Electromagnetic stress-energy tensor

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In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field.

Definition

SI units

In free space in SI units, the electromagnetic stress-energy tensor is (in flat space-times)

$T^{\mu\nu} = \frac{1}{\mu_0}[ F^{\mu \alpha}F^{\nu}{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}] \,.$

where F^μν is the electromagnetic tensor. Note: The tensor T^μν is a symmetric tensor.
And in explicit matrix form:

$T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}(\epsilon_0 E^2+\frac{1}{\mu_0}B^2) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}$,

with

Poynting vector $\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}$,

Electromagnetic field tensor $F_{\mu\nu}\!$,

Minkowski metric tensor $\eta_{\mu\nu}\! = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$, and

Maxwell stress tensor $\sigma_{ij} = \epsilon_0 E_i E_j + \frac{1} {{\mu _0 }}B_i B_j - \frac{1} {2}\left( {\epsilon_0 E^2 + \frac{1} {{\mu _0 }}B^2 } \right)\delta _{ij}$.

Note that $c^2=\frac{1}{\epsilon_0 \mu_0}$ where c is light speed.

CGS units

In free space in cgs-Gaussian units, we simply substitute $\epsilon_0\,$ with $\frac{1}{4\pi}$ and $\mu_0\,$ with $4\pi\,$ :

$T^{\mu\nu} = \frac{1}{4\pi} [ F^{\mu\alpha}F^{\nu}{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}] \,.$

And in explicit matrix form:

$T^{\mu\nu} =\begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}$

where Poynting vector becomes the form:

$\vec{S}=\frac{c}{4\pi}\vec{E}\times\vec{B}$.

The stress-energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the unresolved Abraham-Minkowski controversy (however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)).

The element, $T^{\mu\nu}\!$, of the energy momentum tensor represents the flux of the μ^th-component of the four-momentum of the electromagnetic field, $P^{\mu}\!$, going through a hyperplane x^ν = constant. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space-time) in general relativity.

Conservation laws

The electromagnetic stress-energy tensor allows a compact way of writing the conservation laws of linear momentum and energy by electromagnetism.

$\partial_{\nu}T^{\mu \nu} + \eta^{\mu \rho} \, f_{\rho} = 0 \,$

where f_ρ is the density of the (3D) Lorentz force on matter.

This equation is equivalent to the following 3D conservation laws

$\frac{\partial u_{em}}{\partial t} + \vec{\nabla} \cdot \vec{S} + \vec{J} \cdot \vec{E} = 0 \,$

$\frac{\partial \vec{p}_{em}}{\partial t} - \vec{\nabla}\cdot \sigma + \rho \vec{E} + \vec{J} \times \vec{B} = 0 \,$

where

Electromagnetic energy density (joules/meter³) is $u_{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 \,$

Poynting vector (watts/meter²) is $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \,$

Density of electric current (amperes/meter²) is $\vec{J} \,$

Electromagnetic momentum density (newton·seconds/meter³) is $\vec{p}_{em} = {\vec{S} \over c^2} \,$

Maxwell stress tensor (newtons/meter²) is $\sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}{{\mu _0 }}B_i B_j - \frac{1}{2}\left( {\epsilon_0 E^2 + \frac{1}{{\mu _0 }}B^2 } \right)\delta _{ij} \,$

Density of electric charge (coulombs) is $\rho \,.$

See also

• Stress-energy tensor
• Covariant formulation of classical electromagnetism