Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group

TL;DR

This paper generalizes Chasles' theorem to special relativity, showing that certain spacetime isometries can be generated by observers with constant acceleration and angular velocity, and classifies conjugacy classes of the inhomogeneous Lorentz group.

Contribution

It provides a relativistic extension of Chasles' theorem, classifies conjugacy classes of the inhomogeneous Lorentz group, and links these to physical significance in Minkowski spacetime.

Findings

Proves exponentiality of the proper orthochronous inhomogeneous Lorentz group.

Identifies a causal semigroup and Lie cone within the group.

Classifies conjugacy classes with physical relevance.

Abstract

This work is devoted to the relativistic generalization of Chasles' theorem, namely to the proof that every proper orthochronous isometry of Minkowski spacetime, which sends some point to its chronological future, is generated through the frame displacement of an observer which moves with constant acceleration and constant angular velocity. The acceleration and angular velocity can be chosen either aligned or perpendicular, and in the latter case the angular velocity can be chosen equal or smaller than than the acceleration. We start reviewing the classical Euler's and Chasles' theorems both in the Lie algebra and group versions. We recall the relativistic generalization of Euler's theorem and observe that every (infinitesimal) transformation can be recovered from information of algebraic and geometric type, the former being identified with the conjugacy class and the latter with some additional geometric ingredients (the screw axis in the usual non-relativistic version). Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in detail. We prove its exponentiality and identify a causal semigroup and the corresponding Lie cone. Through the identification of new Ad-invariants we classify the conjugacy classes, and show that those which admit a causal representative have special physical significance. These results imply a classification of the inequivalent Killing vector fields of Minkowski spacetime which we express through simple representatives. Finally, we arrive at the mentioned generalization of Chasles' theorem.