Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

Dyonic RN as adopted Einstein–Maxwell model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources