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SI and geometric solution parameter identifications

1 · Prerequisites

2 · Summary

Scope

This page fixes a model-local conversion between selected SI mass, angular-momentum, electric-charge and magnetic-flux inputs and the geometric length parameters of the chosen RN and KN solution families. It then states conditional source-free equation-based model deductions for those data and converts selected RN/KN horizon, escape, redshift and duality results to SI readings. The identified parameters are not automatically ADM or Komar quantities, and mathematical draft suppliers are used only through their exact current interfaces and proof status.

The page distinguishes adopted clock and geometric-optics assumptions from derived model predictions. It supplies no measured black-hole values, observational report, empirical precision, charge quantization, or claim that the selected solution families describe an observed object. Claims about global potentials, horizons and escape are restricted to their explicitly specified domains and conventions.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

SI and geometric solution parameter identifications

Physical scope

A model-local map from SI mass, angular-momentum, electric-charge, and magnetic-flux inputs to the length parameters in the selected four-dimensional, zero-cosmological-constant Einstein–Maxwell solution families. It fixes no global ADM or Komar identification, magnetic-charge convention, charge quantization, empirical value, observed black-hole claim, or field law.

Definition

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 $μ_{0} > 0$ as the gravitational and electromagnetic constants used by this coupled model, with $ε_{0} = (μ_{0} c^{2})^{- 1}$ . Let $m_{*} > 0$ be a selected SI mass parameter in kg, $J \in R$ a selected angular-momentum parameter in kg m $^{2}$ s $^{- 1}$ , $Q_{e} \in R$ a selected electric charge parameter in C, and $Φ_{B} \in R$ 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 ${A C}_{ω}$ convention explicitly stated by Geometric electrovac equations and flux normalization and defined by The Axiom of Countable Choice ( ${A C}_{ω}$ ). No arbitrary-index choice principle or magnetic-charge quantization is added.

Set

$α : = \sqrt{\frac{4 π G_{N}}{μ_{0} c^{4}}}, F : = α F,$

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

$q_{e} : = \frac{1}{4 π} \int_{S} ⋆_{g} F, q_{m} : = \frac{1}{4 π} \int_{S} F .$

The SI electric-flux convention is $Q_{e} = ε_{0} c \int_{S} ⋆_{g} F$ , and the magnetic quantity used here is the flux $Φ_{B} = \int_{S} F$ . Thus

$q_{e} = \frac{α}{4 π ε_{0} c} Q_{e} = \sqrt{\frac{G_{N}}{4 π ε_{0} c^{4}}} Q_{e}, q_{m} = \frac{α}{4 π} Φ_{B} .$

The second equality in the electric formula uses the exact SI relation $ε_{0} μ_{0} c^{2} = 1$ . 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 $Q_{m} : = Φ_{B} / μ_{0}$ with unit A m, then the same definition reads $q_{m} = \sqrt{G_{N} μ_{0} / (4 π c^{4})} Q_{m}$ ; this optional relabeling adds no new physical assumption or quantization rule.

For the metric parameters, set

$M : = \frac{G_{N} m_{*}}{c^{2}}, a : = \frac{J}{m_{*} c}, Q_{g e o m}^{2} : = q_{e}^{2} + q_{m}^{2} .$

Here $M$ , $a$ , $q_{e}$ , and $q_{m}$ are derived length parameters, while $m_{*}$ , $J$ , $Q_{e}$ , $Φ_{B}$ , $G_{N}$ , $c$ , and $μ_{0}$ are the selected inputs of this map. The dimensionless metric coefficients use $M / r$ , $a / r$ , and $Q_{g e o m}^{2} / r^{2}$ . 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 $μ_{0}$ , 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.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Dyonic RN as adopted Einstein–Maxwell model

Physical scope

Conditional source-free RN model deduction for the equation-based smooth Einstein–Maxwell branch with zero cosmological constant and no material stress or current, on positive-radius static coordinate domains away from the roots. It excludes an asserted source at the removed center, a global potential in the magnetic-flux sector, an ADM/Komar identification, any maximal-extension result, observations, and empirical validity.

Statement

Let $m_{*} > 0$ be an SI mass parameter, $Q_{e} \in R$ an SI electric charge parameter, and $Φ_{B} \in R$ an SI magnetic flux through an oriented sphere. Use SI and geometric solution parameter identifications to set

$M = \frac{G_{N} m_{*}}{c^{2}}, q_{e} = \sqrt{\frac{G_{N}}{4 π ε_{0} c^{4}}} Q_{e}, q_{m} = \frac{\sqrt{4 π G_{N} / (μ_{0} c^{4})}}{4 π} Φ_{B} .$

Thus $M, q_{e}, q_{m}$ are real length parameters. Let $J \subset (0, \infty)$ be a nonempty open set on which $f (r) = 1 - 2 M / r + (q_{e}^{2} + q_{m}^{2}) / r^{2}$ has no zero, and use the exact dyonic RN metric and smooth two-form of Dyonic Reissner-Nordström metric and field domains on $R_{x^{0}} \times J \times S^{2}$ , with $x^{0} = c t$ . Write $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ , $F = F / α$ , and

$T_{a b}^{E M} : = \frac{1}{μ_{0}} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}), κ_{E} : = \frac{8 π G_{N}}{c^{4}} .$

In the equation-based physical model of Equation-based Einstein–Maxwell model, specialize to $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ on this domain. Conditional on the exact current draft statements and hypotheses of Full dyonic Reissner-Nordström electrovac verification and Sourced Maxwell sign and current conservation, the RN datum satisfies the adopted source-free equations

$d F = 0, \nabla_{a}^{g} F^{a b} = 0, G_{a b} = κ_{E} T_{a b}^{E M} .$

This is a conditional model deduction, not a claim that nature contains an RN object. The coordinate domain excludes $r = 0$ ; the conclusion does not assign a regular source there or require a global potential when $q_{m} \neq 0$ .

Facts & Assumptions

Given: Positive SI constants $G_{N}, c, μ_{0}$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ , a positive SI mass parameter $m_{*}$ , SI flux parameters $Q_{e}, Φ_{B}$ , the smooth dyonic RN data on a nonempty positive-radius domain with $f \neq 0$ , and the stated zero-source specialization.

[F1]

SI and geometric solution parameter identifications: $M = G_{N} m_{*} / c^{2}$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $q_{m} = α Φ_{B} / (4 π)$ , and $F = α F$ with $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})} > 0$ .

[F2]

Dyonic Reissner-Nordström metric and field domains: on $R_{x^{0}} \times J \times S^{2}$ , $f = 1 - 2 M / r + (q_{e}^{2} + q_{m}^{2}) / r^{2}$ , $g = - f (d x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ , and $F = - q_{e} r^{- 2} d x^{0} \land d r + q_{m} ω_{S^{2}}$ , with $r > 0$ and $f \neq 0$ .

[A1]

Equation-based Einstein–Maxwell model: select its zero-matter instance $M_{0}$ , with configuration set $C_{0} [g, F] = {*}$ , no matter-field variables, equations or additional constraints, and maps $J_{0} [g, F; *] = 0$ and $T_{0} [g, F; *] = 0$ to the smooth zero vector field and smooth zero symmetric covariant two-tensor on the stated domain. Thus $Ψ = *$ is admissible and $j = 0$ , $T^{m} = 0$ . With $Λ = 0$ , the adopted equations are $d F = 0$ , $\nabla_{a}^{g} F^{a b} = - μ_{0} j^{b}$ , and $G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m})$ .

[L1]

Full dyonic Reissner-Nordström electrovac verification: for the exact datum in [F2] and every formal $G_{N}, c, μ_{0} > 0$ , its current Statement gives $d F = d (⋆_{g} F) = 0$ , $S = 0$ , and $G_{a b} = κ_{E} T_{a b}^{E M}$ for $F = F / α$ . Its proof and mathematical suppliers remain draft and are not independently accepted; the current Step-1 record is authoring-ready, so this conclusion is used only conditionally.

[L2]

Sourced Maxwell sign and current conservation: its current Statement assumes the sourced divergence equation before stating the form-equation equivalence. Its Facts [F1]–[F3] specify the smooth oriented Lorentzian geometry, torsion-free metric connection and Hodge conventions. Its Proof 2.1 derives the pointwise identity $d (⋆_{g} F) = - ι_{v} {v o l}_{g}$ , $v^{b} = \nabla_{a}^{g} F^{a b}$ , using only smoothness and those geometric conventions; the equivalence clause is not used as a black-box implication here. The supplier is a draft proof without independent acceptance.

