Stationary trajectories in Minkowski spacetimes

We determine the conjugacy classes of the Poincaré group ${\mathrm{I}\mathrm{S}\mathrm{O}}^{+}(n,1)$ 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.