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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c85df07621e236ed30b2d84de127540516e9dff7)
where
is the electromagnetic field tensor,
is the metric tensor and
is the permeability of free space
And in explicit matrix form:
,
with
- Poynting vector
,
- electromagnetic field tensor
,
- metric tensor
, and
- Maxwell stress tensor
.
Note that
where c is light speed.
In cgs units, we simply substitute
with
and
with
:
.
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/450645233fbc84d8135d9c9ea4c42b12ef6482a7)
where Poynting vector becomes the form:
.
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,
, of the energy momentum tensor represents the flux of the αth-component of the four-momentum of the electromagnetic field,
, going through a hyperplane xβ = constant. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space-time) in general relativity.