Electromagnetic stress-energy tensor

Electromagnetic stress-energy tensor
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "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. In free space, it is (SI units):

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

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 $F_{\mu\nu}\!$ ,

Lorentzian metric tensor $g_{\mu\nu}\!$ , 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.

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

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

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:

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

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

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

$\partial_{\nu}T^{\mu\nu}= - k^{\mu}$

where $k μ$ is the four-force density.

From this equation, the following conservation laws can be derived:

$\frac{\partial }{\partial t} (u_{em}+u_{mech}) + \vec{\nabla}\cdot \vec{S}= 0$

$\frac{\partial }{\partial t} (\vec{p}_{em}+\vec{p}_{mech})-\vec{\nabla}\cdot \sigma= 0$

with

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

Electromagnetic energy density $u_{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2$

Mechanical energy density $u_{mech}=\vec{E}\cdot\vec{J}$

Electromagnetic momentum density $\vec{p}_{em} = \mu_0\epsilon_0 \vec{S}$

Mechanical momentum density $\vec{p}_{mech}$

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

See also