[L3]

Lorentzian exterior calculus and Hodge star: in signature $(-, +, +, +)$ and the stated orientation, $(⋆_{g} F)_{a b} = \frac{1}{2} ({v o l}_{g})_{a b c d} F^{c d}$ for smooth two-forms. The metric-selected connection is torsion-free and metric-compatible under the geometric conventions of [L2]. The identity $\nabla {v o l}_{g} = 0$ used below is derived locally in step 3.1.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the geometric flux convention inherited through [F1] uses only ${A C}_{ω}$ , meaning each specified countable family of nonempty sets has a choice function. The finite tensor normalization and equation conversion below make no additional choice and use no full AC.

Proof

1.1givenF1F2A2algebra

By [F1], $M, q_{e}, q_{m}$ are real lengths and $α > 0$ . The chosen inputs therefore give exactly the geometric parameters and rescaling in [F2], with no inference about an asymptotic conserved mass or angular momentum. The only inherited choice convention is [A2]; this finite substitution makes no choice.

2.1F2L1step 1.1

Apply the exact conditional statement [L1] to the RN datum [F2]. It gives $d F = 0$ , $d (⋆_{g} F) = 0$ , zero scalar curvature, and $G_{a b} = κ_{E} T_{a b}^{E M}$ on every component of the stated $f \neq 0$ domain.

3.1L2L3step 2.1algebra

Put $V^{b} : = \nabla_{a}^{g} F^{a b}$ . In an oriented coordinate chart write ${v o l}_{g} = w d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ with $w = \sqrt{∣ \det (g_{a b}) ∣}$ ; this coefficient follows from the defining value of the metric volume form on a positively oriented orthonormal basis and the determinant change-of-basis formula. Differentiating the determinant expansion gives $\partial_{c} w = \frac{1}{2} w g^{a b} \partial_{c} g_{a b}$ , while the Christoffel formula gives $Γ^{a}_{c a} = \frac{1}{2} g^{a b} \partial_{c} g_{a b}$ because the other two terms cancel after interchanging their dummy indices. Hence $(\nabla_{c} {v o l}_{g})_{0123} = \partial_{c} w - Γ^{a}_{c a} w = 0$ , so $\nabla {v o l}_{g} = 0$ . At any point $p$ , choose a positively oriented orthonormal frame with metric matrix $η = diag (- 1, 1, 1, 1)$ . Torsion-freeness gives $(d ⋆_{g} F)_{a b c} = 3 \nabla_{[a} (⋆_{g} F)_{b c]}$ . The component formula in [L3] has $(⋆_{g} F)_{i j} = - ϵ_{i j k} F_{0 k}$ and $(⋆_{g} F)_{0 i} = \frac{1}{2} ϵ_{i j k} F_{j k}$ , where $ϵ_{i j k}$ is alternating and $ϵ_{123} = + 1$ . In this frame $V^{0} = \sum_{i} \nabla_{i} F_{0 i}$ and $V^{k} = - \nabla_{0} F_{0 k} + \sum_{i} \nabla_{i} F_{i k}$ . Substituting the displayed Hodge components into $3 \nabla_{[a} (⋆_{g} F)_{b c]}$ gives $(d ⋆_{g} F)_{123} = - V^{0}, (d ⋆_{g} F)_{0 i j} = ϵ_{i j k} V^{k} .$ In the same oriented frame, $(ι_{V} {v o l}_{g})_{123} = V^{0}$ and $(ι_{V} {v o l}_{g})_{0 i j} = - ϵ_{i j k} V^{k}$ , so these components establish $d (⋆_{g} F) ∣_{p} = - ι_{V (p)} {v o l}_{g} ∣_{p}$ . Step 2.1 gives $d (⋆_{g} F) = 0$ . Since ${v o l}_{g} ∣_{p}$ is a nonzero four-form, $V (p) \mapsto ι_{V (p)} {v o l}_{g} ∣_{p}$ is an isomorphism $T_{p} M \to Λ^{3} T_{p}^{*} M$ , so $V (p) = 0$ . As $p$ was arbitrary, $\nabla_{a}^{g} F^{a b} = 0$ throughout the stated domain.

4.1A1step 2.1step 3.1∎

Under the zero-source specialization in [A1], the equations established in steps 2.1 and 3.1 are exactly the adopted equation-based Einstein–Maxwell system. They establish the claimed conditional solution on the stated domain, without asserting a source at the omitted center or an empirical realization.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Verified rotating electric KN physical model

Physical scope

Conditional solution of the smooth equation-based Einstein–Maxwell model with zero cosmological constant, no material stress or current, and a purely electric KN parameter on the stated pole-free Boyer–Lindquist domains and full-sphere exterior. It makes no horizon-extension, maximality, uniqueness, formation, observational, or empirical-validity claim.

Statement

Let $m_{*} > 0$ be an SI mass parameter, $J \in R$ an SI angular-momentum parameter, and $Q_{e} \in R$ an SI electric-charge parameter. Use SI and geometric solution parameter identifications to set

$M = \frac{G_{N} m_{*}}{c^{2}}, a = \frac{J}{m_{*} c}, q = \sqrt{\frac{G_{N}}{4 π ε_{0} c^{4}}} Q_{e}, α = \sqrt{\frac{4 π G_{N}}{μ_{0} c^{4}}} .$

Set the magnetic flux parameter to zero. Let $Σ = r^{2} + a^{2} \cos^{2} θ$ and $Δ = r^{2} - 2 M r + a^{2} + q^{2}$ . Use the exact metric and potential of Kerr-Newman metric and potential on the exact Boyer-Lindquist domain on any open pole-free Boyer–Lindquist domain with $Σ > 0$ , $Δ \neq 0$ , and $0 < θ < π$ . Also use its smooth full-sphere exterior extension of Full-sphere exterior extension of the Kerr-Newman datum on $X = R_{x^{0}} \times (R_{0}, \infty) \times S^{2}$ for any $R_{0} > 4 ∣ M ∣$ , with geometric field $F = d A$ and $F = F / α$ . Define

$T_{a b}^{E M} : = \frac{1}{μ_{0}} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}), κ_{E} : = \frac{8 π G_{N}}{c^{4}} .$

In the equation-based physical model of Equation-based Einstein–Maxwell model, specialize to $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ . Conditional on the exact current draft statements and hypotheses of Fully verified electric Kerr-Newman electrovacuum and Sourced Maxwell sign and current conservation, these data satisfy

$d F = 0, \nabla_{a}^{g} F^{a b} = 0, G_{a b} = κ_{E} T_{a b}^{E M} .$

The claim is a conditional solution on the stated domains. It does not extend the field equations across $Δ = 0$ or assert that the selected model describes any observed object.

Facts & Assumptions

Given: $G_{N}, c, μ_{0} > 0$ , $m_{*} > 0$ , SI parameters $J, Q_{e}$ , zero magnetic flux, the stated coordinate domains, and the zero-source equation-based specialization.

[F1]

SI and geometric solution parameter identifications: $M = G_{N} m_{*} / c^{2}$ , $a = J / (m_{*} c)$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $F = α F$ , and $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})} > 0$ .

[F2]

Kerr-Newman metric and potential on the exact Boyer-Lindquist domain: on a pole-free chart, $Σ = r^{2} + a^{2} \cos^{2} θ$ , $Δ = r^{2} - 2 M r + a^{2} + q^{2}$ , $g = - \frac{Δ}{Σ} (d x^{0} - a \sin^{2} θ d ϕ)^{2} + \frac{Σ}{Δ} d r^{2} + Σ d θ^{2} + \frac{\sin^{2} θ}{Σ} ((r^{2} + a^{2}) d ϕ - a d x^{0})^{2}$ , and $A = - q r Σ^{- 1} (d x^{0} - a \sin^{2} θ d ϕ)$ with $F = d A$ .

[F3]

Full-sphere exterior extension of the Kerr-Newman datum: for $R_{0} > 4 ∣ M ∣$ , the full-sphere exterior has a smooth global metric and potential agreeing with [F2] away from the axes, with $r > R_{0}$ , $Σ > 0$ , and $Δ > 0$ .

[A1]

Equation-based Einstein–Maxwell model: select its zero-matter instance $M_{0}$ , with configuration set $C_{0} [g, F] = {*}$ , no matter-field variables, equations or additional constraints, and maps $J_{0} [g, F; *] = 0$ and $T_{0} [g, F; *] = 0$ to the smooth zero vector field and smooth zero symmetric covariant two-tensor on each stated domain. Thus $Ψ = *$ is admissible and $j = 0$ , $T^{m} = 0$ . With $Λ = 0$ , the adopted equations are $d F = 0$ , $\nabla_{a}^{g} F^{a b} = - μ_{0} j^{b}$ , and $G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m})$ .

[L1]

Fully verified electric Kerr-Newman electrovacuum: for the exact pole-free KN datum and its stated full-sphere exterior, the current mathematical Statement gives $d F = d (⋆_{g} F) = 0$ , $Ric = 2 Q_{g} (F)$ , and, for $F = F / α$ , $G_{a b} = κ_{E} T_{a b}^{E M}$ . Its current proof derives the Hodge-divergence identity locally and uses the exact present observer-stress interface; its Step-1 record is authoring-ready, but the mathematical theorem remains draft and has no independent proof acceptance, so the conclusion is used only conditionally.

