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In special relativity, events happen in the Minkowski spacetime, with signature $(3,1)$ or $(1,3)$. I was wondering about the need of different sign for temporal and spatial coordinates. Looking online I found a possible explanation: spacetime is a Lorentizian manifold (a pseudo-Riemannian manifold with signature $(1,n-1)$ or $(n-1,1)$) because otherwise we could not define null-, space- and time-like distances based on the value of $ds^2$. Indeed in a Riemannian manifold metric signature is $(0,n)$ so distances are always positive. Is this a good explanation or am I missing something?

Qmechanic
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john
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    I think that's pretty much it. – tonetillo 4 Mar 21 '22 at 16:04
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    Btw, "is my understanding correct?" is not a proper way to ask here, as "Yes/No" would suffice as an answer, and mono-sylabic posts are not accepted here. – DanielC Mar 21 '22 at 19:02
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    The laws of physics (in particular, The Principle of Relativity and the Speed of Light Principle) lead to Lorentzian signature Riemannian manifold... as I show (following an argument by Bondi) in my answer https://physics.stackexchange.com/a/508251/148184 to https://physics.stackexchange.com/questions/107443/minkowski-metric-signature . From a physics point of view, notions of "timelike", "null", "spacelike" follow as consequences. Only in hindsight (or with tremendous insight or foresight or luck) could we ask that a pseudo-riemannian metric be used to model spacetime structure. – robphy Mar 21 '22 at 20:00

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