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摘要

描述
English: Radar-separation hypersurfaces[1] for an accelerated traveler, defined as the average proper-time of sending/receiving a light signal to/from an event i.e. τevent = (τsendreceive)/2. Using the axis-labeled units, this is basically a one-gee (~1[ly/y2]) constant proper-acceleration round-trip which takes 4 traveler years to complete.

Four purple dots mark engine on/off events as well as the two mid-voyage points at which the direction of acceleration is reversed. Null trajectories extending forward and back from these points define twenty-five separate event-regions in space-time.

These zones (marked off by faint dotted lines) have blue radar-isochrons, and radar-distance grid-lines, drawn in to characterize for events from the accelerated traveler perspective. Separations in this case are at intervals of 0.2 (years or lightyears) from the traveler's radar-perspective.

The two vertical dashed lines (brown) represent the world lines of home and destination sites that are stationary in the x-ct coordinate system here. Following the separation between start and return isochrons into the flat-space region to the right will allow you to measure the triptime elapsed on the traveler's clocks using the ct-axis calibration here.
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来源 自己的作品
作者 P. Fraundorf

Added notes

The figure below compares the radar-time definition of extended-simultaneity with the tangent free-float-frame defintion for a simpler case, in which the traveler (red trajectory) is undergoing constant proper-acceleration with no start or end in sight.

Radar-time vs. tangent free-float frame simultaneity for constant proper-acceleration.

See also

  • Intro kinematics note[2] on differential-aging caused by acceleration and/or by gravity.

Footnotes

  1. (December 2001). "On radar time and the twin “paradox”". American Journal of Physics 69 (12): 1257–1261. DOI:10.1119/1.1407254. (arXiv:gr-qc/0104077)
  2. P. Fraundorf (2012) "A traveler-centered intro to kinematics", arXiv:1206.2877 [physics.pop-ph] mobile.

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