Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 311 / 2 / d / ii / Solution

In an orthonormal frame, $F_{0i}=-E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. The electromagnetic stress-energy tensor givesThe sign of $T_{0i}$ follows directly from $F_{0j}{F}_i{}^j=-E_j{ϵ}_{ijk}{B}_k$. Thusand $T_{00}\ge 0$. These formulas hold in every local Lorentz frame, so the criterion just proved establishes the dominant energy condition. Equality occurs for a nonzero null Maxwell field with $E^2=B^2$ and $\mathbf{E}\cdot \mathbf{B}=0$, whose energy current is null vector.