File:3p1dAcceleration.jpg
Original file (700 × 700 pixels, file size: 96 KB, MIME type: image/jpeg)
Captions
Captions
Summary
[edit]
| Description |
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].
Licensing
[edit]
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| 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 |
You cannot overwrite this file.
File usage on Commons
There are no pages that use this file.
