Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / b / Solution

The symmetric stress-energy tensor obtained by metric variation isIt is conserved on shell. With signature $(+---)$ its energy density isThe canonical stress-energy tensor differs from this symmetric tensor by the divergence of $F^{\mu \rho }{A}^{\nu }$ plus terms proportional to the field equation. Thus its integrated energy agrees after discarding the corresponding total spatial derivative, and positivity holds on shell.