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I know that the classical mechanics stays valid under Galilean transformation. The same argument applies to relativistic equations and Lorentz transformation. My question is, can a set of equations dependent on both spatial coordinates and time be valid under both transformations at the same time?

Edit: I guess the wave equation satisfies this.

In that case, what other examples can you provide, or under which assumptions is this true?

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If we consider an infinitesimal displacement in space, $(dx, dy, dz)$, and and infinitesimal displacement in time, $dt$, then in Galilean transformations the quantities:

$$ dx^2 + dy^2 + dz^2 \\ dt $$

are both invariants. In special relativity neither of these are invariants. Instead the invariant is the proper distance:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$

So the two geometries are fundamentally different.

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