Electromagnetic stress–energy tensor

Remarks

In terms of the electric field $E$ and the magnetic field $B$:

1. $T^{00} = \frac{\varepsilon_0}{2} \abs{E}^2 + \frac{1}{2 \mu_0} \abs{B}^2$ is the energy density.
2. $T^{0i} = T^{i0} = S_i / c$, where $S$ is the Poynting vector.
3. $T^{ij} = -\sigma_{ij}$, where $\sigma$ is the Maxwell stress tensor.

Example: for a static uniform field $E = (0, 0, E_0)$, $B = 0$, the tensor is $\frac{\varepsilon_0 E_0^2}{2} \operatorname{diag}(1, 1, 1, -1)$; the last entry is a tension along the field.

The tensor is traceless, $\eta_{\mu\nu} T^{\mu\nu} = 0$, as a direct computation shows in four dimensions. For solutions of Maxwell's equations without sources, $\partial_\nu T^{\mu\nu} = 0$; the component $\mu = 0$ is Poynting's theorem and the components $\mu = i$ are the momentum balance.

Non-example: the tensor $\frac{1}{\mu_0} \eta_{\alpha\beta} F^{\mu\alpha} F^{\beta\nu}$ without the trace term gives $T^{00} = \varepsilon_0 \abs{E}^2$ when $B = 0$, which is twice the correct energy density.