#### PeterDonis

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True in their respective geometries, yes. But they are different geometries. Euclidean 3-space is not the same geometry as 4-D Minkowski spacetime. Each one has its own formula for ##ds^2##. It makes no sense to say the formula for ##ds^2## in Euclidean 3-space is "true" in Minkowski spacetime.if it is not the same quantity then both equations are true

Euclidean 3-space is not spacetime. It's Euclidean 3-space.It is still the same Spacetime

No, this is not correct. ##dx^2 + dy^2 + dz^2## is the invariant length in Euclidean 3-space. ##- c^2 dt^2 + dx^2 + dy^2 + dz^2## is the invariant length in 4-D Minkowski spacetime. This length is called a "spacetime interval" in the Minkowski spacetime, but that's just nomenclature.Surely a2+b2+c2 is still the aggregate length in Minkowski Spacetime; while -ct2+a2+b2+c2 is the Spacetime interval and both are invariant intervals.