SI and geometrized Einstein–Maxwell parameters

Use the smooth oriented, time-oriented Lorentzian four-manifold, metric signature $(-,+,+,+)$, and arbitrary smooth real two-form $F$ of Oriented Lorentzian fields, observers and coordinate conventions. In a chart using the coordinate convention $x^0=ct$ with all four coordinates assigned metres, assign the SI component unit tesla to $F_{ab}$; the two-form $F=\frac{1}{2}{F}_{ab}d{x}^a\wedge d{x}^b$ then has unit weber.

For this conversion dictionary, take $c>0$, $G>0$, and ${\mu }_0>0$ as numerical input parameters, with SI dimensions

$[c]=\mathrm{m}{\mathrm{s}}^{-1},[G]={\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2},[{\mu }_0]=\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{C}}^{-2}\mathrm{.}$

Define the derived parameters

${\epsilon }_0:=\frac{1}{{\mu }_0{c}^2},\kappa :=\frac{8\pi G}{c^4},\alpha :=\sqrt{\frac{\kappa }{2{\mu }_0}},\mathcal{F}:=\alpha F\mathrm{.}$

Thus ${\epsilon }_0$ has unit ${\mathrm{C}}^2{\mathrm{s}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{m}}^{-3}$, $\kappa$ has unit $\mathrm{m}{\mathrm{J}}^{-1}$, $\alpha$ has unit $\mathrm{C}\mathrm{s}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{m}}^{-1}$, and the components ${\mathcal{F}}_{ab}$ have unit ${\mathrm{m}}^{-1}$. In modern SI, $c$ is fixed exactly by the definition of the metre; ${\mu }_0$ is not fixed exactly and its numerical value is determined experimentally. Here the equations use only the positive symbolic parameters above, not a selected numerical value.

For a physical mass $M_{\text{\hspace{0.17em}⁣}p}>0$ in kilograms, signed charge $Q_e$ in coulombs, and angular momentum $J_{\mathrm{a}\mathrm{n}\mathrm{g}}$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, define the geometric length parameters

$m_{\mathrm{g}\mathrm{e}\mathrm{o}}:=\frac{G{M}_{\text{\hspace{0.17em}⁣}p}}{c^2},q_{\mathrm{g}\mathrm{e}\mathrm{o}}:=\sqrt{\frac{G}{4\pi {\epsilon }_0{c}^4}}{Q}_e,a_{\mathrm{g}\mathrm{e}\mathrm{o}}:=\frac{J_{\mathrm{a}\mathrm{n}\mathrm{g}}}{M_{\text{\hspace{0.17em}⁣}p}c}\mathrm{.}$

For a supplied magnetic-flux parameter ${\mathrm{\Phi }}_B$ in webers, also define $p_{\mathrm{g}\mathrm{e}\mathrm{o}}:=\alpha {\mathrm{\Phi }}_B\mathrm{/}(4\pi )$ in metres. This is only a conversion of an already specified flux value; it asserts neither a global potential nor a magnetic source. The inverse parameter maps are $M_{\text{\hspace{0.17em}⁣}p}=c^2{m}_{\mathrm{g}\mathrm{e}\mathrm{o}}\mathrm{/}G$, $Q_e=\sqrt{4\pi {\epsilon }_0{c}^4\mathrm{/}G}{q}_{\mathrm{g}\mathrm{e}\mathrm{o}}$, $J_{\mathrm{a}\mathrm{n}\mathrm{g}}=M_{\text{\hspace{0.17em}⁣}p}c{a}_{\mathrm{g}\mathrm{e}\mathrm{o}}$, and ${\mathrm{\Phi }}_B=4\pi p_{\mathrm{g}\mathrm{e}\mathrm{o}}\mathrm{/}\alpha$.

The mass, charge, angular momentum and flux are inputs to this dictionary; their primitive or derived status in any physical model is framework-specific. No field equation, Maxwell postulate, matter law, or empirical value is part of this definition.

The Lorentzian geometry supplier records the inherited tangent-bundle background under Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$; this dictionary uses no further choice and does not assume the full Axiom of Choice.