File:XctauRoundTrip3.png

Note that one-gee of acceleration is about 1.03 ≈ 1 lightyear/year², so that a 4 traveler-year roundtrip at one-gee (for human comfort) would correspond to our ατ_q/c = 1 profile, while an 8 traveler year 1-gee roundtrip corresponding to our ατ_q/c = 2 profile (with the blue labels) would take you τ_q×2×1.38 ≈ 5.5 light-years (well past α-Centauri) even though about τ_q×4×1.81 ≈ 14.5 years would elapse on map-clocks of folks who stayed at home. This effect becomes even more pronounced with longer roundtrips, as illustrated by the pileup of map-time isochrons (and hence reddening of top-half trajectories) when ατ_q/c is larger.

In contrast to more familiar x-ct plots, plots of map-distance versus traveler (i.e. proper) time take advantage of the flat-space metric-equation's Pythagorean relationship, which defines map-time increment cδt as the hypotenuse of a right triangle whose orthogonal sides are map-distance δx and proper-time cδτ. For constant speed travelers, such a plot illustrates graphically how motion-through-time (dτ/dt) is traded for motion-through-space (dx/dt) as one's coordinate-velocity approaches spacetime constant c.

In general x-cτ plots are excellent for considering problems involving single travelers (or families of single travelers) in context of one map-frame of yardsticks & synchronized clocks^[1]. Its special advantage in flat (Minkowski) spacetime is that all three variables (x, ct & cτ) share a common scale even though for map-time ct this applies only along the selected traveler's world line. This is not true for x-ct diagrams, which however are needed for visualizing world-lines of multiple travelers as well as the relationship between events in context of two or more coordinate-frames of yardsticks & synchronized clocks.

The fundamental equation for x-cτ plots in general is the (1+1)D flat-space metric equation re-arranged in the form of a Pythagorean expression relating the hypotenuse to the sides of a right triangle i.e.:

${\displaystyle (c\delta t)^{2}=(c\delta \tau )^{2}+(\delta x)^{2}\,}$.

Here c is the space-time constant known as lightspeed, δx and δt are map-distance δx and map-time δt increments measured in a single map-frame of yardsticks & synchronized clocks, between separated events along the world line of a traveling object during the proper-time interval δτ between those same two events on traveler clocks.

1. ↑ P. Fraundorf (2011/2012) "Metric-first & entropy-first surprises", arXiv:1106.4698 [physics.gen-ph].

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