Stationary trajectories in Minkowski spacetimes

TL;DR

This paper classifies stationary trajectories in Minkowski spacetimes by analyzing Poincaré group conjugacy classes and extends Frenet-Serret equations to higher dimensions, providing explicit solutions in 4+1 spacetime.

Contribution

It extends the classification of stationary trajectories and Frenet-Serret equations to arbitrary dimensions, including explicit solutions in 4+1 Minkowski spacetime.

Findings

Classified conjugacy classes of the Poincaré group.

Extended Frenet-Serret equations to higher dimensions.

Explicit stationary trajectories in 4+1 Minkowski spacetime.

Abstract

We determine the conjugacy classes of the Poincar\'e group $\mathrm{ISO}^+(n,1)$ and apply this to classify the stationary trajectories of Minkowski spacetimes in terms of timelike Killing vectors. Stationary trajectories are the orbits of timelike Killing vectors and, equivalently, the solutions to Frenet-Serret equations with constant curvature coefficients. We extend the $3+1$ Minkowski spacetime Frenet-Serret equations due to Letaw to Minkowski spacetimes of arbitrary dimension. We present the explicit families of stationary trajectories in $4+1$ Minkowski spacetime.