Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 4; Problems 4.8, 4.9.
The electromagnetic field tensor is
If we contract with the metric tensor in flat space, we get
In the first line, we used ; in the second, and in the last line, we swapped the dummy indexes and . Thus the original quantity is equal to its negative, so
Now consider . First, since and for :
This is because the only sign change occurs when , which is when , so all elements change sign.
This time, the sign change occurs when , so elements change sign. Thus lowering both indices in flat spacetime in rectangular coordinates changes the sign of all the electric field entries, but leaves the magnetic field unchanged.
Combining them, we get
To work this out, note that if we first formed the matrix product , then is the sum of the diagonal elements of this matrix product. That is