Metric-based approaches to mechanics allow one to accomplish a great deal in context of a single reference frame of yardsticks and synchronized clocks, before moving to considerations that involve more than one such frame. One of these things is the quantitative treatment of accelerated motion. This is especially clear if one plots velocity-parameters derived from the metric equation e.g. coordinate-velocity v=dx/dt, proper-velocity w=dx/dτ, and gamma γ=dt/dτ as a function of map-position. Rather complex scenarios, like that of an accelerated-twin adventure, can be illustrated in very quantitative and concrete terms. Such plots can be made dimensionless in terms of a fixed constant proper-acceleration α. Since these apply locally to all accelerated motions (even in curved spacetime), we refer to them here as "universal acceleration plots".
In this context, below find an interactive plot of velocities (green is coordinate-velocity or v=dx/dt; blue is proper-velocity or w=dx/dτ) and gamma (red is speed-of-map-time or γ=dt/dτ=E/mc2) for a system undergoing constant proper-acceleration α, versus displacement parallel and perpendicular to that acceleration. All distances (and simultaneity) are defined with respect to a single reference "map-frame". The velocity-axis is in the "facing funnel" direction, while perpendicular displacements lie along the "butterfly symmetry axis". The section of the universal plot provided here covers values of coordinate-velocity v (green) for map-times t between -2 and 2, and of proper-velocity w (blue) for proper-times τ between -2 and 2, with all times measured relative to the "turn-around" event (vx=0) in units of c/α (or years at 1 gee). Values of transverse coordinate-velocity (a constant for a given trajectory) range from -0.9c to 0.9c (i.e. around 2 lightyears/traveler year). The red surface represents γ for x-distance traveled relative to rest of 0 to +2, in units of c2/α (or lightyears at 1 gee).
Here comes the applet...
On slower computers it may take some time until the image appears. Dragging the mouse across the image can rotate or spin the plot, while dragging the mouse up or down with s-down on your keyboard can zoom in or out. Left click enables many javaview pulldown dialogs. See if you can determine, with help from your browser, how deep into the funnel the red γ "cap" extends.
Universal acceleration plots are useful, for example, if one wants to visualize an accelerated-twin adventure in concrete terms. A (1+1)-dimensional example of this is shown below. Please excuse the arcane "one-map three-clock" nomenclature, which in addition to the above velocities plots map-time t and proper-time τ in dimensionless form, as well as "chase-plane" time T and "Galilean-kinematic" velocity V=dx/dT, for which the chase-plane trajectory has been chosen so that Galileo's constant acceleration equations describe the motion of our accelerated traveler. The most interesting times and velocities, of course, are the ones mentioned above: map-time t and coordinate-velocity v=dx/dt, as well as the less frame-variant quantitites: proper-time τ, proper-velocity w=dx/dτ, and gamma γ=dt/dτ.
Note in the changing plot on the right that as the round trip duration and maximum velocity decreases, the time-lapse disagreements between traveler and map get smaller. As the trip becomes longer, traveler time elapsed can become almost negligible in comparison to map time.
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