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Galilean transform of the Laplacian

  • #1

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.

Homework Equations



x'=x-vt, t'=t

The Attempt at a Solution



Just consider the second derivative wrt x in cartesian coordinates:

[tex]\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}[/tex]

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??
 
Last edited:

Answers and Replies

  • #2
Matterwave
Science Advisor
Gold Member
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How did you go from your 3rd step to the 4th one?
 
  • #3
How did you go from your 3rd step to the 4th one?
[tex]\frac{\partial x}{\partial x'} = 1[/tex] since x'=x-vt according to Lorentz transform, and v is constant since we are talking about an inertial reference frame.
 
  • #4
Can anybody help?
 
  • #5
213
8
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
 
Last edited:

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