Galilean transform of the Laplacian

Homework Statement

I'm trying to show that the wave equation is not invariant under Galilean transform. To do that I need to figure out how the Laplacian transforms from S to S'. I seem to have trouble understanding why the laplacian actually changes.

x'=x-vt, t'=t

The Attempt at a Solution

Just consider the second derivative wrt x in cartesian coordinates:

$$\frac{\partial^2}{\partial x'^2}=\frac{\partial}{\partial x'} \frac{\partial}{\partial x'} = (\frac{\partial x}{\partial x'} \frac{\partial}{\partial x}) (\frac{\partial x}{\partial x'} \frac{\partial}{\partial x}) = \frac{\partial}{\partial x} \frac{\partial}{\partial x} = \frac{\partial^2}{\partial x^2}$$

So if you do the same for all three components, it looks like the Laplacian just transforms as it is. But I know that this is not right. Any help??

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Matterwave
Gold Member
How did you go from your 3rd step to the 4th one?

How did you go from your 3rd step to the 4th one?

$$\frac{\partial x}{\partial x'} = 1$$ since x'=x-vt according to Lorentz transform, and v is constant since we are talking about an inertial reference frame.

Can anybody help?

the Laplacian doesn't change the invariance comes from the function you apply it to i.e. from f(x,t) -> f(x',t')

and aren't you supposed to be considering the d'Alembertian for a wave equation

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