User:Maschen/Electromagnetic displacement tensor

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

:$$

\mathsf{D}^{\mu\nu} =

\begin{pmatrix}

0 & - D_xc & - D_yc & - D_zc \\

D_xc & 0 & - H_z & H_y \\

D_yc & H_z & 0 & - H_x \\

D_zc & - 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

:$\backslash mathsf\{D\}^\{\backslash alpha\backslash beta\}\; =\; E^\{\backslash alpha\backslash beta\}\{\}\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash nu\}$

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

:$\backslash mathsf\{D\}^\{\backslash alpha\backslash gamma\}\; =\; \backslash Lambda^\{\backslash alpha\}\{\}\_\{\backslash beta\}\backslash Lambda^\{\backslash gamma\}\{\}\_\{\backslash delta\}\backslash mathsf\{D\}^\{\backslash beta\backslash delta\}$

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

:$\backslash mathsf\{D\}^\{\backslash alpha\backslash beta\}\; =\; \backslash partial^\backslash alpha\; \backslash mathsf\{A\}^\backslash beta\; -\; \backslash partial^\backslash beta\; \backslash mathsf\{A\}^\backslash alpha$

allowing for a covariant formulation using potentials in matter.