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Nonzero null-wave field and curvature with four specified scalar contractions vanishing

Physical scope

Hypothetical exact null-wave solution of the selected classical SI Einstein–Maxwell model, specialized to one nonzero constant transverse profile. It illustrates a pointwise tensor distinction only and makes no observation, finite-energy, asymptotic-flatness, or empirical-validity claim.

Statement

Choose a constant $ν > 0$ with units $m^{- 1}$ and the profiles $a (u) = ν$ , $b (u) = 0$ , $h (u, x, y) = 0$ in Smooth null electrovac wave candidate data. Then $P = - ν^{2} (x^{2} + y^{2}), g = - 2 d u d v + d x^{2} + d y^{2} - ν^{2} (x^{2} + y^{2}) d u^{2},$ $F = ν d u \land d x, F = α^{- 1} F .$ Use orientation $d u \land d v \land d x \land d y$ and future unit field $N = \partial_{u} + (P + 1) \partial_{v} / 2$ from the null-wave definition. This is a smooth source-free solution of the adopted SI model of Equation-based Einstein–Maxwell model by Exact coupled electromagnetic null-wave model. Its electromagnetic field is nonzero. Define the scalar curvature by $S : = g^{a b} {Ric}_{a b}$ , using the curvature convention of Geometric electrovac equations and flux normalization. Then $F_{a b} F^{a b} = 0, S = 0, {Ric}_{a b} {Ric}^{a b} = 0, R_{a b c d} R^{a b c d} = 0.$ At the same time $R_{u x u x} = ν^{2} \neq 0$ , so the spacetime is not flat. These zero scalar contractions therefore do not imply a vanishing electromagnetic field or flat spacetime. This is a hypothetical model calculation, not an observation.

Facts & Assumptions

Given: A positive inverse-length constant $ν$ , the exact null-wave definitions, the selected equation-based physical postulate, and the current mathematical invariant proposition.

[F1]

Smooth null electrovac wave candidate data: the smooth null-wave metric, potential and two-form are determined by $a (u), b (u), h (u, x, y)$ and the prescribed transverse harmonicity.

[F2]

Exact coupled electromagnetic null-wave model: every datum of that profile class is an exact source-free solution of the selected SI model after $F = α^{- 1} F$ . This current physical theorem is draft with a complete authored proof and Step 1 authoring-ready record, not independent approval.

[F3]

Null wave curvature and zero quadratic invariants: for the null-wave datum, $F_{a b} F^{a b} = S = {Ric}_{a b} {Ric}^{a b} = R_{a b c d} R^{a b c d} = 0$ , its exact coordinate curvature formula is $R_{u i u j} = - \frac{1}{2} P_{i j}$ for $i, j \in {x, y}$ , and if $a^{2} + b^{2} > 0$ then the Riemann tensor is nonzero. This mathematical proposition is draft with a complete current proof and Step 1 authoring-ready record; the result is used conditionally.

[F4]

SI and geometric solution parameter identifications: $α > 0$ is constant, so $F = α^{- 1} F$ and $F_{a b} F^{a b} = α^{- 2} F_{a b} F^{a b}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): only the inherited ${A C}_{ω}$ context is used; the constant-profile substitution and tensor scaling require no choice.

Proof

Given: The hypotheses in [F1]–[F4] and [A1], with $ν > 0$ .

1.1givenF1A1algebra

With $a = ν$ , $b = h = 0$ , the potential in [F1] is $A = - ν x d u$ and direct exterior differentiation gives $F = ν d u \land d x$ . This is a smooth datum on $R^{4}$ , with nonzero $F$ because $ν > 0$ .

2.1givenF2F4step 1.1

Apply [F2] to the datum in step 1.1. The geometric two-form is scaled by the positive constant $α^{- 1}$ , so the SI field $F$ remains nonzero and the metric-field pair solves the adopted source-free equations.

3.1givenF1F3F4step 1.1algebra∎

Since $P = - ν^{2} (x^{2} + y^{2})$ , direct differentiation gives $P_{x x} = - 2 ν^{2}$ . The exact supplier formula in [F3] therefore gives $R_{u x u x} = - \frac{1}{2} P_{x x} = ν^{2} \neq 0$ . The same supplier gives $F_{a b} F^{a b} = 0$ , $S = 0$ , ${Ric}_{a b} {Ric}^{a b} = 0$ , and $R_{a b c d} R^{a b c d} = 0$ ; by [F4], $F_{a b} F^{a b} = α^{- 2} F_{a b} F^{a b} = 0$ . Thus this exact model datum has nonzero electromagnetic field and spacetime curvature although all four stated scalar contractions vanish.

Nonzero null-wave field and curvature with four specified scalar contractions vanishing

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources