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With reference to the above diagram, we can see that if we used a ruler to measure the segments on the XT chart and added up all the segment lengths, that this ruler measured length is longer for the craft on the right which experiences the least elapsed proper time. Now unlike the regular spacetime distance (as correctly depicted in the picture) this XT chart ruler length is not invariant and different inertial observers will disagree on its length. However, from a quick check using a few examples in different reference frames, it seems that if one ruler measured chart path is longer than another in a given reference frame, then all observers will agree on which object has the longer XT chart ruler measured distance (Is there a formal name for this measurement?). Is it possible to show that this always the case or are there counter examples?

Just to be clear, I am talking about using the normal Pythagoras method of using $$ \sqrt{T^2 + X^2} $$ to calculate the segment lengths instead of the normal method of using$$\sqrt{T^2-X^2}$$ to calculate the spacetime distance segments.

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