Lorentz Invariance and Non-Galilean Invariance of Maxwell's Equations

[Dahaka14]

I am having trouble going about proving the Lorentz invariance and non-Galilean invariance of Maxwell's equations. Can someone help me find a simple way to do it? I've looked online and in textbooks, but they hardly give any explicit examples.

 

[robphy]

You have to specify how the fields transform.

To do it in general, it's easiest to do it tensorially.
You could do it vectorially... or possibly less elegantly component-wise.

Can you show some of your attempts so far?

 

[Dahaka14]

I've tried transforming the coordinates of the wave equations for Maxwell's equations into Lorentz transformed equations via the x and t components, excluding the y and z components of the wave equation for simplicity. I figuredsince the equations are homogeneous, the x and t components should be either equal to each other or each equal to zero when taking the second derivatives of each component (since the x - t components equal zero). I received a very messy x components after partially differentiating it twice, and noticed that the electric field doesn't have a time component in it, so it should equal zer, but I didn't see how my differentiated x part could equal zero too. Is this a good way to go about it? With the wave equations, substitute in the transformed coordinates? Otherwise, I've started the tensor formation that you said, with the field strength and the dual tensors, I derived Maxwell's equations via the four-vectors of current and potential. I figured I could simply transform the field strength tensor and the dual tensor each by Lorentz transformation matrices, then take those transformed tensors and try to derive Maxwell's equations by the same previous method, and receive the same result. But, I was confused as to what transformation matrices to use on the tensors, since they are second-rank tensors. What matrices would I use? Which way is better, if either of them are good?

 

[robphy]

You can show that the 1+1 wave equation is not invariant under a Galilean-boost. [Take care with the Chain Rule.]
It is invariant under a Lorentz-boost (as suggested by the d'Alembert form of the solution). [Use the d'Alembert form and light-cone coordinates.]

The calculations in terms of components are tedious. It's worth doing explicitly... then doing it tensorially.

I don't have the patience right now to $$\LaTeX$$ the steps in this exercise. It might be best if you show your explicit steps, which we can comment on. You might find some help from
http://farside.ph.utexas.edu/teaching/jk1/lectures/node6.html
http://www2.maths.ox.ac.uk/~nwoodh/sr/index.html

 

[Mentz114]

Dahaka,

But, I was confused as to what transformation matrices to use on the tensors, since they are second-rank tensors. What matrices would I use?

Have a look here. I managed to boost (Lorentz transform) the F tensor after some help.

https://www.physicsforums.com/showthread.php?t=197942&goto=newpost