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?
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