Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.8.
We’ve seen that Newton’s law for a particle with mass and charge moving in an electromagnetic field given by the field tensor is
Moore offers 5 other options for this equation. The options , , and are all invalid since the tensor indices don’t balance on each side of the equation. Remember that an implied sum occurs only if one repeated index is upper and the other is lower.
The option is invalid, since the time in the derivative is not an invariant between inertial frames. We must use the proper time .