This question is motivated by a desire to understand the math required for an ATM theory I am working on. Thus the methods of solution are of more interest to me than any actual calculations.
From Wikipedia I find that the Reissner–Nordström metric is expressed as:
I understand that this represents a 4x4 diagonal array, .
Does this mean the inverse metric is given by?
I assume this includes the weight of the electric field, which is defined from the potential. This article also gives the electromagnetic potential as
Would ? If so, why did we not define , or even split the difference with and ?