proper accelerometry

Let's start with a high speed version of a popular introductory physics problem, shifted to an interstellar superhighway setting something like that discussed in Douglas Adams' The Hitchiker's Guide to the Galaxy. The solution uses unidirectional acceleration equations from tables on the next page:

A speeding space jalopy (C) traveling at a constant proper speed of w = γv = 1 [lightyear/traveler-year] passes a space trooper (T) hidden behind a billboard on an interstellar superhighway. As the jalopy passes the billboard, the trooper sets off in chase with a constant proper acceleration of α = 1 gee ≈ 1 [ly/y²].

(a) Write an equation for the position x_C of the jalopy as a function of map-time .

(b) Write an equation for the position x_T of the trooper as a function of map-time.

(c) Write an equation for the map-time t* when the trooper overtakes the jalopy, and solve it if you can.

Setting x_C[t*] = x_T[t*] gives:

which may be solved for t* to get:

(d) How would you get the distance from the billboard to where the jalopy was overtaken?

Substituting (c) into (a) gives

One might also ask about the times elapsed, during the chase, on the clocks of both the jalopy and the trooper. How might one estimate those values, and is it really true that the trooper spends less time during the chase than does his target?

The chase starts at the green dot and ends at the red dot, when the space-jalopy is overtaken.

The basic result is illustrated in the figure at left. Does that make sense?

This problem and solution might at first glance seem to gloss over the "relativity of everything" impression left by early introductions to spacetime. Hence we might want to preface it by pointing out that, although map-time intervals, object-lengths, and forces at high speed, like rates of kinetic energy increase at any speed, take on a frame-dependent character, we are here using specially-chosen variables. In particular, the frame-invariance of traveler time τ and proper-acceleration's magnitude α means that all observers, regardless of their choice of bookkeeper metric or their metric's accelerated and/or curved nature, will agree as to the SI-unit value of these parameters. Moreover, given information on the metric yardsticks by which our traveler is measuring distance, such observers will also agree on our traveler's direction of acceleration and proper velocity vector (which makes no assumptions about synchronizable clocks). Finally, the assertions about map time intervals between events are unambiguous (if frame dependent) because they are embedded in a single definition of simultaneity defined by the map-frame's bookkeeper metric itself.

If the map-frame's metric is accelerated or gravitationally curved, then "geometric forces/accelerations" along with proper forces will also contribute to the causes of motion, but that is a subject to discuss elsewhere. Here we can use Newton's (inadvertant) trick, by noticing that geometric forces like gravity or centrifugal (even though invisible to your cell-phone's accelerometer) may be approximated locally as proper forces.