The symmetric <stress-energy tensor> obtained by metric variation is
$$
T^{\mu\nu}=-F^{\mu\rho}F^\nu{}_{\rho}
+\frac14\eta^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}
+m^2\left(A^\mu A^\nu-\frac12\eta^{\mu\nu}A^\rho A_\rho\right).
$$
It is conserved on shell. With signature $(+---)$ its energy density is
$$
T^{00}=\frac12(\mathbf E^2+\mathbf B^2)
+\frac{m^2}{2}\{(A^0)^2+\mathbf A^2\}\geq0.
$$
The <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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-301/1/b/solution