Electromagnetic stress–energy tensor

In physics, the Energy momentum tensor for the electromagnetic field is given by

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

And in explicit matrix form:

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

where

${\overrightarrow {S}}={\frac {c}{4\pi }}{\overrightarrow {E}}\times {\overrightarrow {H}}$ is Poynting vector,

$F_{\alpha \beta }$ is the Electromagnetic field tensor,

$g_{\alpha \beta }$ is the metric tensor and

$\sigma _{\alpha \beta }$ is Maxwell stress tensor