Digression on the Longitudinal Coordinate Definition

A word is in order about the definition of $s$, which we've described as the total, equivalent distance traveled. First, by total distance we mean that $s$ is not measured relative to the bunch center or a fiducial particle. It is entirely a property of the individual particle and its path through the accelerator.

To explain what we mean by equivalent distance, note that the relationship between $s$ and arrival time $t$ at the observation point is, for each particle, $s = \beta c t$, where $\beta c$ is the instantaneous velocity of the particle. Whenever a particle's velocity changes, elegant recomputes $s$ to ensure that this relationship holds. $s$ is thus the ``equivalent'' distance the particle would have traveled at the present velocity to arrive at the observation point at the given time. This book-keeping is required because elegant was originally a matrix-only code using $s$ as the longitudinal coordinate.

Users should keep the meaning of $s$ in mind when viewing statistics for $s$, for example, in the sigma or watch point output files. A quantity like Ss is literally the rms spread in $s$. It is not defined as $\sigma_t/(\langle \beta \rangle c)$. A nonrelativistic beam with velocity spread will show no change in Ss in a drift space, because the distance traveled is the same for all particles.