Electromagnetic stress–energy tensor

In free space and flat space–time, the electromagnetic stress–energy tensor in SI units is[2]

${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,.}$

where ${\displaystyle F^{\mu \nu }}$ is the electromagnetic tensor and where ${\displaystyle \eta _{\mu \nu }}$ is the Minkowski metric tensor of metric signature (− + + +). When using the metric with signature (+ − − −), the expression on the right of the equation will have opposite sign.

Explicitly in matrix form:

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

where

${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,}$

is the Poynting vector,

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

is the Maxwell stress tensor, and c is the speed of light. Thus, ${\displaystyle T^{\mu \nu }}$ is expressed and measured in SI pressure units (pascals).

The permittivity of free space and permeability of free space in cgs-Gaussian units are

${\displaystyle \epsilon _{0}={\frac {1}{4\pi }},\quad \mu _{0}=4\pi \,}$

then:

${\displaystyle T^{\mu \nu }={\frac {1}{4\pi }}[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }]\,.}$

and in explicit matrix form:

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

where Poynting vector becomes:

${\displaystyle \mathbf {S} ={\frac {c}{4\pi }}\mathbf {E} \times \mathbf {B} .}$

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.[3]

The element ${\displaystyle T^{\mu \nu }\!}$ of the stress–energy tensor represents the flux of the μth-component of the four-momentum of the electromagnetic field, ${\displaystyle P^{\mu }\!}$, going through a hyperplane (${\displaystyle x^{\nu }}$ is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space–time) in general relativity.