User:JRSpriggs/EM in GR

Here are the correspondences which I, J.R.Spriggs, prefer when relating electromagnetism in general relativity to its classical equivalents.

From Maxwell's equations in curved spacetime, we have the general relativistic version of the equations of electromagnetism in a vacuum:
 * $$F_{\alpha \beta} \, = \, \partial_{\alpha} A_{\beta} \, - \, \partial_{\beta} A_{\alpha} \,$$
 * $$\mathcal{D}^{\mu\nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu\alpha} \, F_{\alpha\beta} \, g^{\beta\nu} \, \frac{\sqrt{-g}}{c} \,$$
 * $$J^{\mu} \, = \, \partial_\nu \mathcal{D}^{\mu \nu} \,$$
 * $$f_\mu \, = \, F_{\mu\nu} \, J^\nu \,$$

In an inertial frame of reference, I use the following correspondences. Each component is in SI units. For square matrices, the first index is the row and the second is the column.

location in spacetime
 * $$x^{\mu} = (t, x, y, z)$$

partial derivative (used in gradient, curl, or divergence)
 * $$\partial_{\mu} = \left(\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$$

metric
 * $$g_{\mu \nu} = \eta_{\mu \nu}=\begin{pmatrix} -c^2 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & +1 \end{pmatrix}$$

inverse metric
 * $$g^{\mu \nu} = \eta^{\mu \nu}=\begin{pmatrix} -\frac{1}{c^2} & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & +1 \end{pmatrix}$$

electromagnetic potential
 * $$A_{\mu} = (-\phi, A_x, A_y, A_z)$$

electromagnetic field
 * $$F_{\mu \nu} = \left( \begin{matrix}

0  & -E_x & -E_y & -E_z \\ E_x & 0   &  B_z & -B_y \\ E_y & -B_z & 0   & B_x \\ E_z & B_y & -B_x & 0 \end{matrix} \right)\,$$ electromagnetic displacement
 * $$\mathcal{D}^{\mu \nu} = \left( \begin{matrix}

0   &  D_x &  D_y &  D_z \\ -D_x & 0   &  H_z & -H_y \\ -D_y & -H_z & 0   & H_x \\ -D_z & H_y & -H_x & 0 \end{matrix} \right)\,$$ electric current density
 * $$J^{\mu} = (\rho, J_x, J_y, J_z)$$

density of Lorentz force
 * $$f_{\mu} = (-\text{ power density }, f_x, f_y, f_z)$$

in materials where the magnetization or polarization are non-zero
 * $$\mathcal{D}^{\mu \nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu \alpha} \, F_{\alpha \beta} \, g^{\beta \nu} \, \frac{\sqrt{-g}}{c} \, - \, \mathcal{M}^{\mu \nu} \,.$$
 * $$\mathcal{M}^{\mu \nu} = \left( \begin{matrix}

0   & -P_x & -P_y & -P_z \\ P_x & 0   &  M_z & -M_y \\ P_y & -M_z & 0   &  M_x \\ P_z & M_y & -M_x & 0 \end{matrix} \right)\,$$