[L2]

[L3]

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the geometric flux convention inherited through [F1] uses only ${A C}_{ω}$ , meaning each specified countable family of nonempty sets has a choice function. The finite field-equation and tensor-rescaling argument below uses no further choice and does not use full AC.

Proof

1.1givenF1F2F3A2algebra

By [F1], $M, a, q, α$ are the exact derived length and field-normalization parameters for these SI inputs. With zero magnetic flux, $q_{m} = 0$ , so the electric KN parameter in [F2] is precisely $q = q_{e}$ . The open-domain restrictions in the Statement are the nonsingular chart hypotheses of [F2], and $R_{0} > 4 ∣ M ∣$ gives the smooth exterior domain [F3]. The inherited flux convention assumes only [A2].

2.1F2F3L1step 1.1

Apply [L1] to the fields in [F2] and [F3]. It gives the source-free geometric equations and the normalized tensor equation $G_{a b} = κ_{E} T_{a b}^{E M}$ on each stated domain, with the full-sphere equations obtained by continuity from the pole-free chart.

3.1L2L3step 2.1algebra

Put $V^{b} : = \nabla_{a}^{g} F^{a b}$ . In an oriented coordinate chart write ${v o l}_{g} = w d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ with $w = \sqrt{∣ \det (g_{a b}) ∣}$ ; this coefficient follows from the defining value of the metric volume form on a positively oriented orthonormal basis and the determinant change-of-basis formula. Differentiating the determinant expansion gives $\partial_{c} w = \frac{1}{2} w g^{a b} \partial_{c} g_{a b}$ , while the Christoffel formula gives $Γ^{a}_{c a} = \frac{1}{2} g^{a b} \partial_{c} g_{a b}$ because the other two terms cancel after interchanging their dummy indices. Hence $(\nabla_{c} {v o l}_{g})_{0123} = \partial_{c} w - Γ^{a}_{c a} w = 0$ , so $\nabla {v o l}_{g} = 0$ . At any point $p$ of a stated domain, choose a positively oriented orthonormal frame with metric matrix $η = diag (- 1, 1, 1, 1)$ . Torsion-freeness gives $(d ⋆_{g} F)_{a b c} = 3 \nabla_{[a} (⋆_{g} F)_{b c]}$ . The component formula in [L3] has $(⋆_{g} F)_{i j} = - ϵ_{i j k} F_{0 k}$ and $(⋆_{g} F)_{0 i} = \frac{1}{2} ϵ_{i j k} F_{j k}$ , where $ϵ_{i j k}$ is alternating and $ϵ_{123} = + 1$ . In this frame $V^{0} = \sum_{i} \nabla_{i} F_{0 i}$ and $V^{k} = - \nabla_{0} F_{0 k} + \sum_{i} \nabla_{i} F_{i k}$ . Substituting the displayed Hodge components into $3 \nabla_{[a} (⋆_{g} F)_{b c]}$ gives $(d ⋆_{g} F)_{123} = - V^{0}, (d ⋆_{g} F)_{0 i j} = ϵ_{i j k} V^{k} .$ In the same oriented frame, $(ι_{V} {v o l}_{g})_{123} = V^{0}$ and $(ι_{V} {v o l}_{g})_{0 i j} = - ϵ_{i j k} V^{k}$ , so these components establish $d (⋆_{g} F) ∣_{p} = - ι_{V (p)} {v o l}_{g} ∣_{p}$ . Step 2.1 gives $d (⋆_{g} F) = 0$ . Since ${v o l}_{g} ∣_{p}$ is a nonzero four-form, $V (p) \mapsto ι_{V (p)} {v o l}_{g} ∣_{p}$ is an isomorphism $T_{p} M \to Λ^{3} T_{p}^{*} M$ , so $V (p) = 0$ . As $p$ was arbitrary, $\nabla_{a}^{g} F^{a b} = 0$ throughout each stated domain.

4.1A1step 2.1step 3.1∎

Under the zero-source specialization of [A1], the equations established in steps 2.1 and 3.1 are exactly the adopted equation-based Einstein–Maxwell system. Therefore the stated KN normalization is a conditional solution of that physical model on the stated domains. The proof makes no horizon-crossing or empirical claim.

Physics theoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Dyonic rotating model and gauge obstruction

Physical scope

Conditional source-free equation-model solution after a constant dual rotation of the electric KN seed, with geometric flux lengths and the corresponding SI electric charge and magnetic flux. Nonzero magnetic flux obstructs a global real one-form potential on sphere-containing model regions; no magnetic-charge quantization or empirical realization is asserted.

Statement

Take a pure-electric KN seed with SI inputs $m_{*} > 0$ , $J \in R$ and $Q_{0} \in R$ , zero magnetic flux, and parameters $M = \frac{G_{N} m_{*}}{c^{2}}, a = \frac{J}{m_{*} c}, q_{0} = γ_{e} Q_{0}, α = \sqrt{\frac{4 π G_{N}}{μ_{0} c^{4}}}, γ_{e} : = \sqrt{\frac{G_{N}}{4 π ε_{0} c^{4}}} .$ For a fixed real duality angle $β$ , rotate its geometric field by $F_{β} : = \cos β F_{0} + \sin β ⋆_{g} F_{0},$ where $F_{0} = α F_{0}$ is the seed's geometric field. On the seed's stated Boyer–Lindquist domains, the dual-rotated geometric field is source-free and has unchanged geometric stress, conditional on Classical electrovac dual rotation; thus $F_{β} : = F_{β} / α$ and the same metric solve the same equation-based SI model, conditional on Verified rotating electric KN physical model. On the full-sphere exterior $X : = R_{x^{0}} \times (R_{0}, \infty)_{r} \times S^{2}$ of Full-sphere exterior extension of the Kerr-Newman datum, with $R_{0} > 4 ∣ M ∣$ , the resulting geometric flux parameters and matching SI inputs are $q_{e} = q_{0} \cos β, q_{m} = q_{0} \sin β, Q_{e}^{o u t} = Q_{0} \cos β, Φ_{B}^{o u t} = μ_{0} c Q_{0} \sin β .$ In particular $q_{e}^{2} + q_{m}^{2} = q_{0}^{2}$ , so the metric is unchanged. On each sphere-containing full-sphere exterior region, $\int_{S_{r}} F_{β} = 4 π q_{m}$ ; hence a global smooth real one-form potential for $F_{β}$ there is impossible if $q_{m} \neq 0$ , by finite-chain Stokes. In the nonrotating case $a = 0$ , the corresponding dyonic RN field also has the explicit north/south patch potentials and exact global-potential criterion of Dyonic RN gauges and global exactness obstruction. No charge quantization, ADM/Komar identification or observed object is asserted.

Facts & Assumptions

Given: $G_{N}, c, μ_{0} > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ , the pure-electric seed inputs, a fixed real $β$ , and the named model domains.

[F1]

SI and geometric solution parameter identifications: $q_{e} = γ_{e} Q_{e}$ , $q_{m} = α Φ_{B} / (4 π)$ , $γ_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})}$ , $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ , and $ε_{0} μ_{0} c^{2} = 1$ .

[F2]

Verified rotating electric KN physical model: for a pure-electric KN seed, the SI-rescaled data satisfy the adopted $Λ = 0$ , $j = 0$ , $T^{m} = 0$ equations conditionally on the exact draft verification and current-form interfaces; the mathematical proof is draft and not independently accepted.

[F3]

Fully verified electric Kerr-Newman electrovacuum: for the exact pole-free KN datum and its stated full-sphere exterior, $d F = d (⋆_{g} F) = 0$ and $Ric = 2 Q (F)$ ; its current proof remains draft and is not independently accepted.

[L1]

Classical electrovac dual rotation: for any smooth geometric electrovac field and constant $β$ , $F_{β} = \cos β F + \sin β ⋆_{g} F$ has $d F_{β} = d (⋆_{g} F_{β}) = 0$ and $Q (F_{β}) = Q (F)$ ; hence the same metric satisfies the same geometric Einstein–Maxwell equation. This mathematical lemma is draft and not independently accepted.

[F4]

Sourced Maxwell sign and current conservation: its current Statement assumes the sourced divergence equation before stating the form-equation equivalence. Its Facts [F1]–[F3] specify the smooth oriented Lorentzian geometry, torsion-free metric connection and Hodge conventions. Its Proof 2.1 derives the pointwise identity $d (⋆_{g} F) = - ι_{v} {v o l}_{g}$ , $v^{b} = \nabla_{a} F^{a b}$ , from smoothness and those geometric conventions; the equivalence clause is not used as a black-box implication here. The supplier proof is draft and not independently accepted.

