Proving Maxwell's Equations are Lorentz Invariant

[Bakali Thendo]

I want to know how can i prove that Maxwell's equations for the propagation of electromagnetic wave are Lorentz invariant.

 

[Mentz114]

Transform the wave equation and observe that it remains unchanged. I don't have the details to hand but I can find them if necessary.

I did a search and came up with this. It looks good.

http://resonanceswavesandfields.blogspot.co.uk/2011/07/invariance-of-electromagnetic-wave.html

 

[Bakali Thendo]

Yes, this give me a clear understanding on both the lorentz and maxwell. Thank you

 

[vanhees71]

Puh, that looks complicated ;-)). It's much easier to reformulate Maxwell's equations in manifestly covariant form with four-vectors and four-tensors. Then you immideately see, without to preform the pretty time-consuming Lorentz transformations, because then it's clear that the equations are covariant by construction!

 

[Bakali Thendo]

Can you elaborate on what you are talking about...

 

[Mentz114]

Can you elaborate on what you are talking about...

Here is an introduction https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism.

When equations are written in tensor form then invariance under certain transformations is 'built-in'.

Transformed tensor contractions eg $v^a v_a \rightarrow \Lambda v^a {\Lambda}^{-1} v_a$ do not change because contravariant components transform with the inverse of the transformation of the covariant ones.

For instance $f_{\alpha} = F_{\alpha\beta}J^{\beta}$ is manifestly covariant because $F$ and $J$ are tensors. The contraction $f^\alpha f_\alpha$ is unaffected by a Lorentz transformation.