User:Maschen/Electromagnetic displacement tensor

The electromagnetic displacement tensor (no standard name) combines the D and H vector fields

${\displaystyle {\mathsf {D}}^{\mu \nu }={\begin{pmatrix}0&-D_{x}c&-D_{y}c&-D_{z}c\\D_{x}c&0&-H_{z}&H_{y}\\D_{y}c&H_{z}&0&-H_{x}\\D_{z}c&-H_{y}&H_{x}&0\end{pmatrix}}.}${\displaystyle {\mathsf {D}}^{\mu \nu }={\begin{pmatrix}0&-D_{x}c&-D_{y}c&-D_{z}c\\D_{x}c&0&-H_{z}&H_{y}\\D_{y}c&H_{z}&0&-H_{x}\\D_{z}c&-H_{y}&H_{x}&0\end{pmatrix}}.}

It is used for covariant formulations of Maxwell's equations in media (sources are free charges and currents), as well as constitutive equations

${\displaystyle {\mathsf {D}}^{\alpha \beta }=E^{\alpha \beta }{}_{\mu \nu }F^{\mu \nu }}${\displaystyle {\mathsf {D}}^{\alpha \beta }=E^{\alpha \beta }{}_{\mu \nu }F^{\mu \nu }}

where F is the electromagnetic field tensor, and E a fourth order tensor to account for anisotropy in the media.

The Lorentz transformations of the D and H fields are readily obtained from

${\displaystyle {\mathsf {D}}^{\alpha \gamma }=\Lambda ^{\alpha }{}_{\beta }\Lambda ^{\gamma }{}_{\delta }{\mathsf {D}}^{\beta \delta }}${\displaystyle {\mathsf {D}}^{\alpha \gamma }=\Lambda ^{\alpha }{}_{\beta }\Lambda ^{\gamma }{}_{\delta }{\mathsf {D}}^{\beta \delta }}

compared to the tedious transformations of the 3d vector fields.

Just as the electromagnetic field tensor, the displacement tensor can be derived from an appropriate potential

${\displaystyle {\mathsf {D}}^{\alpha \beta }=\partial ^{\alpha }{\mathsf {A}}^{\beta }-\partial ^{\beta }{\mathsf {A}}^{\alpha }}${\displaystyle {\mathsf {D}}^{\alpha \beta }=\partial ^{\alpha }{\mathsf {A}}^{\beta }-\partial ^{\beta }{\mathsf {A}}^{\alpha }}

allowing for a covariant formulation using potentials in matter.