File:DifferentialAgingAndFalling.JPG
DifferentialAgingAndFalling.JPG (595 × 307 pixels, file size: 45 KB, MIME type: image/jpeg)
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DescriptionDifferentialAgingAndFalling.JPG |
English: This plot illustrates Epstein's x-cτ-plot argument[1], in his sketches 9-9 and 10-4, for how differential aging causes gravitational acceleration. Note that x-cτ plots are not "global plots of events in space-time" because the cτ axis is specific to each trajectory.
The green trajectories show two siblings who spend the same amount of map-time Δt (line-length) between start (solid circle) and finish (open circles), one up in the attic (dotted-trajectory) and the other down in the basement (dashed-trajectory). Little proper-time (azimuth-angle) is spent by either sibling in changing floors (radial-position). As you can see, a curved x-cτ plot lets us show how the siblings return to the ground floor with less proper-time Δτ having elapsed on (and hence less aging of) the sibling who stayed in the basement. The red straight and therefore free-float-frame "fly-ball" trajectory shows how spacetime, bent as above to allow differential-aging on the two traveler clocks, as a consequence also requires that an object in free fall experiences a downward acceleration. This is because straight free-fall trajectories naturally change motion always "at lightspeed" through proper time (i.e. when stopped so that cδτ/dt = c) to motion which is also partly through space (i.e. dx/dt ≤ c), and vice versa, when spacetime is curved by gravity's differential aging. Light of course to first order experiences that same motion largely through space and not time, so that dx/dt ≈ c and light trajectories on the plot above will be largely in the radial direction. |
Date | |
Source | Own work |
Author | P. Fraundorf |
Further notes
[edit]The figures below illustrate how x-cτ trajectory plots compare to more familiar x-ct spacetime plots[2], and to perhaps less-familiar ρ-cτ extended-simultaneity plots[3], in flat spacetime. The first of these figures shows how these might work for constant-velocity observers e.g. to illustrate time-dilation and length-contraction.
The figure below illustrates how these work in flat spacetime to show what happens for a traveler undergoing constant proper-acceleration.
In principle these x-cτ plots can also work (schematically at least) to illustrate, again following Epstein, the way that extreme gravitational curvature hinders the far-away observation of events near the event-horizon of a static black hole.
Footnotes
[edit]- ↑ Epstein, Lewis Carroll (1895/1994). Relativity Visualized, (Insight Press, San Fransisco), ch. 9-12.
- ↑ Minkowski, Hermann (1907/1915) "Das Relativitätsprinzip", Annalen der Physik 47 352 (15): 927–938, prepared by A. Sommerfeld long after Minkowski's death. link
- ↑ Carl E. Dolby and Stephen F. Gull (2001) "On radar time and the twin paradox", Amer. J. Phys. 69 (12) 1257-1261 abstract.
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:21, 2 August 2014 | 595 × 307 (45 KB) | Unitsphere (talk | contribs) | User created page with UploadWizard |
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Author | Fraundorf, Philip |
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Date and time of data generation | 20:11, 1 August 2014 |
Date and time of digitizing | 20:11, 1 August 2014 |
DateTimeOriginal subseconds | 83 |
DateTimeDigitized subseconds | 83 |