Electromagnetism in curved spacetime

serali

1. The problem statement, all variables and given/known data
I'm trying to solve problem 4.1 in Sean Carroll's general relativity, but the continuity equation does not check. The problem, relevant equations and my solution attempt is in the attachment. Any suggestion is welcome.


2. Relevant equations
See atttachment.


3. The attempt at a solution
See atttachment.

Ps: I feel like I'm missing smt trivial here. :)
PPs: I tried with/without symmetrizing and Lorenz gauge.

Attached Files

a.pdf (26.6 KB, 14 views)