# Galilean transform of the Laplacian

cartonn30gel

## 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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## Answers and Replies

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

cartonn30gel
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.

cartonn30gel
Can anybody help?

sgd37
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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