User:Mpatel/sandbox/Electromagnetic stress-energy tensor

In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field. In free space (vacuum), it is given in SI units by:

$T_{ab}=\,{\frac {1}{\mu _{o}}}(F_{a}{}^{s}F_{sb}+{1 \over 4}F_{st}F^{st}g_{ab})$

where $F_{ab}$ is the electromagnetic field tensor, $g_{ab}$ is the metric tensor and $\mu _{o}$ is the permeability of free space

And in explicit matrix form:

$T^{\alpha \beta }={\begin{bmatrix}{\frac {1}{2}}(\epsilon _{o}E^{2}+{\frac {1}{\mu _{0}}}B^{2})&S_{x}&S_{y}&S_{z}\\S_{x}&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{bmatrix}}$,

with

Poynting vector ${\vec {S}}={\frac {1}{\mu _{o}}}{\vec {E}}\times {\vec {B}}$,

electromagnetic field tensor $F_{\alpha \beta }\!$,

metric tensor $g_{\alpha \beta }\!$, and

Maxwell stress tensor $\sigma _{ij}=\epsilon _{o}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left({\epsilon _{o}E^{2}+{\frac {1}{\mu _{0}}}B^{2}}\right)\delta _{ij}$.

Note that $c^{2}={\frac {1}{\epsilon _{o}\mu _{0}}}$ where c is light speed.

In cgs units, we simply substitute $\epsilon _{o}\,$ with ${\frac {1}{4\pi }}$ and $\mu _{o}\,$ with $4\pi \,$ :

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

And in explicit matrix form:

$T^{\alpha \beta }={\begin{bmatrix}{\frac {1}{8\pi }}(E^{2}+B^{2})&S_{x}/c&S_{y}/c&S_{z}/c\\S_{x}/c&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}/c&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}/c&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{bmatrix}}$

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.

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

See also

• Stress-energy tensor