[L3]

Dyonic RN gauges and global exactness obstruction: on the nonrotating dyonic RN product the field has the stated smooth north and south patch potentials, and a global smooth real one-form potential exists exactly when $q_{m} = 0$ ; this draft proposition assumes no bundle normalization or quantization.

[F6]

Kerr-Newman metric and potential on the exact Boyer-Lindquist domain and Dyonic Reissner-Nordström metric and field domains: setting $a = 0$ in the electric KN potential gives the static RN electric field $- q_{0} r^{- 2} d x^{0} \land d r$ ; the dyonic RN datum uses $- q_{e} r^{- 2} d x^{0} \land d r + q_{m} ω$ with the outward sphere orientation.

[F7]

Lorentzian exterior calculus and Hodge star: the Lorentzian Hodge map on two-forms is determined by the metric and oriented volume form and has components $(⋆_{g} F)_{a b} = \frac{1}{2} ({v o l}_{g})_{a b c d} F^{c d}$ in an oriented chart.

[F8]

Full-sphere exterior extension of the Kerr-Newman datum: on $r > R_{0} > 4 ∣ M ∣$ the pure-electric seed has the smooth global potential $A_{0} = - q_{0} r Σ^{- 1} (d x^{0} - a λ)$ and $F_{0} = d A_{0}$ on the full-sphere exterior.

[F5]

Stokes theorem for smooth singular chains: for a finite smooth singular chain $c$ and smooth form $η$ , $\int_{c} d η = \int_{\partial c} η$ .

[A1]

$β$ is a fixed real number and the duality rotation acts on the geometric two-form, not on the metric. The seed's magnetic flux is zero.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the geometric flux and metric conventions inherited through [F1] and [F2] use only ${A C}_{ω}$ , meaning every specified countable family of nonempty sets has a choice function. The finite trigonometric and SI-conversion calculation adds no choice and uses no full AC.

Proof

1.1givenF1F2F3A1A2

By [F1] and the seed's zero magnetic flux, its geometric electric parameter is $q_{0} = γ_{e} Q_{0}$ and its geometric magnetic parameter is zero. The exact seed statement [F3] supplies the geometric electrovac hypotheses for [L1]; the conditional physical-model statement [F2] supplies the source-free SI solution on its stated pole-free and full-sphere exterior domains. Each cited carrier is used at its current draft interface only.

2.1F1F2F3F4F7L1step 1.1algebra

Let $F_{0} = F_{0} / α$ be the SI seed field. By [L1], the geometric dual-rotated field is closed with closed Hodge dual and has the same quadratic stress as $F_{0}$ ; since $α > 0$ is constant, $d F_{β} = d (⋆_{g} F_{β}) = 0$ for $F_{β} = F_{β} / α$ . Put $V^{b} : = \nabla_{a} F_{β}^{a b}$ . In an oriented coordinate chart write ${v o l}_{g} = w d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ with $w = \sqrt{∣ \det (g_{a b}) ∣}$ ; this coefficient follows from the defining value of the metric volume form on a positively oriented orthonormal basis and the determinant change-of-basis formula. Differentiating the determinant expansion gives $\partial_{c} w = \frac{1}{2} w g^{a b} \partial_{c} g_{a b}$ ; the Christoffel formula and symmetry of $g$ give $Γ^{a}_{c a} = \frac{1}{2} g^{a b} \partial_{c} g_{a b}$ . Hence $(\nabla_{c} {v o l}_{g})_{0123} = \partial_{c} w - Γ^{a}_{c a} w = 0$ , so $\nabla {v o l}_{g} = 0$ . At each point $p$ , choose a positively oriented orthonormal frame with metric matrix $η = diag (- 1, 1, 1, 1)$ . Torsion-freeness of the metric-selected connection [F4] gives $(d ⋆_{g} F_{β})_{a b c} = 3 \nabla_{[a} (⋆_{g} F_{β})_{b c]}$ . The Hodge component formula [F7] has $(⋆_{g} F_{β})_{i j} = - ϵ_{i j k} (F_{β})_{0 k}$ and $(⋆_{g} F_{β})_{0 i} = \frac{1}{2} ϵ_{i j k} (F_{β})_{j k}$ , where $ϵ_{i j k}$ is alternating with $ϵ_{123} = + 1$ . In this frame $V^{0} = \sum_{i} \nabla_{i} (F_{β})_{0 i}$ and $V^{k} = - \nabla_{0} (F_{β})_{0 k} + \sum_{i} \nabla_{i} (F_{β})_{i k}$ . Substituting the displayed Hodge components into $3 \nabla_{[a} (⋆_{g} F_{β})_{b c]}$ gives $(d ⋆_{g} F_{β})_{123} = - V^{0}, (d ⋆_{g} F_{β})_{0 i j} = ϵ_{i j k} V^{k} .$ In the same oriented frame, $(ι_{V} {v o l}_{g})_{123} = V^{0}$ and $(ι_{V} {v o l}_{g})_{0 i j} = - ϵ_{i j k} V^{k}$ , so these components establish $d (⋆_{g} F_{β}) ∣_{p} = - ι_{V (p)} {v o l}_{g} ∣_{p}$ . Its left side is zero. Contraction with the nonzero volume form is an isomorphism $T_{p} M \to Λ^{3} T_{p}^{*} M$ , so $V (p) = 0$ ; hence $\nabla_{a} F_{β}^{a b} = 0$ throughout every stated domain. Thus the source-free SI Maxwell equations hold without using the equivalence clause of [F4]. Its electromagnetic stress is unchanged because $Q (F_{β}) = α^{- 2} Q (F_{β}) = α^{- 2} Q (F_{0}) = Q (F_{0})$ . The Einstein tensor and metric are unchanged, so the same zero-source SI equation-based model equations hold on every base domain of [F2].

2.2F2F5F7F8step 1.1algebra

On the full-sphere exterior [F8], let $y = \cos θ$ , $Σ = r^{2} + a^{2} y^{2}$ , and $R = r^{2} + a^{2}$ . Differentiating the exact seed potential in [F2] gives $(F_{0})_{r 0} = q_{0} (r^{2} - a^{2} y^{2}) / Σ^{2}$ and $(F_{0})_{r ϕ} = - a \sin^{2} θ (F_{0})_{r 0}$ . Raising indices with the KN inverse metric and applying [F7] yields $(⋆_{g} F_{0})_{θ ϕ} = q_{0} R (r^{2} - a^{2} y^{2}) \sin θ / Σ^{2}$ . Since $\frac{d}{d y} (y / Σ) = (r^{2} - a^{2} y^{2}) / Σ^{2}$ , integration over the outward sphere gives $\frac{1}{4 π} \int_{S_{r}} ⋆_{g} F_{0} = \frac{q_{0} R}{2} \int_{- 1}^{1} \frac{r^{2} - a^{2} y^{2}}{(r^{2} + a^{2} y^{2})^{2}} d y = \frac{q_{0} R}{2} {[\frac{y}{Σ}]}_{- 1}^{1} = q_{0} .$ The global potential in [F8] and Stokes' theorem [F5] give $\int_{S_{r}} F_{0} = \int_{S_{r}} d A_{0} = 0$ . By [F7], $⋆_{g}^{2} = - 1$ on two-forms, so integrating $F_{β} = \cos β F_{0} + \sin β ⋆_{g} F_{0}$ and its Hodge dual directly yields $q_{e} = q_{0} \cos β$ and $q_{m} = q_{0} \sin β$ . Therefore $q_{e}^{2} + q_{m}^{2} = q_{0}^{2} (\cos^{2} β + \sin^{2} β) = q_{0}^{2}$ , and every KN metric coefficient depending on charge through this sum is unchanged. This local integral calculation proves the stated flux values.

3.1F1step 2.2algebra

The electric conversion in [F1] gives $Q_{e}^{o u t} = q_{e} / γ_{e} = Q_{0} \cos β$ , using the flux parameter from step 2.2. For the magnetic flux, [F1] gives $Φ_{B}^{o u t} = 4 π q_{m} / α$ , with $q_{m}$ from step 2.2. From $ε_{0} μ_{0} c^{2} = 1$ , ${(\frac{α μ_{0} c}{4 π})}^{2} = \frac{G_{N} μ_{0}}{4 π c^{2}} = γ_{e}^{2} .$ Both $α, μ_{0}, c, γ_{e}$ are positive, hence $α μ_{0} c / (4 π) = γ_{e}$ . Substituting $q_{m} = γ_{e} Q_{0} \sin β$ gives $Φ_{B}^{o u t} = μ_{0} c Q_{0} \sin β$ . These conversions preserve the stated SI flux convention and introduce no magnetic-charge relabeling.

