Reading a somewhat long and argumentative thread here inspired the following unrelated question in my mind: Where does a 2 dimensional flat Lorentzian geometry depart from Euclidean geometry as axiomatized by Euclid? I.e. Euclid's axioms (in modern language) can be taken to be: We can interpret straight lines in Euclidean geometry as "geodesics" in the Lorentzian geometry. I believe the difficulties start to arise in axioms 3 and 4. A meaningful notion of a "circle" becomes difficult, perhaps we'd replace it with a hyperbola? The concept of rapidity between time-like geodesics is promissing, but doesn't seem to be a sufficiently general replacement for "angle".