PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Diagonal-source shear equation and isotropic-source decay

Statement

For a diagonal Bianchi I datum, let $Λ$ be a constant in $m^{- 2}$ and let $Q_{i} : I \to R$ be continuous functions in $m^{- 2}$ . Suppose the spatial diagonal equations

$G^{i}_{i} + Λ = Q_{i} (t) (i = 1, 2, 3; no sum over i)$

hold, where $G^{i}_{i}$ is defined by the local $C^{2}$ curvature convention on this page. With $Δ_{i j} : = H_{i} - H_{j}$ and $H_{Σ} : = 3 H$ , subtracting the equations gives

${\dot{Δ}}_{i j} + 3 H Δ_{i j} = c^{2} (Q_{i} - Q_{j}) .$

If the diagonal source is isotropic, meaning $Q_{1} = Q_{2} = Q_{3}$ , then $V Δ_{i j}$ is constant for each pair, and there is a constant $C \geq 0$ with units $m^{6} s^{- 2}$ such that $σ^{2} = C V^{- 2}$ throughout $I$ . For unequal diagonal sources the right-hand side remains; this proposition makes no unsourced $V^{- 2}$ claim in that case.

Statement references

Diagonal Bianchi I model

C2 curvature for the diagonal Bianchi I quotient metric

Curvature components of the diagonal Bianchi I metric

Facts & Assumptions

Given: A diagonal Bianchi I datum on $I$ , its $H_{i}$ , $H$ , $V$ and $σ^{2}$ , a constant $Λ$ , and the stated spatial diagonal equations with functions $Q_{i}$ .

[A1]

The Bianchi I product and quotient construction carries ${A C}_{ω}$ and no stronger choice principle (Diagonal Bianchi I model).

[A2]

The diagonal equations $G^{i}_{i} + Λ = Q_{i} (t)$ are explicit mathematical hypotheses of this proposition; when $Q_{i} \neq Q_{j}$ , their difference is retained as a source term.

[F1]

$H_{i} = {\dot{a}}_{i} / a_{i}$ , $H = \frac{1}{3} \sum_{i} H_{i}$ , $V = a_{1} a_{2} a_{3}$ , $σ^{2} = \frac{1}{2} \sum_{i} (H_{i} - H)^{2}$ , and all $a_{i}$ are positive $C^{2}$ functions (Diagonal Bianchi I model).

[F2]

The local Einstein components are defined by $G_{a b} = {Ric}_{a b} - \frac{1}{2} R g_{a b}$ and the stated raising convention (C2 curvature for the diagonal Bianchi I quotient metric).

[F3]

${Ric}_{i i} = a_{i}^{2} ({\dot{H}}_{i} + H_{i} H_{Σ}) / c^{2}$ and the off-diagonal Ricci components vanish (Curvature components of the diagonal Bianchi I metric).

[F4]

If a real-valued function is continuous on an interval and differentiable at every interior point with zero derivative there, then it is constant on the interval (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

technique · subtract the diagonal equations, apply the product rule, and use the zero-derivative constancy result

1.1givenA1A2F2F3algebra

Work under the inherited ${A C}_{ω}$ context [A1]; no choice is used in the component subtraction. Raising the first spatial index in [F3] gives ${Ric}^{i}_{i} = c^{- 2} ({\dot{H}}_{i} + H_{i} H_{Σ})$ . By [F2], $G^{i}_{i} = {Ric}^{i}_{i} - \frac{1}{2} R$ ; therefore $G^{i}_{i} - G^{j}_{j} = c^{- 2} ({\dot{H}}_{i} - {\dot{H}}_{j} + H_{Σ} (H_{i} - H_{j}))$ .

2.1step 1.1A2F1algebra

Subtract the $j$ equation in [A2] from the $i$ equation. The constant $Λ$ and common scalar-curvature term cancel, so step 1.1 gives ${\dot{Δ}}_{i j} + H_{Σ} Δ_{i j} = c^{2} (Q_{i} - Q_{j})$ . Since $H_{Σ} = 3 H$ , this is the displayed sourced shear-difference equation. In particular the source vanishes exactly when the two diagonal source functions agree.

3.1step 2.1F1F4algebra

Differentiating $V = a_{1} a_{2} a_{3}$ gives $\dot{V} = V \sum_{k} {\dot{a}}_{k} / a_{k} = V H_{Σ}$ . Multiplying the equation in step 2.1 by $V$ and applying the product rule yields $\frac{d}{d t} (V Δ_{i j}) = c^{2} V (Q_{i} - Q_{j})$ . Each $H_{i}$ is $C^{1}$ because $a_{i}$ is positive and $C^{2}$ , so $V Δ_{i j}$ is continuous on $I$ and differentiable at every point of $I$ . When the diagonal source is isotropic its derivative vanishes; [F4] then makes $V Δ_{i j}$ constant on $I$ . For unequal sources the displayed derivative retains the stated forcing term.

4.1step 3.1F1algebra∎

Put $d_{i} : = H_{i} - H$ . Then $\sum_{j} Δ_{i j} = 3 H_{i} - H_{Σ} = 3 d_{i}$ , so $V d_{i} = \frac{1}{3} \sum_{j} V Δ_{i j}$ is constant in the isotropic-source case by step 3.1. Consequently $V^{2} σ^{2} = \frac{1}{2} \sum_{i} (V d_{i})^{2}$ is a constant $C \geq 0$ with units $m^{6} s^{- 2}$ , which is equivalent to $σ^{2} = C V^{- 2}$ because $V > 0$ .

Depends on

Dependency tree · 2 levels

Sources

Diego L. Cáceres, Leonardo Castañeda, and Juan M. Tejeiro, Shear Dynamics in Bianchi I Cosmology, arXiv:1003.3491v1 (2010), Section 2, equations (2.1)–(2.2), and Section 2.1, equations (1.6), (2.8)–(2.17), printed pp. 2–4 (PDF pp. 3–5)