3.2F5step 2.2algebra

Assume a global smooth real one-form $A_{β}$ on a sphere-containing exterior with $d A_{β} = F_{β}$ and $q_{m} \neq 0$ . Represent the fixed sphere $S_{r}$ by radially projecting the eight oriented triangular faces of the octahedron onto the unit sphere and taking their signed sum. Each face is disjoint from the origin, so its radial projection is smooth; common edges cancel with opposite orientations. This is a finite smooth singular $2$ -cycle, so $\partial S_{r} = 0$ . By [F5] and step 2.2, $4 π q_{m} = \int_{S_{r}} F_{β} = \int_{S_{r}} d A_{β} = \int_{\partial S_{r}} A_{β} = 0,$ a contradiction. Thus no such global potential exists on the rotating sphere-containing exterior when its magnetic flux is nonzero.

3.3F6F7L3step 2.2algebra

If $a = 0$ , the KN potential in [F6] reduces to $- q_{0} r^{- 1} d x^{0}$ , so its exterior derivative is $F_{0} = - q_{0} r^{- 2} d x^{0} \land d r$ . On the sphere-containing exterior $f > 0$ , the positively oriented orthonormal coframe $e^{0} = \sqrt{f} d x^{0}$ , $e^{1} = f^{- 1 / 2} d r$ , $e^{2} = r d θ$ , $e^{3} = r \sin θ d ϕ$ gives $d x^{0} \land d r = e^{0} \land e^{1}$ and $⋆_{g} (e^{0} \land e^{1}) = - e^{2} \land e^{3}$ by the Hodge component convention [F7]. Hence $⋆_{g} F_{0} = q_{0} ω$ . Therefore the dual rotation is $F_{β} = - q_{0} \cos β r^{- 2} d x^{0} \land d r + q_{0} \sin β ω,$ which is exactly the dyonic RN field of [F6] with the flux parameters in step 2.2. The exact current interface [L3] supplies the two patch potentials and the criterion for a global real potential in this nonrotating case.

4.1A2step 1.1step 2.1step 2.2step 3.1step 3.2step 3.3∎

Steps 1.1-3.3 establish the conditional dual-rotated model, full-sphere flux identifications and global-potential obstruction. The conventions use only the inherited ${A C}_{ω}$ in [A2]; they impose no full AC, flux quantization, or empirical claim.

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Stationary RN observer frequency ratio

Physical scope

Conditional phase-frequency ratio for a local future null eikonal phase crossing two static observers in a source-free RN model region. Ideal-clock and geometric-optics readings are adopted assumptions; no emitted or received measurement, apparatus model, or empirical agreement is asserted.

Statement

Let $m_{*} > 0$ , $Q_{e}, Φ_{B} \in R$ , and use SI and geometric solution parameter identifications and Dyonic Reissner-Nordström metric and field domains for the dyonic RN metric on $M_{J} = R_{x^{0}} \times J_{r} \times S^{2}$ , where $J$ is a connected open interval with $f (r) = 1 - 2 M / r + (q_{e}^{2} + q_{m}^{2}) / r^{2} > 0$ . The coordinate $x^{0} = c t$ and $r$ have units of metres. Put $ξ = \partial_{x^{0}}$ ; it is the future static Killing field on this region. Let $I_{γ} \subseteq R$ be a nonempty open interval, let $γ : I_{γ} \to M_{J}$ be a $C^{2}$ curve, and let $W$ be an open neighborhood of $γ (I_{γ})$ carrying a dimensionless $C^{2}$ phase $θ : W \to R$ . Set $k = d θ$ and $K = k^{♯}$ . Assume the eikonal equation $g^{- 1} (d θ, d θ) = 0$ holds throughout $W$ , and that $K$ is nowhere zero and future directed along $γ (I_{γ})$ . Parameterize the phase ray by $γ^{'} (λ) = K (γ (λ))$ , which fixes its affine parameter normalization. For each spatial point $(r, ω) \in J \times S^{2}$ and nonempty open interval $I \subset R$ , define the static observer curve $σ_{r, ω} : I \to M_{J}$ by $σ_{r, ω} (s) = (s, r, ω)$ , where $s = x^{0}$ has units of metres. It is smooth and regular with tangent $ξ$ , and it is future timelike because $g (ξ, ξ) = - f (r) < 0$ . Its proper-time parameter satisfies $d τ / d s = \sqrt{f (r)} / c$ under the metric proper-time definition, so its normalized four-velocity is $U (r) : = \frac{c}{\sqrt{f (r)}} ξ, g (U, U) = - c^{2}, ν_{U} : = - \frac{k (U)}{2 π} .$ Here $U = d σ_{r, ω} / d τ$ is normalized by the SI proper-time convention of SI unit and coordinate conventions of the classical gravitational framework, and $ν_{U}$ is the phase frequency in cycles per unit ideal-clock time under Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework. If $γ$ intersects static observer curves $σ_{r_{E}, ω_{E}}$ and $σ_{r_{R}, ω_{R}}$ at events $E$ and $R$ , respectively, with $r_{E}, r_{R} \in J$ , then the conditional model prediction is $\frac{ν_{R}}{ν_{E}} = \sqrt{\frac{f (r_{E})}{f (r_{R})}} .$ The phase data and source-free RN model are conditional inputs; the formula is not a reported observation or a claim of measured precision.

Facts & Assumptions

Given: Positive $G_{N}, c, μ_{0}$ , the SI RN parameters and metric on a connected $f > 0$ interval, the adopted classical clock/light-ray postulate, and a local $C^{2}$ dimensionless phase satisfying $g^{- 1} (d θ, d θ) = 0$ on its open domain $W$ , with nowhere-zero future-directed differential along the ray.

[F1]

SI and geometric solution parameter identifications and Einstein–Maxwell primitive quantities and SI conventions: $M = G_{N} m_{*} / c^{2}$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $q_{m} = α Φ_{B} / (4 π)$ , $ε_{0} μ_{0} c^{2} = 1$ , and $x^{0} = c t$ is a length coordinate.

[F2]

Dyonic Reissner-Nordström metric and field domains: on $f > 0$ , $g = - f (d x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ and the chosen future time orientation is $ξ = \partial_{x^{0}}$ .

[F3]

Dyonic RN as adopted Einstein–Maxwell model: the exact dyonic RN datum on each stated positive-radius static domain with $f \neq 0$ is a conditional solution of the adopted zero-source Einstein–Maxwell model; the carrier's proof remains draft and is not independently accepted.

[A1]

Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework: ideal clocks read proper time, and in the geometric-optics approximation ideal light rays are future null geodesics of the metric-selected connection. These are adopted physical assumptions, not deductions from the Einstein–Maxwell equations.

[F4]

Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields and The inverse metric, musical isomorphisms and index raising: the selected connection is torsion-free and metric-compatible, its tensor extension obeys the product rule with $\nabla g^{- 1} = 0$ , and $K = k^{♯}$ is characterized by $g (K, V) = k (V)$ .

[F8]

Clairaut--Schwarz theorem for continuous second partial derivatives: the mixed coordinate partials of a $C^{2}$ scalar field on an open Euclidean chart commute.

[F5]

Killing fields of a Lorentzian metric and Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor: a field is Killing when $L_{ξ} g = 0$ ; for a Killing field and an affinely parametrized geodesic $γ$ , $g (ξ, γ^{'})$ is constant along $γ$ .

[F6]

Classification of future and past causal vectors by a timelike field: a nonzero future null vector has strictly negative metric pairing with every future timelike vector.

[F7]

SI unit and coordinate conventions of the classical gravitational framework: in length coordinates $x^{0} = c t$ , $U = d γ / d τ$ has SI velocity components and the proper-time parameter $τ$ is measured in seconds.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric RN and metric-connection conventions use only ${A C}_{ω}$ . No selection from a family is made by this local calculation; full AC is not used.

Proof

1.1F1F2F3F5F7givenalgebra

The SI map and length-coordinate convention [F1] identify $x^{0} = c t$ and the RN length parameters used in [F2]. The metric in [F2] is independent of $x^{0}$ , so its Lie derivative along $ξ = \partial_{x^{0}}$ vanishes and $ξ$ is Killing by [F5]. For any curve $σ_{r, ω} (s) = (s, r, ω)$ in the Statement, $σ_{r, ω}^{'} = ξ$ and $- g (ξ, ξ) = f (r)$ is constant along the curve, so the metric proper-time definition gives $d τ / d s = \sqrt{f (r)} / c$ and $d σ_{r, ω} / d τ = c ξ / \sqrt{f (r)} = U (r)$ . Thus $g (U, U) = - c^{2}$ and $U$ is future timelike on $J$ . The solution is the conditional physical RN model of [F3]; no observational premise is used.

1.2F4F8A1givenalgebra

The metric dual $K = k^{♯}$ is nonzero and null because $k \neq 0$ and $g^{- 1} (k, k) = 0$ . In coordinates, $k_{a} = \partial_{a} θ$ ; [F8] makes the mixed second partials commute, and torsion freedom in [F4] makes the Christoffel terms symmetric, so $\nabla_{b} k_{a} = \nabla_{a} k_{b}$ . Metric compatibility and the tensor product rule in [F4] then give $K^{b} \nabla_{b} K_{a} = k^{b} \nabla_{b} k_{a} = k^{b} \nabla_{a} k_{b} = \frac{1}{2} \nabla_{a} (g^{b c} k_{b} k_{c}) = 0.$ Thus the integral curves of $K$ are affinely parametrized null geodesics on the phase domain by the geodesic definition; the phase ray $γ^{'} = K$ is one such curve. This is the local eikonal input used for the geometric-optics ray.

2.1F5F6step 1.1step 1.2

Apply the Killing contraction identity [F5] to the affinely parametrized geodesic tangent $K$ from step 1.2. It makes $g (ξ, K) = k (ξ)$ constant along that ray. Since $K$ is future null and $ξ$ is future timelike, [F6] gives $C : = - k (ξ) > 0$ .

3.1A1F7step 2.1algebra

At radius $r$ , the phase frequency is $ν_{U} = - \frac{k (c ξ / \sqrt{f})}{2 π} = \frac{c C}{2 π \sqrt{f}} .$ The positive value is a phase rate; the ideal-clock postulate [A1] supplies its clock-time interpretation. Evaluating at the two events and dividing cancels the common positive constant $c C / (2 π)$ , yielding $ν_{R} / ν_{E} = \sqrt{f (r_{E}) / f (r_{R})}$ .

4.1A1F3step 3.1A2∎

The ratio in step 3.1 is a conditional phase-frequency prediction for the chosen RN domain and local ray. The model postulates and phase assumptions remain explicit; no clock apparatus, data, uncertainty, or empirical confirmation has been supplied.

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RN horizon and escape deductions for specified model

Physical scope

Conditional geometric and causal deductions on the specified smooth ingoing RN model with M>0 and real length flux parameters. It covers charged subextremal/extremal roots, the uncharged simple root, and the superextremal no-root branch. Escape means only the explicitly defined finite piecewise C1 causal-path observable on this product model; no conformal infinity, maximal-extension, formation, uniqueness, censorship, observation, or empirical-validity claim is made.

Statement

Let $m_{*} > 0$ , $Q_{e} \in R$ , and $Φ_{B} \in R$ be the SI mass, electric-charge, and magnetic-flux inputs of Dyonic RN as adopted Einstein–Maxwell model. Put $M = G_{N} m_{*} / c^{2} > 0$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $q_{m} = \sqrt{4 π G_{N} / (μ_{0} c^{4})} Φ_{B} / (4 π)$ , $Q^{2} = q_{e}^{2} + q_{m}^{2}$ , and $f (r) = 1 - 2 M / r + Q^{2} / r^{2}$ . Use the selected ingoing product $M_{i n} = R_{v} \times (0, \infty)_{r} \times S^{2}$ and its metric, field, and future orientation from Selected ingoing RN extension and time orientation.

For $Q^{2} > 0$ and $M^{2} \geq Q^{2}$ , let $r_{\pm} = M \pm \sqrt{M^{2} - Q^{2}}$ . Each positive root is a regular null hypersurface and a Killing horizon generated by $ξ = \partial_{v}$ , with the convention

$κ_{H} : = \frac{f^{'} (r_{H})}{2} = \frac{r_{H} - r_{o t h e r}}{2 r_{H}^{2}} .$

Thus $κ_{+} = 0$ at extremality and $κ_{\pm}$ has unit m $^{- 1}$ ; the associated SI acceleration convention is $c^{2} κ_{\pm}$ . At a simple outer root, $κ_{+} > 0$ , the selected ingoing extension has a local bifurcation chart with dimensionless coordinates $U, V$ satisfying $U V = - K (r)$ and an auxiliary coordinate length $ℓ_{*} > 0$ . At extremality the ingoing chart is regular but this simple-root chart is not asserted.

Use the exact path and relative-boundary convention of Causal solutions, Killing horizons, escape and metric extensions: the path is continuous, future causal, and piecewise $C^{1}$ with a finite subdivision on each compact parameter segment; one such path must reach arbitrarily large $r$ .

Define $E$ on this product to be the set of points from which a future piecewise $C^{1}$ causal path has unbounded radial coordinate. Conditional on the exact draft mathematical interface of Outer RN escape boundary in selected model, for $Q^{2} > 0$ and $M^{2} \geq Q^{2}$ ,

$E = {(v, r, ω) : r > r_{+}}, \partial_{M_{i n}} E = {r = r_{+}} .$

For $Q^{2} = 0$ , the only positive root is $2 M$ , it is a regular simple crossing with $κ = 1 / (4 M)$ , and the same explicit escape observable has boundary $r = 2 M$ . For $Q^{2} > M^{2}$ , there is no positive root and every point of $M_{i n}$ belongs to $E$ . These are statements about the selected metric and escape observable, not an event-horizon assertion requiring a conformal completion.

Facts & Assumptions

Given: $M > 0$ , the real length parameters $q_{e}, q_{m}$ , the smooth ingoing RN datum, its selected future orientation, and the escape observable stated above.

[F1]

Selected ingoing RN extension and time orientation: $M_{i n} = R_{v} \times (0, \infty)_{r} \times S^{2}$ , $g = - f d v^{2} + 2 d v d r + r^{2} γ$ , $F = - q_{e} r^{- 2} d v \land d r + q_{m} ω$ , and $ℓ = - \partial_{r}$ , $n = \partial_{v} + (f / 2) \partial_{r}$ are future null with $g (ℓ, n) = - 1$ .

[F2]

Dyonic RN as adopted Einstein–Maxwell model: on every positive-radius static domain with $f \neq 0$ , the SI-rescaled dyonic RN datum is a conditional solution of the zero-source, $Λ = 0$ equation-based Einstein–Maxwell model.

[L1]

Regular RN crossings and local bifurcation extension: for $Q^{2} > 0$ , $M > 0$ , and $M^{2} \geq Q^{2}$ , the charged RN ingoing block is regular at the roots; for a simple outer root, its current Statement gives a smooth local bifurcation chart with $U V = - K (r)$ and fixed auxiliary length $ℓ_{*} > 0$ . Its current Step-1 record is authoring-ready, but its mathematical proof remains draft and is not independently accepted; its field-equation continuation is used only conditionally.

[L2]

Outer RN escape boundary in selected model: for $Q^{2} > 0$ , $M > 0$ , and $M^{2} \geq Q^{2}$ , its exact current Statement gives $E = {r > r_{+}}$ and $\partial_{M_{i n}} E = {r = r_{+}}$ for the observable in [L3]. Its current dependencies and proof were checked against [L3]; the theorem is draft content with authoring readiness, not independent proof acceptance.

[L3]

Causal solutions, Killing horizons, escape and metric extensions: with $U = M_{i n}$ and radius map $r$ , the selected escape set uses the path class stated in the Statement and its boundary is relative to $U$ . The current item is placed on batch-6 page emg-cp01 and has a ready Step-1 authoring record; its proof remains draft and is not independent approval.

[F3]

Choice-free smooth inverse function theorem in Euclidean space: for $n \geq 1$ , an open $U \subseteq R^{n}$ , a smooth map $H : U \to R^{n}$ , and $a \in U$ , if $D H (a)$ is invertible, then $H$ restricts to a diffeomorphism between open neighborhoods of $a$ and $H (a)$ , with smooth inverse. In dimension one, $D H (a)$ is multiplication by $H^{'} (a)$ , so invertibility is equivalent to $H^{'} (a) \neq 0$ .

[F4]

Selected ingoing RN extension and time orientation: the selected time orientation is fixed by $N = ℓ + n$ , with $g (ℓ, n) = - 1$ ; the sphere tangent space is orthogonal to $ℓ, n$ and has positive metric $r^{2} γ$ . Future timelike vectors lie in the timelike component containing $N$ , and future causal vectors are nonzero vectors in its closure under the selected cone convention. By Timelike, null, spacelike and causal vectors of a Lorentz vector space, causality means $Z \neq 0$ and $g (Z, Z) \leq 0$ ; the zero vector belongs to the cone closure but is not causal.

[A1]

The physical model is the source-free equation-based RN specialization with $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ , as in [F2]. The omitted set $r = 0$ is not included as a source point.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the geometric parameter and horizon suppliers carry only ${A C}_{ω}$ , meaning every specified countable family of nonempty sets has a choice function. No new choice is made in this finite root, chart, or causal-path calculation, and no full AC is used.

Proof

1.1givenF1algebra

If $Q^{2} > 0$ and $M^{2} \geq Q^{2}$ , the quadratic numerator factors as $r^{2} f = (r - r_{+}) (r - r_{-})$ ; both roots are positive because $M > 0$ and $\sqrt{M^{2} - Q^{2}} < M$ . If $Q^{2} = 0$ , then $f = 1 - 2 M / r$ and its only positive root is $2 M$ . If $Q^{2} > M^{2}$ , then $r^{2} f = (r - M)^{2} + (Q^{2} - M^{2}) > 0$ for every $r > 0$ , so there is no positive root.

2.1F1step 1.1algebra

The inverse of the $v, r$ block in [F1] is $(\begin{array}{cc} 0 & 1 \\ 1 & f \end{array})$ , so $g^{- 1} (d r, d r) = f$ . The vector $ξ = \partial_{v}$ is Killing because the metric coefficients do not depend on $v$ , and $ξ^{♭} = - f d v + d r$ equals $d r$ at a root. Therefore each root is null and $ξ$ is its null normal. Since $ξ^{2} = - f$ , $d (ξ^{2}) = - f^{'} (r_{H}) d r = - 2 κ_{H} ξ^{♭}$ there. For two distinct roots, differentiating $f = (r - r_{+}) (r - r_{-}) / r^{2}$ gives $f^{'} (r_{\pm}) = (r_{\pm} - r_{\mp}) / r_{\pm}^{2}$ ; at $Q^{2} = M^{2}$ this derivative is zero. For $Q^{2} = 0$ , $f^{'} (2 M) = 1 / (2 M)$ and $κ = 1 / (4 M)$ . The product definition [F1] keeps the metric nondegenerate at each root.

2.2L1F1F3step 1.1algebraconstruct

For $Q^{2} > 0$ and $M^{2} \geq Q^{2}$ , [L1] gives the claimed root crossing and, when $M^{2} > Q^{2}$ , the local bifurcation chart. If $Q^{2} = M^{2}$ , [L1] gives the regular ingoing crossing but explicitly does not assert that simple-root chart. For $Q^{2} = 0$ , put $r_{H} = 2 M$ , $κ = 1 / (4 M)$ , and write $f = (r - r_{H}) b (r)$ with $b (r) = 1 / r$ and $b (r_{H}) = 2 κ > 0$ . Then $\frac{1}{f} - \frac{1}{2 κ (r - r_{H})} = \frac{2 κ - b (r)}{2 κ b (r) (r - r_{H})} = - \frac{1}{2 κ b (r)} \int_{0}^{1} b^{'} (r_{H} + s (r - r_{H})) d s,$ so this difference extends smoothly across $r_{H}$ . Integrate it to $h$ with $h (r_{H}) = 0$ , and define $K (r) = (r - r_{H}) ℓ_{*}^{- 1} e^{2 κ h (r)}$ . Then $K^{'} (r_{H}) = ℓ_{*}^{- 1} > 0$ and differentiation gives $f K^{'} = 2 κ K$ on both sides and at the root. By [F3], shrink the radial interval so $K^{'} > 0$ and $K$ has a smooth local inverse $ρ$ about zero. On the selected ingoing model define $V = e^{κ v}, U = - K (r) e^{- κ v} .$ Thus $U V = - K (r)$ , with $U < 0$ for $r > r_{H}$ and $U > 0$ for $r < r_{H}$ , and the Jacobian is $κ K^{'} (r) > 0$ across the root. The inverse is $r = ρ (- U V)$ and $v = κ^{- 1} \log V$ on $V > 0$ . Differentiating gives $d v = d V / (κ V)$ and $d r = - (V d U + U d V) / K^{'} (r)$ . Substitution in the original ingoing block $- f d v^{2} + 2 d v d r$ cancels its $d V^{2}$ coefficient by $f K^{'} = 2 κ K$ , leaving $- \frac{1}{κ K^{'} (ρ (- U V))} (d U \otimes d V + d V \otimes d U) .$ This coefficient is smooth and nonzero, and the sphere term is $ρ (- U V)^{2} γ$ . Choose $ε > 0$ so these formulas are defined with $ρ (- U V) > 0$ on $(- ε, ε)^{2}$ . They give a smooth Lorentzian metric on the full rectangle times $S^{2}$ , agreeing with the selected ingoing metric on its entire $V > 0$ half, including both $r < r_{H}$ and $r > r_{H}$ . Glue that half to its old coordinate preimage. Any overlap sequence tending to a new point on $V = 0$ has $v = κ^{- 1} \log V \to - \infty$ , so cannot approach a finite old point; the other rectangle edges are not new points. This gives a Hausdorff local extension. The field is zero in this uncharged case. The old Killing field becomes $κ (V \partial_{V} - U \partial_{U})$ ; it extends smoothly, preserves the displayed metric, is null and normal on either axis away from their intersection, and vanishes on the spacelike sphere $U = V = 0$ . Thus this is the required two-sided local bifurcation chart.

2.3L2L3step 1.1

For $Q^{2} > 0$ and $M^{2} \geq Q^{2}$ , apply [L2] using the exact observable [L3]. Its current Definition and manifest placement now exist and were checked against this use; the escape conclusion remains conditional on the current draft mathematical proof, with no maximal-extension or conformal-completion inference.

2.4F1step 1.1algebra

If $Q^{2} > M^{2}$ , then $f > 0$ by step 1.1 and the future null field $n$ of [F1] satisfies $d r / d v = f / 2 > 0$ . For any starting $r_{0} > 0$ , the function $v (r) = v_{0} + \int_{r_{0}}^{r} 2 / f (s) d s$ is smooth and strictly increasing on $[r_{0}, \infty)$ ; since $f (r) \to 1$ , $v (r) \to + \infty$ and the corresponding integral curve reaches arbitrarily large $r$ . Hence every point belongs to $E$ .

2.5F1F4step 1.1algebra

If $Q^{2} = 0$ , then $f < 0$ below $2 M$ , $f = 0$ at $2 M$ , and $f > 0$ above. The product splitting in [F1] gives a unique decomposition of every tangent as $Z = α ℓ + β n + W$ with $W$ tangent to the sphere. Since the sphere term is orthogonal to $ℓ, n$ and positive, causality gives $g (Z, Z) = - 2 α β + g (W, W) \leq 0$ and hence $α β \geq 0$ . The timelike condition is $- 2 α β + g (W, W) < 0$ , so a timelike vector has $α β > 0$ and its coefficients have the same nonzero sign. Each of the two sign branches is connected: scale $W$ to zero within the timelike cone and then vary the positive (respectively negative) coefficients while keeping them equal. Since $N = ℓ + n$ has coefficients $(1, 1)$ , the selected future timelike component is the positive branch. Every future causal vector lies in its closure, so $α, β \geq 0$ . Therefore $d r (Z) = - α + (f / 2) β \leq 0$ for $r \leq 2 M$ . On a collar of $2 M$ , put $τ = v - r$ ; for nonzero future causal $Z$ , $d τ (Z) = α + (1 - f / 2) β \geq α + β / 2 > 0$ after shrinking the collar so $∣ f ∣ < 1$ . Smoothness and $f (2 M) = 0$ give $C < \infty$ with $f (r) \leq C ∣ r - 2 M ∣$ there, hence $d r (Z) \leq C ∣ r - 2 M ∣ d τ (Z)$ . On each $C^{1}$ path piece in the collar we may use $τ$ as parameter. If $r < 2 M$ , then $w = 2 M - r$ obeys $d w / d τ \geq - C w$ , so $d (e^{C τ} w) / d τ \geq 0$ and $w (τ) \geq w (τ_{0}) e^{- C (τ - τ_{0})} > 0$ for every finite $τ$ ; a path starting below the root cannot reach it. If a path starts at the root, $u = (r - 2 M)_{+}$ obeys $d u / d τ \leq C u$ almost everywhere on the collar: for $r > 2 M$ this is the displayed cone bound, for $r < 2 M$ its derivative is zero, and on the root the cone bound gives $d r / d τ \leq 0$ . Thus $d (e^{- C τ} u) / d τ \leq 0$ and $u (τ_{0}) = 0$ forces $u = 0$ , so no outward excursion starts at the root. The same estimates apply across finitely many piece joins by continuity. Every point with $r > 2 M$ escapes along the future null field $n$ , because $d r / d v = f / 2 > 0$ and $f (r) \to 1$ . Thus the uncharged escape boundary is $r = 2 M$ .

3.1A1A2F2step 1.1step 2.1step 2.2step 2.3step 2.4step 2.5∎

Steps 1.1-2.5 prove the stated geometric roots, signed surface-gravity convention, regular-crossing cases, and selected escape conclusions on the chosen product, conditional on the exact current draft mathematical interfaces [L1]-[L3] and on the equation-based physical-model input [F2]. Their authoring-ready records do not constitute independent proof acceptance. The inherited ${A C}_{ω}$ premise [A2] is stated without invoking full choice; no event-horizon, maximal-extension, or empirical claim follows.

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Rotating model horizons and selected causal escape

Physical scope

Conditional conversion of local horizon geometry and the selected escape boundary of the electric Kerr–Newman solution family to SI parameters. The selected ingoing model does not establish a maximal extension, an event horizon in a conformal completion, a black-hole observation, or empirical validity.

Statement

Let $m_{*} > 0$ , $J, Q_{e} \in R$ be the selected SI mass, angular-momentum and electric-charge inputs, with zero magnetic flux. Use SI and geometric solution parameter identifications to set $M = \frac{G_{N} m_{*}}{c^{2}}, a = \frac{J}{m_{*} c}, q = \sqrt{\frac{G_{N}}{4 π ε_{0} c^{4}}} Q_{e}, α = \sqrt{\frac{4 π G_{N}}{μ_{0} c^{4}}} .$ Assume $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ . In the selected ingoing KN model let $r_{\pm} = M \pm \sqrt{M^{2} - a^{2} - q^{2}}, R_{\pm} = r_{\pm}^{2} + a^{2}, χ_{\pm} = \partial_{v} + \frac{a}{R_{\pm}} Ψ,$ where $Ψ = X \partial_{Y} - Y \partial_{X}$ is the smooth rotation field on the unit sphere and $v$ has length units. For either root $r_{H} = r_{\pm}$ , with $r_{o t h e r}$ the other root, the selected horizon data of KN Killing generators, surface gravity, area and potential give $Area (S_{H}) = 4 π R_{H}, Ω_{H, g e o m} = \frac{a}{R_{H}}, κ_{H, g e o m} = \frac{r_{H} - r_{o t h e r}}{2 R_{H}}, - A_{i n} (χ_{H}) = \frac{q r_{H}}{R_{H}} .$ Thus the area is in m $^{2}$ , $Ω_{H, g e o m}$ and the signed $κ_{H, g e o m}$ are in m $^{- 1}$ , and the associated SI angular rate and signed acceleration convention are $Ω_{H, S I} = c \frac{a}{R_{H}} and κ_{H, a c c, S I} = c^{2} \frac{r_{H} - r_{o t h e r}}{2 R_{H}} .$ With $A_{S I} = A_{i n} / α$ and time-normalized generator $χ_{H, S I} = c χ_{H}$ , the gauge-normalized horizon voltage is $- A_{S I} (χ_{H, S I}) = \frac{c q r_{H}}{α R_{H}};$ This voltage normalization uses the displayed ingoing gauge, whose coordinate limit is specified as follows: for each fixed $v$ and $ω \in S^{2}$ , both coefficient functions in the global one-form expression $A_{i n} = - q r Σ^{- 1} d v + a q r Σ^{- 1} β$ tend to zero as $r \to + \infty$ . At extremality the two roots coincide and the signed surface gravity is zero. Conditional on the current exact draft interface of Outer KN escape boundary on the selected ingoing model and the relative-boundary convention of Causal solutions, Killing horizons, escape and metric extensions, the selected escape set on $M_{i n}$ has relative boundary $r = r_{+}$ . These are properties of the chosen solution family and escape observable, not a maximal-extension or observational claim.

Facts & Assumptions

Given: The selected electric SI inputs, $G_{N}, c, μ_{0} > 0$ , the zero-magnetic-flux KN branch, the parameter inequality in the Statement and the chosen ingoing model.

[F1]

[F2]

Verified rotating electric KN physical model: with zero magnetic flux, the SI-rescaled KN datum on its stated pole-free domains and full-sphere exterior is a conditional solution of the equation-based $Λ = 0$ , $j = 0$ , $T^{m} = 0$ model; the mathematical verification remains a draft premise, not independent proof acceptance.

[F3]

Regular ingoing KN metric, field and sphere axes: the selected product $M_{i n} = R_{v} \times (0, \infty)_{r} \times S^{2}$ carries the smooth ingoing metric and potential $A_{i n} = - q r Σ^{- 1} (d v - a β)$ , with $β = X d Y - Y d X$ and $Σ > 0$ ; its field-equation continuation is conditional on the current KN-verification draft.

[L1]

KN Killing generators, surface gravity, area and potential: for $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ , each root $r_{H}$ has future null Killing generator $χ_{H} = \partial_{v} + (a / R_{H}) Ψ$ , area $4 π R_{H}$ , signed $κ_{H} = Δ^{'} (r_{H}) / (2 R_{H}) = (r_{H} - r_{o t h e r}) / (2 R_{H})$ , and $- A_{i n} (χ_{H}) = q r_{H} / R_{H}$ ; the supplier proof remains draft and is used conditionally.

[L2]

Outer KN escape boundary on the selected ingoing model: on the selected ingoing product and for $M > 0$ , $M^{2} \geq a^{2} + q^{2} > 0$ , the exact current statement gives escape set ${r > r_{+}}$ and its relative boundary ${r = r_{+}}$ for the named path class; its proof is draft and not independently accepted.

[F4]

Causal solutions, Killing horizons, escape and metric extensions: the escape boundary is taken relative to the selected manifold, and the escape set uses continuous future piecewise- $C^{1}$ causal paths with finite $C^{1}$ subdivisions on compact parameter intervals.

[A1]

Geometric time is $t = v / c$ , so the time-normalized generator is $c χ_{H}$ when the angular rate is $c (a / R_{H})$ ; the SI one-form is $A_{S I} = A_{i n} / α$ . The gauge for the voltage is the displayed potential, whose contraction tends to zero at infinity.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric electrovac and intrinsic horizon-area conventions use only ${A C}_{ω}$ , meaning each specified countable family of nonempty sets has a choice function. No new selection is made in these finite parameter conversions, and full AC is not used.

Proof

1.1givenF1F2A2algebra

The exact SI map [F1] gives the displayed real length parameters $M, a, q$ and positive normalization $α$ . Setting magnetic flux to zero makes the KN datum's geometric charge exactly $q = q_{e}$ ; no ADM/Komar identification is involved. The stated inequality gives $D = M^{2} - a^{2} - q^{2} \geq 0$ and $\sqrt{D} < M$ , since $a^{2} + q^{2} > 0$ ; hence both roots are positive.

2.1F2F3L1step 1.1algebra

The factorization $Δ = (r - r_{+}) (r - r_{-})$ identifies these roots on the smooth selected ingoing model [F3]. The area, generator, signed surface gravity and displayed-gauge potential formulas then apply to either root by [L1].

2.2L1step 1.1algebra

When $D = 0$ , $r_{+} = r_{-} = M$ , so the signed difference formula in [L1] gives $κ_{H} = 0$ . When $D > 0$ , it gives positive outer and negative inner values under the displayed sign convention.

2.3L2F3F4step 1.1

Apply [L2] with the exact path class and relative topology in [F4]. Its current statement yields the selected escape boundary $r = r_{+}$ on $M_{i n}$ . This use is conditional on that draft supplier and says nothing about a conformal completion or maximal extension.

3.1F1F3L1A1step 2.1algebra

By [F1] and [A1], changing from length-time generator $χ_{H}$ to the SI time generator multiplies it by $c$ , while the SI potential is $A_{i n} / α$ . Therefore $- A_{S I} (c χ_{H}) = (c / α) (- A_{i n} (χ_{H})) = c q r_{H} / (α R_{H})$ . Also $[a / R_{H}] = m^{- 1}$ , so its SI angular rate is $c a / R_{H}$ ; $[κ_{H}] = m^{- 1}$ , so the stated acceleration convention is $c^{2} κ_{H}$ . The area formula has units of square metres. Finally $Σ = r^{2} + a^{2} \cos^{2} θ \geq r^{2}$ , so $r / Σ \leq 1 / r \to 0$ ; the two coefficient functions $- q r / Σ$ and $a q r / Σ$ of $A_{i n}$ in the $(d v, β)$ expression in the Statement vanish in the specified fixed- $(v, ω)$ limit.

4.1A2step 2.1step 2.2step 2.3step 3.1∎

The horizon formulas in steps 2.1-2.2 and the SI conversions in step 3.1 establish the claimed identifications on the specified model, conditional on the named draft mathematical interfaces. The escape boundary in step 2.3 uses the selected-model path convention. The ${A C}_{ω}$ premise [A2] is inherited as stated; no full choice principle, empirical measurement, or global event-horizon claim has been added.

5 · Examples, experiments and counterexamples

None yet.

SI and geometric solution parameter identifications

1 · Prerequisites

2 · Summary

Scope

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

SI and geometric solution parameter identifications

Definition

Dyonic RN as adopted Einstein–Maxwell model

Statement

Facts & Assumptions

Proof

Verified rotating electric KN physical model

Statement

Facts & Assumptions

Proof

Dyonic rotating model and gauge obstruction

Statement

Facts & Assumptions

Proof

Stationary RN observer frequency ratio

Statement

Facts & Assumptions

Proof

RN horizon and escape deductions for specified model

Statement

Facts & Assumptions

Proof

Rotating model horizons and selected causal escape

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources