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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
α
β
=
∂
α
A
β
−
∂
β
A
α
{\displaystyle F_{\alpha \beta }\,=\,\partial _{\alpha }A_{\beta }\,-\,\partial _{\beta }A_{\alpha }\,}
D
μ
ν
=
1
μ
0
g
μ
α
F
α
β
g
β
ν
−
g
c
{\displaystyle {\mathcal {D}}^{\mu \nu }\,=\,{\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\frac {\sqrt {-g}}{c}}\,}
J
μ
=
∂
ν
D
μ
ν
{\displaystyle J^{\mu }\,=\,\partial _{\nu }{\mathcal {D}}^{\mu \nu }\,}
f
μ
=
F
μ
ν
J
ν
{\displaystyle 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
μ
=
(
t
,
x
,
y
,
z
)
{\displaystyle x^{\mu }=(t,x,y,z)}
partial derivative (used in gradient, curl, or divergence)
∂
μ
=
(
∂
∂
t
,
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
{\displaystyle \partial _{\mu }=\left({\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}
metric
g
μ
ν
=
η
μ
ν
=
(
−
c
2
0
0
0
0
+
1
0
0
0
0
+
1
0
0
0
0
+
1
)
{\displaystyle 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
μ
ν
=
η
μ
ν
=
(
−
1
c
2
0
0
0
0
+
1
0
0
0
0
+
1
0
0
0
0
+
1
)
{\displaystyle 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
μ
=
(
−
ϕ
,
A
x
,
A
y
,
A
z
)
{\displaystyle A_{\mu }=(-\phi ,A_{x},A_{y},A_{z})}
electromagnetic field
F
μ
ν
=
(
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
)
{\displaystyle 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
D
μ
ν
=
(
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
)
{\displaystyle {\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
μ
=
(
ρ
,
J
x
,
J
y
,
J
z
)
{\displaystyle J^{\mu }=(\rho ,J_{x},J_{y},J_{z})}
density of Lorentz force
f
μ
=
(
−
power density
,
f
x
,
f
y
,
f
z
)
{\displaystyle f_{\mu }=(-{\text{ power density }},f_{x},f_{y},f_{z})}
in materials where the magnetization or polarization are non-zero
D
μ
ν
=
1
μ
0
g
μ
α
F
α
β
g
β
ν
−
g
c
−
M
μ
ν
.
{\displaystyle {\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 }\,.}
M
μ
ν
=
(
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
)
{\displaystyle {\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)\,}