Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.10, 7.2.
The electromagnetic field tensor is
The various electromagnetic laws can be expressed by the equation
The lowered version of in the flat metric of special relativity is
Because , if we set two of the indices equal in this equation, the LHS is identically zero. For example if :
Since the three terms are cyclic permutations of each other, setting any other pair of indices equal gives the same result.
If we choose , , , we get
This is the law .
If we now choose , , , we get
This is the component of Faraday’s law .
Other choices give the remaining components of Faraday’s law.