Both x-ct and radar-time ρ-cτ diagrams for constant proper-acceleration of a traveler whose trajectory is marked in red. We've marked isocontours and a fiducial event in green for {x,ct} = {2,1}, and in purple for {ρ,cτ} = {-1,1}, on "both" plots for comparison. Note that this radar-time (instead of the tangent free-float-frame) definition of extended-simultaneity shows clearly that map and traveling observers will legitimately disagree on which of those two fiducial events happened first. This sort of disagreement is of course possible from any two events which are separated by a space-like instead of a time-like interval.