# Proving that Maxwell's equations for the propagation of electromagnetic wave are Lorentz invariant

1. Jul 20, 2015

### Bakali Thendo

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

2. Jul 20, 2015

### Mentz114

3. Jul 20, 2015

### Bakali Thendo

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

4. Jul 20, 2015

### 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!

5. Jul 21, 2015

### Bakali Thendo

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

6. Jul 21, 2015

### Mentz114

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.