Spherical electrovacuum quotient and charged mass

An exact spherical Einstein–Maxwell–$\mathrm{\Lambda }$ datum consists of the orbit geometry specified above, a smooth rotationally invariant real two-form $\mathcal{F}$, a constant $\mathrm{\Lambda }\in \mathbb{R}$, and the equations $d\mathcal{F}=0,d({\star }_g\mathcal{F})=0,\mathrm{Ric}(g{)}_{ab}=\mathrm{\Lambda }{g}_{ab}+2Q_{ab}(\mathcal{F}),$ where $Q_{ab}(\mathcal{F}):={\mathcal{F}}_{ac}{\mathcal{F}}_b{}^c-\frac{1}{4}{g}_{ab}{\mathcal{F}}_{cd}{\mathcal{F}}^{cd}\mathrm{.}$ This is a mathematical system of equations, not a physical postulate. For $\mathrm{\Lambda }=0$ it uses exactly the curvature, Hodge, and Maxwell normalization of the previously defined geometric electrovac equations. For $\mathrm{\Lambda }\mathrm{\ne }0$ the extra constant term is part of this local mathematical definition; it is not supplied by that zero-$\mathrm{\Lambda }$ item.

By the preceding Maxwell reduction lemma, the field has unique constant flux parameters $q_e,q_m\in \mathbb{R}$ and $\mathcal{F}=\frac{q_e}{(r\circ {\pi }_U{)}^2}{\pi }_U^{\ast }{ϵ}_2+q_m{\pi }_S^{\ast }{\omega }_{S^2},Q^2:=q_e^2+q_m^2\mathrm{.}$ In subsequent formulas on $\mathcal{M}$, a tensor or differential form on $U$ or $S^2$ written without a pullback symbol denotes its pullback by ${\pi }_U$ or ${\pi }_S$, respectively; a scalar on $U$ denotes its composition with ${\pi }_U$. Formulas stated on $U$ retain their original base objects. In geometric units, coordinates have length, $r,q_e,q_m$ have length, the metric components are dimensionless, and $\mathrm{\Lambda }$ has units of inverse length squared. Define $z:=q^{-1}(dr,dr),\varpi :=\frac{r}{2}(1-z)+\frac{Q^2}{2r}-\frac{\mathrm{\Lambda }{r}^3}{6}\mathrm{.}$ Thus $z$ is dimensionless and $\varpi$ has length. The scalar $\varpi$ is called the charged spherical mass function; this definition does not assert that it is constant. The field equations are local on $U\times S^2$ and impose no global potential, bundle, horizon, asymptotic, completeness, or extension condition.

The geometric-electrovacuum flux convention inherits only ${\mathrm{A}\mathrm{C}}_{\omega }$ through the supplied geometric electrovac definition. No full axiom of choice is used by this definition or by the subsequent local algebra.

Spherical orbit geometry and rotational invariance

Source-free rotational Maxwell fields have constant fluxes

Geometric electrovac equations and flux normalization

Dyonic Reissner-Nordström metric and field domains