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My argument would be something along the following lines: assume that your spacetime is $\mathbb E^4$ and determine its isometry group. This turns out to be $O(4)\ltimes\mathbb R^4$, which doesn't coincide with the Galilei group (indeed in special relativity $O(1,3)\ltimes\mathbb R^4$ is the full Poincaré group, which is also the isometry group of flat ...

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