Observers with constant proper acceleration, constant proper jerk, and beyond

We discuss in Minkowski spacetime the differences between the concepts of constant proper $n$-acceleration and of vanishing $(n+1)$-acceleration. By $n$-acceleration we essentially mean the higher order time derivatives of the position vector of the trajectory of a point particle, adapted to Minkowski spacetime or eventually to curved spacetime. The $2$-acceleration is known as the Jerk, the $3$-acceleration as the Snap, etc. As for the concept of {\sl proper} $n$-acceleration we give a specific definition involving the instantaneous comoving frame of the observer and we discuss, in such framework, the difficulties in finding a characterization of this notion as a Lorentz invariant statement. We show how the Frenet-Serret formalism helps to address the problem. In particular we find that our definition of an observer with constant proper acceleration corresponds to the vanishing of the third curvature invariant $\kappa_3$ (thus the motion is three dimensional in Minkowski spacetime) together with the constancy of the first and second curvature invariants and the restriction $\kappa_2 < \kappa_1$, the particular case $\kappa_2=0$ being the one commonly referred to in the literature. We generalize these concepts to curved spacetime, in which the notion of trajectory in a plane is replaced by the vanishing of the second curvature invariant $\kappa_2$. Under this condition, the concept of constant proper $n$-acceleration coincides with that of the vanising of the $(n+1)$-acceleration and is characterized by the fact that the first curvature invariant $\kappa_1$ is a $(n-1)$-degree polynomial of proper time. We illustrate some of our results with examples in Minkowski, de Sitter and Schwarzschild spacetimes.