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


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

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


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

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


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

allowing for a covariant formulation using potentials in matter.