Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 4; Problems 4.10.
The electromagnetic field tensor transforms between inertial frames by using 2 Lorentz transformations (we’ll see why in a future post):
Using this, let’s see how the quantity that we considered in the last post transforms. Since the metric tensor is invariant under Lorentz transformations
We can now use
So we substitute in 2
Thus the quantity is invariant under Lorentz transformations. As we saw in the last post, this quantity is