User:Mpatel/sandbox/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 (vacuum), it is given in SI units by:

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

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

And in explicit matrix form:

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

with

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

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

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

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

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

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

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

And in explicit matrix form:

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

where Poynting vector becomes the form:

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

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, ${\displaystyle T^{\alpha \beta }\!}$, of the energy momentum tensor represents the flux of the α^th-component of the four-momentum of the electromagnetic field, ${\displaystyle P^{\alpha }\!}$, 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