ExampleConstruction: AI-adaptedVerification: 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.

Exact finite-frequency transverse Maxwell field

Example

Let $M = R^{4}$ have length coordinates $(x^{0}, x^{1}, x^{2}, z)$ , Minkowski metric $η = diag (- 1, 1, 1, 1)$ , orientation $d x^{0} \land d x^{1} \land d x^{2} \land d z$ , and future direction $\partial_{0}$ , with $x^{0} = c t$ . Choose constants $ν > 0$ with units $m^{- 1}$ and $A_{*} \neq 0$ with potential-component units $T m$ . For $0 < ϵ \leq 1$ , set

$θ = ν (x^{0} - z), a_{0} = A_{*} d x^{1}, a_{1} = 0, A_{ϵ}^{R} : = Re (ϵ e^{i θ / ϵ} a_{0}) .$

Then the real potential and field are

$A_{ϵ}^{R} = ϵ A_{*} \cos (θ / ϵ) d x^{1}, (F_{ϵ})_{01} = - ν A_{*} \sin (θ / ϵ), (F_{ϵ})_{31} = + ν A_{*} \sin (θ / ϵ),$

with all other independent field components zero. Let $Ω = R^{3}$ , $I = R$ , $x = (x^{1}, x^{2}, z)$ and $t = x^{0} / c$ . The corresponding smooth spatial fields $E, B : Ω \times I \to R^{3}$ are

$E (x, t) = c ν A_{*} \sin ⁣ (\frac{ν (c t - z)}{ϵ}) e_{1}, B (x, t) = ν A_{*} \sin ⁣ (\frac{ν (c t - z)}{ϵ}) e_{2},$

where $e_{1} = (1, 0, 0)$ and $e_{2} = (0, 1, 0)$ , with units $V m^{- 1}$ and $T$ , respectively. These are the SI fields of Smooth formal matter, EM tensors and observer quantities. This is an exact source-free Maxwell field for every fixed $ϵ$ . For $A_{*} \neq 0$ it has nonzero electromagnetic stress on an open set, so it is not a fully coupled source-free Einstein–Maxwell solution on unchanged Minkowski space with $Λ = 0$ .

Facts & Assumptions

Given: The Minkowski chart, phase, constant transverse amplitude and parameter ranges in the Example.

[F1]

Smooth formal matter, EM tensors and observer quantities fixes $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ and the smooth-field conventions.

[F2]

Finite WKB Maxwell residual, null rays and polarization quotient proves the exact finite-parameter potential and field expansions, residual identity, and conditional residual bounds under its four stated amplitude constraints.

[F3]

Smooth source/dust scaling and test-backreaction obstructions proves that a nonzero source-free Maxwell field with positive observer energy cannot solve the vacuum Einstein equation on the unchanged Minkowski metric; it also states the qualification against interpreting a test field as a coupled solution.

[F4]

Lorentzian exterior calculus and Hodge star gives the oriented Lorentzian Hodge-star component formula $(⋆_{g} F)_{a b} = \frac{1}{2} ({v o l}_{g})_{a b c d} F^{c d}$ ; in a positively oriented orthonormal coframe its basis formula fixes the signs used in the direct calculation below.

[A1]

The real field is the real part of the complex potential specified in the Example. This is an algebraic construction, not an observation.

Verification

1.1givenA1F1F2algebra

The phase gradient is $k = ν (d x^{0} - d z)$ and $g^{- 1} (k, k) = ν^{2} (- 1 + 1) = 0$ . The constant $a_{0}$ obeys $g^{- 1} (k, a_{0}) = 0$ and $L a_{0} = 0$ ; $a_{1} = 0$ makes $i L a_{1} + P a_{0} = 0$ and $i k^{a} (a_{1})_{a} + \nabla^{a} (a_{0})_{a} = 0$ . Here $\nabla$ is the flat connection and the Ricci tensor vanishes. Thus all four hypotheses of [F2] hold.

1.2givenA1F1algebra

Differentiating $A_{ϵ}^{R}$ in the displayed coordinates gives $(F_{ϵ})_{01} = \partial_{0} (A_{ϵ}^{R})_{1} = - ν A_{*} \sin (θ / ϵ)$ and $(F_{ϵ})_{31} = \partial_{z} (A_{ϵ}^{R})_{1} = + ν A_{*} \sin (θ / ϵ)$ ; no other independent component is produced. The SI component convention in [F1] therefore gives the displayed $E_{1}$ and $B_{2}$ .

2.1F2F3F4step 1.1step 1.2algebra

In [F2], this example has $a_{1} = 0$ , hence $d_{1} = 0$ and $P a_{1} - \nabla d_{1} = 0$ . Its exact residual formula therefore gives $\nabla^{a} (F_{ϵ})_{a b} = 0$ , and its exact closure identity gives $d F_{ϵ} = 0$ . Raising the free index in the first equality gives $\nabla_{a} F_{ϵ}^{a b} = 0$ . To check the other source-free Maxwell equation directly, put $f : = ν A_{*} \sin (θ / ϵ)$ . The displayed components give $F_{ϵ} = - f d x^{0} \land d x^{1} + f d z \land d x^{1}$ . From the metric and orientation in the Example, [F4] yields $⋆ F_{ϵ} = f (d x^{0} - d z) \land d x^{2}$ . Since $d f$ is a scalar multiple of $d θ = ν (d x^{0} - d z)$ , $d (⋆ F_{ϵ}) = d f \land (d x^{0} - d z) \land d x^{2} = 0$ . Therefore $d F_{ϵ} = 0$ and $d (⋆ F_{ϵ}) = 0$ hold exactly for every fixed $ϵ$ , as required by [F3].

3.1givenF3step 2.1algebra∎

Since $ν A_{*} \neq 0$ , there are points with $\sin (θ / ϵ) \neq 0$ . At any such point the field is nonzero and [F3] gives positive electromagnetic observer energy. Minkowski has $G_{a b} = 0$ , so the source-free Einstein equation with unchanged metric and $Λ = 0$ fails there. This caveat concerns coupling only; the Maxwell calculation in step 2.1 remains exact.

Exact finite-frequency transverse Maxwell field

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources