Harmonic static multicenter electrovac model

Given: The smooth oriented and time-oriented Lorentzian data and geometric electrovac equations above; constants $G_N,c,{\mu }_0>0$; the SI-to-geometric normalization from SI and geometric solution parameter identifications; and the zero-source, zero-$\mathrm{\Lambda }$ specialization of the adopted equation-based model.

Geometric electrovac equations and flux normalization: the geometric datum has smooth $\mathcal{F}$, signature $(-,+,+,+)$, and equations $d\mathcal{F}=d({\star }_g\mathcal{F})=0$ and $\mathrm{Ric}=2Q(\mathcal{F})$, with the displayed trace-free quadratic tensor.

SI and geometric solution parameter identifications: $\mathcal{F}=\alpha F$ and ${\alpha }^2=4\pi G_N\mathrm{/}({\mu }_0{c}^4)$; in length coordinates $\mathcal{F}$ has units ${\mathrm{m}}^{-1}$ and $F$ has units T.

Einstein–Maxwell primitive quantities and SI conventions: ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$, ${\kappa }_E=8\pi G_N\mathrm{/}{c}^4$, and the Einstein tensor has units ${\mathrm{m}}^{-2}$ while stress has units ${\mathrm{J}\mathrm{m}}^{-3}$.

Sourced Maxwell sign and current conservation: for $v^b={\mathrm{\nabla }}_a{F}^{ab}$, $d({\star }_{g}F)=-{\iota }_v{\mathrm{v}\mathrm{o}\mathrm{l}}_g$; contraction with the nonvanishing volume form is injective. The current mathematical carrier remains draft; its complete conditional proof and current Step 1 authoring-ready record were inspected, and this use is not independent proof approval.

Equation-based Einstein–Maxwell model: the adopted smooth-field equations are $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, and $G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the inherited geometric convention carries only ${\mathrm{A}\mathrm{C}}_{\omega }$, the assertion that each countable family of nonempty sets has a choice function; the finite pointwise rescaling and contractions below make no choice.

Given: A finite nonempty set of distinct centers, positive SI masses, one common sign $\sigma$, positive SI constants, and the smooth source-free branch of the adopted equation-based postulate.

Positive harmonic conformastatic electrovac data: for smooth positive harmonic $H$, the metric is $g_H=-H^{-2}(d{x}^0{)}^2+H^2{\delta }_{ij}d{x}^{i}d{x}^j$, with ${\mathcal{A}}_{\sigma }=\sigma (H^{-1}-1)d{x}^0$ and ${\mathcal{F}}_{\sigma }=d{\mathcal{A}}_{\sigma }$.

Finite positive multicenter MP exterior and fluxes: for finitely many distinct centers and positive lengths $m_A$, $H=1+{\sum }_A{m}_A\mathrm{/}\mathrm{\mid }x-a_A\mathrm{\mid }$ is positive harmonic on the punctured exterior; its geometric sphere fluxes are $q_e=\sigma m_A$ locally and $q_e=\sigma {\sum }_A{m}_A$ outside all centers, with $q_m=0$. This proposition is draft with a complete current proof and Step 1 authoring-ready record; this consumer uses its statement conditionally, not as independent proof approval.

Full harmonic Majumdar-Papapetrou Einstein-Maxwell verification: the harmonic data satisfy $d{\mathcal{F}}_{\sigma }=d({\star }_g{\mathcal{F}}_{\sigma })=0$ and $\mathrm{Ric}=2Q({\mathcal{F}}_{\sigma })$. This draft theorem has a complete current proof and Step 1 authoring-ready record; its result is consumed conditionally, without claiming independent approval.

Geometric electrovac data solve the selected SI equations: every such geometric electrovac datum, after $F={\alpha }^{-1}\mathcal{F}$, satisfies the zero-source SI equation-based Einstein–Maxwell model. This local theorem is draft with its complete proof authored; it is used conditionally and has no independent approval.

