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Single-center extremal charged model correspondence

Physical scope

Equality of the single-center harmonic MP and electric extremal RN model data on their open exteriors under one explicit SI mass/charge parametrization. No horizon extension, global completeness, source at the omitted center, ADM or Komar identification, observed object, or empirical validity is asserted.

Statement

Let $m_{*} > 0$ be a selected SI mass parameter, choose $σ \in {- 1, + 1}$ , and set $m : = \frac{G_{N} m_{*}}{c^{2}}, Q_{e} : = σ \sqrt{4 π ε_{0} G_{N}} m_{*}, Φ_{B} : = 0.$ Use $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ and the parameter map of SI and geometric solution parameter identifications. On the single-center MP exterior $D = R^{3} ∖ {0}$ with $H = 1 + m / ∣ x ∣$ , the change of radial coordinate $R = ∣ x ∣ + m$ identifies the SI metric and field $F = α^{- 1} F_{σ}$ exactly with the electric RN data of Dyonic Reissner-Nordström metric and field domains on $R > m$ , with $M = m, q_{e} = σ m, q_{m} = 0, f (R) = {(1 - \frac{m}{R})}^{2} .$ Thus $∣ q_{e} ∣ = M$ and the two presentations describe the same source-free solution of the adopted zero- $Λ$ equation-based model on that open exterior. The correspondence does not extend the coordinate map to $R = m$ .

Facts & Assumptions

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

[F1]

Positive harmonic conformastatic electrovac data: the single-center harmonic datum has $H = 1 + m / ρ$ , $g = - H^{- 2} (d x^{0})^{2} + H^{2} (d ρ^{2} + ρ^{2} γ_{S^{2}})$ , and $F_{σ} = d (σ (H^{- 1} - 1) d x^{0})$ .

[F2]

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 $F = α F$ .

[F3]

Single-center MP is extremal Reissner-Nordström exterior: for $H = 1 + m / ρ$ and $R = ρ + m$ , the full MP metric and two-form equal RN data with $M = m$ , $q_{e} = σ m$ , $q_{m} = 0$ , and $f = (1 - m / R)^{2}$ 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.

[F4]

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.

[F5]

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 $f \neq 0$ . This current physical theorem is draft with a complete authored proof and Step 1 authoring-ready record, not independent proof approval.

[F6]

Dyonic Reissner-Nordström metric and field domains: the RN metric and two-form are $g = - f (d x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ and $F = - q_{e} r^{- 2} d x^{0} \land d r + q_{m} ω_{S^{2}}$ .

[F7]

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

[A1]

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

Proof

Given: The hypotheses and exact current interfaces [F1]–[F7] and [A1].

1.1givenF2algebra

Substituting $Q_{e}$ and $Φ_{B} = 0$ into [F2] gives $q_{e} = σ G_{N} m_{*} / c^{2} = σ m$ and $q_{m} = 0$ , while $M = m > 0$ . Thus $∣ q_{e} ∣ = M$ and $f (R) = (1 - m / R)^{2} > 0$ for $R > m$ .

2.1givenF1F2F3F6step 1.1constructalgebra

The radial map $R = ρ + m$ is a smooth diffeomorphism $(0, \infty) \to (m, \infty)$ with inverse $ρ = R - m$ . By [F1] the single-center candidate has $H = 1 + m / ρ$ and the stated metric-field form. Apply [F3]: $H^{- 1} = 1 - m / R$ and $H^{2} ρ^{2} = R^{2}$ , so the metric and two-form agree with [F6] on the full open exterior. Multiplication of both geometric two-forms by the same constant $α^{- 1}$ preserves their equality for the SI fields.

3.1givenF4F5F7A1step 2.1∎

By [F4] and [F5], the identical data on the two sides satisfy the same specialized physical equations from [F7]. This establishes a coordinate correspondence between exact model solutions only; no extension through the excluded surface or center follows.

Single-center extremal charged model correspondence

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources