SI and geometric solution parameter identifications

Use the SI quantities, coordinate convention, and primitive/derived distinction of Einstein–Maxwell primitive quantities and SI conventions, and the zero-cosmological-constant geometric electrovac flux normalization of Geometric electrovac equations and flux normalization. This definition is the model-local parameter map for the RN and KN ansatz families; it is not a claim that nature contains an object with these parameters.

Take $G_N>0$, $c>0$, and ${\mu }_0>0$ as the gravitational and electromagnetic constants used by this coupled model, with ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$. Let $m_{\ast }>0$ be a selected SI mass parameter in kg, $J\in \mathbb{R}$ a selected angular-momentum parameter in kg m${}^2$ s${}^{-1}$, $Q_e\in \mathbb{R}$ a selected electric charge parameter in C, and ${\mathrm{\Phi }}_B\in \mathbb{R}$ a selected magnetic flux in Wb through an oriented sphere. These are the chosen SI inputs of this parametrization; this role is local to the solution families and is not a universal ontology for mass, angular momentum, or charge.

For the geometric flux integrals below, retain the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention explicitly stated by Geometric electrovac equations and flux normalization and defined by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). No arbitrary-index choice principle or magnetic-charge quantization is added.

Set

$\alpha :=\sqrt{\frac{4\pi G_N}{{\mu }_0{c}^4}},\mathcal{F}:=\alpha F,$

where $F$ is the SI electromagnetic two-form of Einstein–Maxwell primitive quantities and SI conventions and $\mathcal{F}$ is its geometric normalization. In length coordinates $F_{ab}$ has unit T, ${\mathcal{F}}_{ab}$ has unit m${}^{-1}$, and $\alpha$ has unit T${}^{-1}$m${}^{-1}$. For an oriented sphere $S$, write the geometric flux parameters as

$q_e:=\frac{1}{4\pi }{\int }_S{\star }_g\mathcal{F},q_m:=\frac{1}{4\pi }{\int }_S\mathcal{F}\mathrm{.}$

The SI electric-flux convention is $Q_e={\epsilon }_{0}c{\int }_S{\star }_{g}F$, and the magnetic quantity used here is the flux ${\mathrm{\Phi }}_B={\int }_{S}F$. Thus

$q_e=\frac{\alpha }{4\pi {\epsilon }_{0}c}{Q}_e=\sqrt{\frac{G_N}{4\pi {\epsilon }_0{c}^4}}{Q}_e,q_m=\frac{\alpha }{4\pi }{\mathrm{\Phi }}_B\mathrm{.}$

The second equality in the electric formula uses the exact SI relation ${\epsilon }_0{\mu }_0{c}^2=1$. Both $q_e$ and $q_m$ have units of length; $q_m$ is a flux parameter and is not assigned a universal SI magnetic-charge convention. If one separately chooses the convention $Q_m:={\mathrm{\Phi }}_B\mathrm{/}{\mu }_0$ with unit A m, then the same definition reads $q_m=\sqrt{G_N{\mu }_0\mathrm{/}(4\pi c^4)}{Q}_m$; this optional relabeling adds no new physical assumption or quantization rule.

For the metric parameters, set

$M:=\frac{G_N{m}_{\ast }}{c^2},a:=\frac{J}{m_{\ast }c},Q_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}^2:=q_e^2+q_m^2\mathrm{.}$

Here $M$, $a$, $q_e$, and $q_m$ are derived length parameters, while $m_{\ast }$, $J$, $Q_e$, ${\mathrm{\Phi }}_B$, $G_N$, $c$, and ${\mu }_0$ are the selected inputs of this map. The dimensionless metric coefficients use $M\mathrm{/}r$, $a\mathrm{/}r$, and $Q_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}^2\mathrm{/}{r}^2$. Horizon radii, areas, angular velocities, and surface gravities are further derived quantities of a specified solution, not additional primitive inputs.

These parameter identifications do not establish that $M$ equals an ADM or Komar mass, or that $J$ equals an ADM or Komar angular momentum; such identifications need their own global hypotheses and proofs. They supply no empirical values for $G_N$ or ${\mu }_0$, no measured black-hole parameters, no global potential in a nonzero magnetic-flux sector, and no claim of charge quantization. The source treatments use differing signatures, charge normalizations, and natural or Gaussian unit conventions; the SI formulas above are fixed by the stated local conventions and are not copied without conversion.