# Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter

The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations.

Say I consider an Euclidean space and a time parameter. How does the Euclidean metric (which in canoncial coordinates is $\delta_{ij}$, with $i,j\in \{1,2,3\}$), transform under a Galilean transformation, more specifically a velocity boost $t\mapsto t,x\mapsto x-vt$? In a way, from the three dimensional pov, which doesn't see $t$, this is just a translation.

So here is what I'd like a derivation for:

$$(t\mapsto t'=t,\ x\mapsto x'=x-vt)\ \Longrightarrow\ h_{ij}\equiv\delta_{ij}\mapsto\ ?$$

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