Digression on the Longitudinal Coordinate Definition

A word is in order about the definition of , which we've described
as the *total, equivalent* distance traveled. First, by *total* distance we mean that 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 and arrival time at the observation point
is, for each particle, , where is the
instantaneous velocity of the particle. Whenever a particle's
velocity changes, `elegant` recomputes to ensure that this
relationship holds. 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 as
the longitudinal coordinate.

Users should keep the meaning of in mind when viewing statistics
for , for example, in the `sigma` or watch point output files.
A quantity like `Ss` is literally the rms spread in . It is
*not* defined as
. 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.