Introduction of the Lie group of Lorentz matrices in Special Relativity. Tangent boost along a worldline and its associated matrix in the Lie algebra. Applications

TL;DR

This paper extends the concept of Lorentz boosts to non-inertial frames and general relativity using Lie groups and algebras, providing new tools for analyzing relativistic systems and phenomena.

Contribution

It introduces a generalized notion of tangent boost along worldlines and explores the Lie algebra structure of Lorentz matrices, including applications to the Thomas rotation and relativistic scenarios.

Findings

Defined tangent boost along a worldline for non-inertial particles.

Showed the Lie group of Lorentz matrices has four one-parameter subgroups.

Applied the framework to examples like electron orbits and the twin paradox.

Abstract

In order to generalize the relativistic notion of boost to the case of non inertial particles and to general relativity, we come back to the definition of Lie group of Lorentz matrices and its Lie algebra and we study how this group acts on the Minskowski space. We thus define the notion of tangent boost along a worldline. This notion very general notion gives a useful tool both in special relativity (for non inertial particles or/and for non rectilinear coordinates) and in general relativity. We also introduce a matrix of the Lie algebra which, together with the tangent boost, gives the whole dynamical description of the considered system (acceleration and Thomas rotation). After studying the properties of Lie algebra matrices and of their reduced forms, we show that the Lie group of special Lorentz matrices has four one-parameter subgroups. These tools lead us to introduce the Thomas rotation in a quite general way. At the end of the paper, we present some examples using these tools and we consider the case of an electron rotating on a circular orbit around an atom nucleus. We then discuss the twin paradox and we show that when the one who made a journey into space in a high-speed rocket returns home he is not only younger than the twin who stayed on Earth but he is also disorientated because his gyroscope has turned with respect to earth referential frame.