English: Two views of a year of one-gee constant proper-acceleration from rest: Metric-first kinematics^[1] and radar-time simultaneity^[2] show that acceleration curves flat space-time from an accelerated-traveler's perspective, and that rigidity is a locally-useful concept whose behavior over extended distances and times is more entertaining than a typical intro-physics course might lead you to believe. These plots are a direct consequence of special relativity i.e. of the flat-space Minkowski metric alone.

Radar-separation ρ-cτ contours (red) used to define simultaneity for the accelerated traveler are plotted in the free-floating map-frame observer's x-ct space at left, while free-float-frame x-ct contours (brown) define simultaneity for map-frame observers in the traveler's ρ-cτ space at right. Co-moving free-float-frame hypersurfaces for the accelerated-traveler in the left panel will be straight lines tangent to the corresponding radar-hypersurface, at the point of contact with the (solid red) traveling observer's world line.

The dashed red curves correspond to objects accelerated from rest in parallel to the primary (red) object, but initially positioned a distance L = ±0.4c²/α ahead and behind. The large red dots correspond to ignition and shutdown events for these parallel-accelerated worldlines. The ends of comparable object exhibiting "smart rigidity" (see discussion below) are marked with red dotted-lines.

The free-float-frame (reference map) origin in both figures is a dashed brown line, while the accelerated traveler trajectory is in red. Two large purple dots mark the start and end of traveler acceleration, while two large green dots mark float-frame-origin light pulses that might trigger and then detect the traveler's acceleration shutdown. Another six events are shown as smaller bright-green or indigo dots in both panels.