 Born-rigid front trajectory from a rear worldline

A front worldline at fixed proper distance $L$ along a rear observer's simultaneity slice is parametrized by the displayed equations, with velocity and Lorentz factor evaluated at the rear event. The front time is locally future-oriented when $1+{\alpha }_{F}L/c^2>0$, where ${\alpha }_F$ is the rear signed proper acceleration. At a zero of that factor the rigid coordinate extension encounters an acceleration-horizon obstruction.