### 4 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 = βct, where βc 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 σ_{t}∕(⟨β⟩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.