Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Geometric electrovac data solve the selected SI equations

Physical scope

Conditional conversion of a smooth zero-cosmological-constant geometric electrovac datum to the source-free branch of the selected SI equation-based Einstein–Maxwell model. It does not assert existence of geometric data, derive or empirically validate a postulate, add material sources, or claim physical realization.

Statement

Let $(M, g, N, o, F)$ be a smooth source-free geometric electrovac datum of Geometric electrovac equations and flux normalization on a nonempty smooth four-manifold. Thus $g$ has signature $(-, +, +, +)$ , $F$ is a smooth real two-form, and $d F = 0, d (⋆_{g} F) = 0, {Ric}_{a b} = 2 Q_{a b} (F), Q_{a b} (X) : = X_{a c} X_{b}^{c} - \frac{1}{4} g_{a b} X_{c d} X^{c d} .$ Choose constants $G_{N}, c, μ_{0} > 0$ in the SI roles of Einstein–Maxwell primitive quantities and SI conventions, set $ε_{0} = (μ_{0} c^{2})^{- 1}, α = \sqrt{\frac{4 π G_{N}}{μ_{0} c^{4}}}, F : = α^{- 1} F,$ and use the same metric and orientation for $F$ . In length coordinates $x^{0} = c t$ , $F_{a b}$ has units T, $F_{a b}$ has units $m^{- 1}$ , and $α$ has units $T^{- 1} m^{- 1}$ by SI and geometric solution parameter identifications. Define $T_{a b}^{E M} : = \frac{1}{μ_{0}} Q_{a b} (F), κ_{E} : = \frac{8 π G_{N}}{c^{4}} .$ In the adopted postulate Equation-based Einstein–Maxwell model, specialize to $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ . Then $d F = 0, \nabla_{a}^{g} F^{a b} = 0, G_{a b} = κ_{E} T_{a b}^{E M} .$ This is a conditional model deduction: it neither derives the postulate nor claims that the model describes nature.

Facts & Assumptions

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

[F1]

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

[F2]

SI and geometric solution parameter identifications: $F = α F$ and $α^{2} = 4 π G_{N} / (μ_{0} c^{4})$ ; in length coordinates $F$ has units $m^{- 1}$ and $F$ has units T.

[F3]

Einstein–Maxwell primitive quantities and SI conventions: $ε_{0} = (μ_{0} c^{2})^{- 1}$ , $κ_{E} = 8 π G_{N} / c^{4}$ , and the Einstein tensor has units $m^{- 2}$ while stress has units ${J m}^{- 3}$ .

[F4]

Sourced Maxwell sign and current conservation: for $v^{b} = \nabla_{a} F^{a b}$ , $d (⋆_{g} F) = - ι_{v} {v o l}_{g}$ ; 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.

[F5]

Equation-based Einstein–Maxwell model: the adopted smooth-field equations are $d F = 0$ , $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , and $G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m})$ .

[A1]

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

Proof

Given: The hypotheses and definitions in [F1]–[F5], [A1], $α > 0$ , and $F = α^{- 1} F$ .

1.1givenF1F4algebra

Since $α$ is constant, $d F = α^{- 1} d F = 0$ and $d (⋆_{g} F) = α^{- 1} d (⋆_{g} F) = 0$ by [F1]. Apply the sign identity [F4] to $F$ : the vanishing of $d (⋆_{g} F)$ gives $ι_{v} {v o l}_{g} = 0$ , hence $v^{b} = \nabla_{a} F^{a b} = 0$ at every point.

1.2givenF1F2algebra

Contracting the definition of $Q$ in [F1] gives $g^{a b} Q_{a b} (X) = X_{a c} X^{a c} - (4 / 4) X_{c d} X^{c d} = 0$ . Therefore [F1] implies the scalar curvature is zero and $G_{a b} = {Ric}_{a b} = 2 Q_{a b} (F)$ . Since $Q$ is quadratic and $F = α F$ , this is $G_{a b} = 2 α^{2} Q_{a b} (F)$ .

2.1givenF2F3F5A1step 1.1step 1.2algebra∎

Substituting $α^{2} = 4 π G_{N} / (μ_{0} c^{4})$ from [F2] yields $G_{a b} = (8 π G_{N} / c^{4}) Q_{a b} (F) / μ_{0} = κ_{E} T_{a b}^{E M}$ . Together with step 1.1 and the specialization of [F5], these are exactly the adopted source-free equations. No equation is asserted at a removed point or outside the datum's domain, and the finite rescaling uses no choice beyond [A1].

Geometric electrovac data solve the selected SI equations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources