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Although beyond the scope of an introductory physics class, the curvature of spacetime coordinates for an accelerated frame in flat-spacetime can be visualized via that accelerated traveler's radar-time simultaneity ρ-cτ isocontours, as shown in the figure at right. Note that the radar-time of an event is simply halfway between the last time you might have affected it & the earliest time you might have seen it. This allows one to examine differential-aging, from a traveler's perspective, associated with event-points on arbitrary world lines in spacetime. The map-frame ct versus x radar-time simultaneity plot in the figure at right shows how acceleration, in this case of a 1-gee proper-acceleration round-trip lasting 4 traveler-years, distorts distances (blue vertical mesh-lines) and simultaneity (blue horizontal mesh-lines) experienced by that accelerated observer. For objects that are extended along the line of their acceleration, these distortions in space and time will occur even across an accelerated-object’s own length.
For example, in addition to the metric-equation’s motion-related time-dilation in which δτtraveler/δtmap = Sqrt[1-(v/c)2], for accelerated objects of length L in the direction of proper-acceleration α, one finds an acceleration-related time-dilation of the form: δτtrailing/δτleading = e-αL/c2. Here the leading-edge of the object is in the direction of the acceleration α, not necessarily in the direction of travel.
Example Problem 4.2.6a: Imagine that standing on the earth's surface is like undergoing an upward but constant proper-acceleration for an extended period of time. What estimate would this give for the differential aging between your head and your feet as a 1.8 meter tall adult, if you spent 25% of your life standing during the last 10 years?
This calculation, using Dolby and Gull's radar-time metric (Dolby 2001), gives essentially the same differential-aging as does the Schwarzschild metric for this ``stationary-acceleration" problem, with a simpler approximation. In fact, in this case one gets simply an increased tick rate of 109 attoseconds per second for each meter of height. This of course is only a local calculation, in comparison to the Schwarzschild one, since the acceleration due to gravity falls off with height h above the surface unless h<<R. However this sort of calculation may not work quite as advertized for spaceship problems. That's because a ``rigid spaceship" at 1-gee acceleration might to first order force its leading and trailing portions into something like a tangent free-float-frame but differentially-accelerated Rindler coordinate-system (Rindler 2006), unlike e.g. a ``convoy" of ships maintaining fixed radar-time distances apart. Regardless, the effect on time-dilation from one end to the other should be small for reasonably-sized assemblies, as a close look at the comparison in the panels at right will show.
For the 1-gee proper-acceleration of a standing human in the vertical direction, this differential-aging between head and foot becomes δtfoot/δthead ≃ 1−2×10−16. This means that if you stand up (or sit tall) for a sizeable fraction of your lifetime, your head may be a few-hundred nanoseconds older than your feet. This is a small effect for humans, but as discussed earlier it’s quite significant for global-positioning satellites for which nano-second timing-errors give rise to macroscopic errors in position.
The differential-aging of accelerated objects is linked to the gravitational time-dilation experienced by shell-frame objects in a planet's gravitational field, because a proper-acceleration is needed to keep objects in such a field from following a rain-frame trajectory.
Related references:
Carl E. Dolby and Stephen F. Gull (2001) "On radar time and the twin paradox", Amer. J. Phys. 69 (12) 1257-1261 abstract.
Wolfgang Rindler (2006) Relativity : Special, General, and Cosmological (Oxford U. Press, UK).
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