SI and geometric solution parameter identifications: $m_A=G_N{m}_{\ast ,A}\mathrm{/}{c}^2$, $q_e=\alpha (4\pi {\epsilon }_{0}c{)}^{-1}{Q}_e$, $q_m=\alpha (4\pi {)}^{-1}{\mathrm{\Phi }}_B$, and ${\alpha }^2=4\pi G_N\mathrm{/}({\mu }_0{c}^4)$.

Equation-based Einstein–Maxwell model: the adopted field equations use $G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$; this item sets all three sources to zero.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the geometric flux convention inherits only ${\mathrm{A}\mathrm{C}}_{\omega }$; the finite center family and its finite flux sums require no stronger choice principle.

Given: $m_{\ast }>0$, positive SI constants, $\sigma \in \{-1,+1\}$, zero magnetic flux, and the exact MP and RN coordinate conventions.

Positive harmonic conformastatic electrovac data: the single-center harmonic datum has $H=1+m\mathrm{/}\rho$, $g=-H^{-2}(d{x}^0{)}^2+H^2(d{\rho }^2+{\rho }^2{\gamma }_{S^2})$, and ${\mathcal{F}}_{\sigma }=d(\sigma (H^{-1}-1)d{x}^0)$.

SI and geometric solution parameter identifications: $M=G_N{m}_{\ast }\mathrm{/}{c}^2$, $q_e=\sqrt{G_N\mathrm{/}(4\pi {\epsilon }_0{c}^4)}{Q}_e$, $q_m=\alpha {\mathrm{\Phi }}_B\mathrm{/}(4\pi )$, and $\mathcal{F}=\alpha F$.

Single-center MP is extremal Reissner-Nordström exterior: for $H=1+m\mathrm{/}\rho$ and $R=\rho +m$, the full MP metric and two-form equal RN data with $M=m$, $q_e=\sigma m$, $q_m=0$, and $f=(1-m\mathrm{/}R{)}^2$ on $R>m$, with the positive $x^0$ time direction and the orientation-preserving radial map of the mathematical supplier. This is a draft mathematical proposition with a complete current proof and Step 1 authoring-ready record; the coordinate identity is used conditionally.

Harmonic static multicenter electrovac model: the single-center specialization satisfies the selected SI source-free equations on its punctured exterior. This current physical theorem is draft with a complete authored proof and Step 1 authoring-ready record, not independent proof approval.

Dyonic RN as adopted Einstein–Maxwell model: the RN datum with these parameter values satisfies the same selected SI source-free equations wherever $r>0$ and $f\mathrm{\ne }0$. This current physical theorem is draft with a complete authored proof and Step 1 authoring-ready record, not independent proof approval.

Dyonic Reissner-Nordström metric and field domains: the RN metric and two-form are $g=-f(d{x}^0{)}^2+f^{-1}d{r}^2+r^2{\gamma }_{S^2}$ and $\mathcal{F}=-q_e{r}^{-2}d{x}^0\wedge dr+q_m{\omega }_{S^2}$.

Equation-based Einstein–Maxwell model: the equation-based model is adopted, not derived; the source-free conclusion specializes to $\mathrm{\Lambda }=0$, $j=0$, and $T^m=0$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the inherited geometric and flux convention is at most ${\mathrm{A}\mathrm{C}}_{\omega }$; the radial diffeomorphism and substitutions are choice-free.

Given: $L>0$, one of the two open product domains and functions $f$, positive SI constants, the geometric product data, and the zero-source branch of the adopted postulate.

Constant-radius Bertotti-Robinson product data: $g=L^2(-fd{\tau }^2+f^{-1}d{z}^2+{\gamma }_{S^2})$ and ${\mathcal{F}}_L=-Ld\tau \wedge dz$ on the two stated open chart types.

Product electrovacuum equations from full factor Ricci: for each such datum, $d{\mathcal{F}}_L=d({\star }_g{\mathcal{F}}_L)=0$ and $\mathrm{Ric}=2Q({\mathcal{F}}_L)$; the full factor Ricci tensors are computed in its current proof. This mathematical theorem is draft with a complete proof and Step 1 authoring-ready record; its result is used conditionally, not as independent approval.

