The speed of a procession

I recently saw a parade – lots of motorbikes rode up Vogt's gate at a steady, comfortable urban traffic speed, compete with moderate amounts of variation of speed within the parade; since the parade tookup its side of the road, the bikers had room to manoeuvre around one another – and did so, in moderation; slight differences in speed between bikers chagned the shapes of the spaces between them, sometimes leaving room for one to manoeuvre to a different position in relation to the others.

Now, the main impact of a procession on the bits of town it passes through is that it obliges other traffic to wait for the procession to pass (or for a break in it; there were none in this); thanks to the comfortable urban traffic speed they kept up, this didn't take more than about two cycles of the traffic lights (which the parade ignored; they were gone soon enough that this wasn't a problem, beyond the extent to which some might be unsympathetic to the parade itself); it (only) briefly disrupts everyone else's business.

So pause to consider: how long does a parade take to pass ? It's the time from when the front bike passes a given point on its journey to when the back part passes. In some sense, this is the parade's length divided by its speed; but pause to look at a parade and realise it's not a rigid body and its parts move at various speeds; so what are its length and speed ? (The same considerations may equally be applied to the movement of any flow made up of many parts, such as a current within some larger body of mostly still water, or the stars of a cluster moving within a galaxy.)

For any one biker, we understand that biker as having, at each moment, a speed indicated on a speedometer. We can average this over any interval of time or segment of the biker's trajectory; we can associate such an average with some point in the interval and wind the interval or segment through time, keeping fixed the biker's position relative to the interval. For example, we can at any time identify the front and back bikers of the procession and average any given biker's speed over the interval from when this biker passed where the back one now is to when it later shall pass where the front one now is. The position of the present within the time-interval a particular biker is averaging over roughly corresponds to (a time-averaged characterisation of) the biker's position within the procession, which typically varies only slowly, compared to the speed of any one biker. This would associate an averaged speed with each biker at each moment, that mixes some past with some future.

The street's frame of reference shows each biker follow a trajectory that, on a suitably scaled space-time diagram with future time upwards, departs from straight up to a degree that increases with speed. The trajectories of the bikers form a bundle, winding upwards at a varying slope. A helicopter over-head might be able to see the front and back of the procession; the resulting snap-shot of the parade corresponds to a space-like (horizontal, in the diagram) slice through the bundle, the intersection of the bundle with the observer's now. A street-side observer watching the bikers all pass gets a time-ward slice through the bundle, intersecting it with the volume of space the observer can see, which is a volume at each time, extruded into a tube (that may change shape if the observer moves around) by the observer's experience of time passing. The volume at any time, technically, is really the part of the observer's incoming light cone from which information is arriving at that time, so it's also sloped, but propagating it through time gives us a tube little different from the one using this observer's now of the volume, at least as long as the observer and bikers aren't moving at relativistic speeds relative to the street.

If we were watching a space-parade passing at speeds varying by a noticeable (albeit perhaps rather small, but not tiny) fraction of the speed of light, each individual in the procession has a speed-dependent frame of reference that determines where the front and back currently are, using that frame's time to determine simultaneity. Individuals (whether in the procession or watching it) moving at different speeds use different time co-ordinates, for which their now is a differently-sloped spacelike cut through the weave of trajectories, giving them different spatial intervals to average over; but this is not so hugely different, when I was already using a speed averaged over a different interval of time for each biker.

TODO: continue what I began writing on 2018-05-05.

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