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?
