, we considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Linear transformation of Minkowski space mapping at least one orthonormal basis into orthonormal basis is called motion. The set of motions of Minkowski space V generates not complete group SO(V) which acts single transitive on the basis manifold. Passive transformation of Minkowski space mapping at least one orthonormal basis into orthonormal basis is called quasimotion of Minkowski space. The set of passive transformations of Minkowski space generates passive representation of not complete group SO(V) on basis manifold. Since twin representations (active and passive) of not complete group SO(V) on basis manifold are single transitive, then we may consider definition of geometric object.