- #106

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One crucial point of relativistic spacetime models is that they are not Euclidean/Riemannian affine spaces/manifolds but pseudo-Euclidean/pseudo Riemannian affine space (SR/GR). Uncounted textbooks and paper indeed call corresponding the fundamental forms a "scalar product" or ##\mathrm{d}s^2## "a metric", but I've really never seen that they introduce ##\|\mathrm{d} x\| =\sqrt{\mathrm{d} s^2}##. It's already confusing to call the indefinite fundamental forms scalar product, but to introduce a symbol ##\| \cdot \|## for something that's definitely not a norm, is too much.

At least, I'd not accept such a thing for any textbook I'd recommend to my students. It's hard enough for them to learn to read a Minkowski diagram correctly, i.e., not mistaken it as if it could be read as having "distances" and/or even "angles" as in the Euclidean plane with a Cartesian basis.