proper speedometry
The strengths and weaknesses of the traveler-point perspective may be concretely explored by taking a close look at speedometers that measure distance traveled on the road (e.g. in units of the tire's circumference) as a function of time on the clocks of the traveler. It may be easiest to do this in context of a dream that George Gamow's Mr. Tompkins [1] had when he fell asleep in a physics lecture by his future wife Maud's father, and imagined a bicycle riding down the road in a world where the speed of light was only a few miles per hour. For those who want to skip this and simply solve some problems, this Wolfram language calculator lets you solve relative velocity addition problems at any speed.
Speedometers, like the one in your car which uses the at-rest tire circumference (or GPS distance) per unit time on the traveler clock, instead of on externally synchronized "gate clocks", measure proper-velocity i.e. distance traveled down the road per unit time on the traveler's clock. This is a quantity that's proportional to your car's momentum at any speed, and has no upper limit as long as the wheel itself manages to hold together. In particular, if lightspeed were on the order of a mile per hour then we can limit the bad effects of both length contraction at the perimeter, and centrifugal expansion of the diameter, by making the wheel from say 16 spokes (with separate rim segments) of material having a speed of sound (as well as light) in the mph range.
The animations below show different views of the same 26" bicycle wheel rolling past a 1-foot square flower pot at proper velocity (map distance per unit traveler time) of about w/c = 4/3 lightspeed, which has been set to c = 0.435 mph. The coordinate speed is v/c = 4/5, and the differential aging factor is gamma = 5/3 (whose reciprocal also shows up as length contraction in the shrinking segments along the wheel's perimeter). The left and right animations have been recorded with a telephoto full-frame camera using detectors essentially equidistant from all objects shown, so that they picture instantaneous positions [2] in the map and axle frames, respectively. The center animation superposes a local "traveler-point" model of the moving wheel on the same stationary map field shown in the animation at left.
Map-frame view
Traveler-point view
Axle-frame view
Each frame in the left panel corresponds to 1/16 second on map clocks with distances and simultaneity measured in the map frame. Each frame in the right panel corresponds to 1/16 second on axle clocks with distances and simultaneity measured in an extended axle frame.
Each frame in the center panel corresponds to 1/16 second on axle clocks with distances and simultaneity measured in the map frame. In other words, the center panel uses a hybrid kinematic [3] which we find simplifies dynamical calculations, but is only good locally for the traveler and not useful e.g. for calculating extended-object overlaps or collisions.
Thus only the left and right animations constitute movies recordable with full-frame "equal-traversal time" cameras. Hence they should predict overlaps correctly, but each uses different definitions of simultaneity so that "synchronization" between them in general is not possible.
The center animation superposes an axle-frame view of the wheel and its contact with the ground at right, onto the stationary map frame at left. This traveler point perspective therefore drops a local view of the traveler into a global view of the map, limiting the value of extended-traveler overlaps while simplifying models of traveler motion in the map frame.
The still images below, with previous-frame afterimages, may be useful for measuring distances, velocities, coincident overlaps, etc.
Map-frame view
map distances & simultaneity
Traveler-point view
map distance & axle time
Axle-frame view
axle distances & simultaneity
The usefulness of proper velocity in the matter of road safety doesn't end with its ease of measurement. It turns out that proper velocity is connected directly to momentum, kinetic energy, and driver/pedestrian reaction times by simple power-law relations. On the other hand all of these things, surprisingly, are quite oblivious to nature's self-imposed limit on coordinate velocity!
To be specific (in dimensionless terms) momentum p for a mass m vehicle is p/mc = w/c, kinetic energy K is K/mc2 = Sqrt[1+(w/c)2]-1, driver reaction time τ available following detection at map-distance R is cτ/R = c/w, and pedestrian reaction time t is ct/R = Sqrt[(c/w)2+1]-1. The expressions with square roots in them transition between slope 1 and slope 2 on a log-log (power-law) plot, as proper velocity transitions between much less and much more than 1 lightyear per traveler-year.
The reaction-time relations are illistrated at right in the x-ct plot of a driver (blue line coming in from the left at about 4/3 or 1.33 feet per traveler-nanosecond or lightyears/traveler-year) approaching a stationary intersection (vertical black line in the center). The curved green iso-contours represent driver-time elapsed before/after arrival, separated by 10 "traveler-nanosecond" intervals. The gray horizontal lines represent map-time elapsed, at 10 "map-nanosecond" intervals. The dotted magneta arrows show light paths from and to the intersection, in connection with the "appearance event" (represented by a blue dot) which happens at "visibility-range" Δx = 35 feet (perhaps there is a bit of fog) to the left of the intersection.
An x-ct plot of an intersection approach, showing how at high speeds the pedestrian gets much less time to react than does the driver.
References:
- George Gamow (1940) Mr. Tompkins in Wonderland (Cambridge U. Press, NY) preview.
- V. N. Matvejev, O. V. Matvejev, and O. Gron (2016) "A relativistic trolley paradox", Amer. J. Phys. 84, 419 pdf.
- P. Fraundorf (1995) "Three self-consistent kinematics in (1+1)D special relativity", arxiv:gr-qc/9512012.