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#Summary, 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

${\mathcal {D}}^{\mu \nu }\,=\,{\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\frac {\sqrt {-g}}{c}}\,-\,{\mathcal {M}}^{\mu \nu }\,.$

${\mathcal {M}}^{\mu \nu }=\left({\begin{matrix}0&-P_{x}&-P_{y}&-P_{z}\\P_{x}&0&M_{z}&-M_{y}\\P_{y}&-M_{z}&0&M_{x}\\P_{z}&M_{y}&-M_{x}&0\end{matrix}}\right)\,$