SR, electromagnetic waves in moving reference frames.

Lavabug

1. The problem statement, all variables and given/known data
Not really a homework/coursework problem, I'm just trying to make sense of some class notes from our chapter on special relativity. I'm trying to find the expression for electromagnetic wave propagation in a reference frame S' that is moving at a constant velocity with respect to an inertial reference frame S.

Where capital phi is either the B or E vector (see relevant equations).

2. Relevant equations


or more generally:


Just need to find the 2nd order partials with respect to x, y, z for the Laplacian and the 2nd order partial with respect to t.
3. The attempt at a solution

I have no trouble finding the first order partial derivatives of phi with respect to all 4 variables (note the red checkmark), nor the 2nd order partial derivatives with respect to x, y and z. But I don't know what happened with the 2nd derivative with respect to time, I don't know where the two middle terms came from. (see arrows)

2nd question: What does the solution imply? I didn't understand/am missing the bit of theory that came afterwards (I'm not studying in my native language and my lecturer speaks blazingly fast lol), supposedly the solution for this equation in a moving reference frame gives an answer of 0, which implies that their is no preferential reference frame.

Sorry for posing a question so convoluted but I'd appreciate any help.

JesseC

The transformation you've written down there [tex] x^\prime = x - vt [/tex] is a Galilean transformation. The relativistic transformation is known as the Lorentz transformation.

I haven't looked over your maths in detail, but if my memory serves me right, the EM wave equation is not invariant under a Galilean transformation but it is under a Lorentz transformation. You've basically shown there that the Galilean transformation doesn't work for EM waves.