File:3p1dAcceleration.jpg
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Summary
[edit]Description3p1dAcceleration.jpg |
English: An example of high-speed constant proper-acceleration of an object that began with a (fixed) coordinate-speed perpendicular to the proper-acceleration direction. By defining a plane in (3+1)D containing the proper-acceleration and any fixed-transverse coordinate-speed, one can use this (2+1)D plot to describe constant accelerations in (3+1)D spacetime as well.
In the lower-left and upper-right panels, which represent two views of the same motion, velocity and acceleration vectors are drawn at tenth-of-a-year intervals from both Galilean-kinematic (t,v,a on left) and traveler-kinematic (τ,w,α on right) points of view. The rate of velocity-change vectors (indigo & magenta) are scaled to the same corresponding interval. The upper-left panel represents low-speed motion in which the Galilean and traveler kinematics show the same thing. The lower-right panel shows how the traveler-kinematic changes shape further as one moves into the hyper-relativistic regime. |
Date | |
Source | Own work |
Author | P. Fraundorf |
Discussion
[edit]Note above on the lower-left that the (orange) coordinate-acceleration a ≡ dv/dt is not constant because (green) coordinate-speed v ≡ dx/dt cannot keep increasing as our traveler's rocket continues to exert a constant proper-acceleration/force on her[1], even though the coordinate-speed in the y-direction does remain the same.
The indigo arrows represent dv/dt at their point of origin along the trajectory, multiplied by that 10th year time interval, and hence represent estimates for change in coordinate-velocity over such an interval.
Note on the upper-right that changes per unit proper-time τ in (blue) proper-velocity w ≡ dx/dτ (and hence momentum) are not constant in either x or y, even though proper-acceleration α is. That's because proper-acceleration is defined in the rest frame of our traveler, while proper-velocity and momentum are defined in the traveler-kinematic of the map-frame.
The magenta arrows represent dw/dτ (frame-variant force divided by mass) at their point of origin along the trajectory, multiplied by that 10th year time interval, and hence represent estimates for change in proper-velocity (hence momentum per unit mass) over such an interval. Note that the "frame-variant forces" seen by map-frame observers are not in the proper-acceleration direction experienced by the traveler in her frame.
Footnotes
[edit]- ↑ P. Fraundorf (2012) "A fun intro to 1D kinematics", arxiv:1206.2877 [physics.pop-ph].
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 16:52, 12 February 2014 | 700 × 700 (96 KB) | Unitsphere (talk | contribs) | We add two more panels (top left and bottom right) to illustrate how these change in the low and high speed limits. | |
08:00, 11 February 2014 | 705 × 373 (62 KB) | Unitsphere (talk | contribs) | Vertical scaling correction. | ||
18:37, 10 February 2014 | 712 × 405 (72 KB) | Unitsphere (talk | contribs) | User created page with UploadWizard |
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