# Is Lorentz and Galilean invariance mutually exclusive?

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?

• Since Galilean transformations are the isometries of flat space with Euclidean signature ("Euclidean space") and Lorentzian transformations are the isometries of flat space with Lorentzian signature ("Minkowski space"), what space are these hypothetical equations that are invariant under both supposed to live on? – ACuriousMind Apr 19 '19 at 18:49

If we consider an infinitesimal displacement in space, $$(dx, dy, dz)$$, and and infinitesimal displacement in time, $$dt$$, then in Galilean transformations the quantities:
$$dx^2 + dy^2 + dz^2 \\ dt$$
$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$