Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 7; Problems 7.1.
Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Problem 12.48.
To apply this to Griffiths problem 12.48, use
in place of in what follows. The results are the same.
The electromagnetic field tensor is
We can use the usual tensor transformation rules to see how the electric and magnetic fields transform under a Lorentz transformation. We get
where the Lorentz transformation matrix is
As we saw when discussing the inertia tensor, we can write this transformation as a matrix equation
The first product is
The final product is
Using we get
From this, we see that
Unlike lengths, the components of and in the direction of motion are unchanged, while those perpendicular to the motion are altered.