Geometric electrovac data solve the selected SI equations: a source-free geometric electrovac datum scales by $F={\alpha }^{-1}\mathcal{F}$ to solve the selected SI equation model. This local theorem is draft with its complete proof authored and is used conditionally, not as independent proof approval.

SI and geometric solution parameter identifications: ${\alpha }^2=4\pi G_N\mathrm{/}({\mu }_0{c}^4)$ and the geometric field has units ${\mathrm{m}}^{-1}$ while the SI two-form has units T in length coordinates.

Equation-based Einstein–Maxwell model: the equation-based law is an adopted physical assumption; this item specializes it to $\mathrm{\Lambda }=0$, $j=0$, and $T^m=0$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the inherited geometric background convention is ${\mathrm{A}\mathrm{C}}_{\omega }$; this pointwise product calculation makes no choice.

Given: $L>0$, $\lambda >0$, the extremal SI parameter values, the exact RN and product definitions, and the adopted zero-source equation model.

Extremal RN to Bertotti-Robinson in C-infinity local scaling: for $M=q_e=L$, $q_m=0$, the exact pullbacks and geometric fields converge smoothly on compact subsets of $R>0$ to the $f=z^2$ Bertotti–Robinson product after $T=L\tau$, $R=Lz$. This mathematical theorem is draft with a complete current proof and Step 1 authoring-ready record; it is used conditionally.

Dyonic RN as adopted Einstein–Maxwell model: the dyonic RN data with the parameter map of SI and geometric solution parameter identifications solve the adopted SI equations on the open domain where $r>0$ and $f\mathrm{\ne }0$. This physical theorem is draft with a complete current proof and Step 1 authoring-ready record; its solution claim is used conditionally.

Constant-radius charged product model: the $f=z^2$ product with $L>0$ solves the same source-free SI model on $z>0$. This physical theorem is draft with a complete proof authored in this batch and is used conditionally, without independent approval.

SI and geometric solution parameter identifications: $q_e=\sqrt{G_N\mathrm{/}(4\pi {\epsilon }_0{c}^4)}{Q}_e$, $M=G_N{m}_{\ast }\mathrm{/}{c}^2$, ${\mathrm{\Phi }}_B=0$ gives $q_m=0$, and $\alpha$ is a positive constant.

Equation-based Einstein–Maxwell model: the same adopted postulate is used with $\mathrm{\Lambda }=0$, $j=0$, and $T^m=0$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the inherited geometric convention is at most ${\mathrm{A}\mathrm{C}}_{\omega }$; the scaling map and compact convergence do not make a choice.

Given: The length-coordinate null-wave data, smoothness and harmonicity hypotheses, positive SI constants, and the source-free specialization of the adopted postulate.

Smooth null electrovac wave candidate data: $g=-2dudv+d{x}^2+d{y}^2+Pd{u}^2$, $\mathcal{A}=-(ax+by)du$, and $\mathcal{F}=adu\wedge dx+bdu\wedge dy$, with arbitrary smooth profiles and $h_{xx}+h_{yy}=0$.

Exact electrovac plane-wave family: the defined data satisfy $d\mathcal{F}=d({\star }_g\mathcal{F})=0$ and $\mathrm{Ric}=2Q(\mathcal{F})$ for the full stated profile class. This mathematical theorem is draft with a complete current proof and Step 1 authoring-ready record; it is used conditionally.

Geometric electrovac data solve the selected SI equations: constant rescaling by ${\alpha }^{-1}$ turns any source-free geometric electrovac datum into a solution of the selected SI model. This local theorem is draft with its complete proof authored and is used conditionally, not as independent proof approval.

SI and geometric solution parameter identifications: $\alpha >0$ is constant, $\mathcal{F}=\alpha F$, and the two-form components have units ${\mathrm{m}}^{-1}$ geometrically and T in the SI convention.

Equation-based Einstein–Maxwell model: the equation-based model is an adopted physical assumption, specialized here to $\mathrm{\Lambda }=0$, $j=0$, and $T^m=0$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the inherited geometry and Hodge convention carries ${\mathrm{A}\mathrm{C}}_{\omega }$; the local field rescaling makes no choice.