File:XctauConstantSpeed.png

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English: An x-cτ diagram for constant speed travelers, with distance units arbitrary. One nice feature of this plot is that all 3 variables (x, ct, and ) share a common scale, although technically the ct scale only has meaning along the (green) world line of a specific traveler. The blue example point corresponds to the 300 keV electrons (with proper velocities near 1.23 ly/ty) used for high resolution transmission electron microscopy (TEM) by researchers in the UM-StL Center for NanoScience (CNS) Microscope Image & Spectroscopy Tech (MIST) Lab.
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Author P. Fraundorf

Note on x-cτ plots

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Speeds through time versus space.
x-cτ diagram for roundtrips.

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 & ) 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.:

.

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.

Footnotes

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

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current12:02, 20 September 2013Thumbnail for version as of 12:02, 20 September 2013362 × 372 (69 KB)Unitsphere (talk | contribs)User created page with UploadWizard

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