File:XctauConstantSpeed.png

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 rearranged 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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