Electromagnetic stress–energy tensor

Algebraic properties

The electromagnetic stress–energy tensor has several algebraic properties:

• It is a symmetric tensor: $${\displaystyle T^{\mu \nu }=T^{\nu \mu }}$$
• The tensor ${\displaystyle T^{\nu }{}_{\alpha }}$ is traceless (in 4D): $${\displaystyle T^{\alpha }{}_{\alpha }=0.}$$ Proof

  Starting with $${\displaystyle T^{\mu }{}_{\mu }={\eta }_{\mu \nu }{T}^{\mu \nu }}$$

  Using the explicit form of the tensor, $${\displaystyle T^{\mu }{}_{\mu }=\frac{1}{{\mu }_0}\left[{\eta }_{\mu \nu }{F}^{\mu \alpha }{F}^{\nu }{}_{\alpha }-{\eta }_{\mu \nu }{\eta }^{\mu \nu }\frac{1}{4}{F}^{\alpha \beta }{F}_{\alpha \beta }\right]}$$

  Lowering the indices and using the fact that ⁠${\displaystyle {\eta }^{\mu \nu }{\eta }_{\mu \nu }={\delta }_{\mu }^{\mu }}$⁠, $${\displaystyle T^{\mu }{}_{\mu }=\frac{1}{{\mu }_0}\left[F^{\mu \alpha }{F}_{\mu \alpha }-{\delta }_{\mu }^{\mu }\frac{1}{4}{F}^{\alpha \beta }{F}_{\alpha \beta }\right]}$$

  Then, using ⁠${\displaystyle {\delta }_{\mu }^{\mu }=4}$⁠, $${\displaystyle T^{\mu }{}_{\mu }=\frac{1}{{\mu }_0}\left[F^{\mu \alpha }{F}_{\mu \alpha }-F^{\alpha \beta }{F}_{\alpha \beta }\right]}$$

  Note that in the first term, ${\displaystyle \mu }$ and ${\displaystyle \alpha }$ are dummy indices, so we relabel them as ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ respectively. $${\displaystyle T^{\alpha }{}_{\alpha }=\frac{1}{{\mu }_0}\left[F^{\alpha \beta }{F}_{\alpha \beta }-F^{\alpha \beta }{F}_{\alpha \beta }\right]=0}$$

• The energy density is positive-definite: $${\displaystyle T^{00}\ge 0}$$

The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The trace of the energy–momentum tensor is a Lorentz scalar; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy–momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the photon.

Conservation laws

The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is: $${\displaystyle {\mathrm{\partial }}_{\nu }{T}^{\mu \nu }+{\eta }^{\mu \rho }{f}_{\rho }=0}$$ where ${\displaystyle f_{\rho }}$ is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 3D conservation laws $${\displaystyle \begin{array}{cc}\frac{\mathrm{\partial }{u}_{\mathrm{e}\mathrm{m}}}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{S}+\mathbf{J}\cdot \mathbf{E}& =0\\ \frac{\mathrm{\partial }{\mathbf{p}}_{\mathrm{e}\mathrm{m}}}{\mathrm{\partial }t}-\mathrm{\nabla }\cdot \sigma +\rho \mathbf{E}+\mathbf{J}\times \mathbf{B}& =0\iff {ϵ}_0{\mu }_0\frac{\mathrm{\partial }\mathbf{S}}{\mathrm{\partial }t}-\mathrm{\nabla }\cdot \sigma +\mathbf{f}=0\end{array}}$$ respectively describing the electromagnetic energy density $${\displaystyle u_{\mathrm{e}\mathrm{m}}=\frac{1}{2}\left({ϵ}_0{\mathbf{E}}^2+\frac{1}{{\mu }_0}{\mathbf{B}}^2\right)}$$ and electromagnetic momentum density $${\displaystyle {\mathbf{p}}_{\mathrm{e}\mathrm{m}}=\frac{\mathbf{S}}{c^2},}$$ where ${\displaystyle \mathbf{J}}$ is the electric current density, ${\displaystyle \rho }$ the electric charge density, and ${\displaystyle \mathbf{f}}$ is the Lorentz force density.