Summary

The concept of "radar distance" for illustrated in the Rindler chart

${\displaystyle ds^{2}=-x^{2}\,dt^{2}+dx^{2}+dy^{2}+dz^{2},\;\;0<x<\infty ,\;-\infty <t,y,z<\infty }$

for the Minkowski vacuum (flat spacetime). The vertical red line at far left is the Rindler horizon ${\displaystyle x=0}$. Also shown (two vertical navy lines) are two Rindler observers with ${\displaystyle x=1,2}$ respectively, who are accelerating with constant magnitude ${\displaystyle 1,{\frac {1}{2}}}$ respectively. The leftmost observer sends a radio signal which bounces off the rightmost observer and returns. He measures the time required by this round trip radar signal, using an ideal clock which he carries, and divides by two to obtain the radar distance to the other observer. This notion of distance agrees very nearly with the ruler distance for small distances, but in general gives a smaller value. However, both notions of distance do give the crucial result that the Rindler observers maintain constant distance from one another.

In the figure, the light cones are correctly scaled to suggest the relative time dilation of the two observers, due to their acceleration; recall that the leftmost observer is accelerating twice as hard.

This png image was converted using eog from a jpg image created by User:Hillman using Maple.

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