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

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Kasner vacuum constraints

Statement

Let $a_{i} (t) = A_{i} (t / t_{0})^{p_{i}}$ on a nonempty open interval $I \subseteq (0, \infty)$ , where $A_{i} > 0$ are lengths in metres, $t_{0} > 0$ is a time in seconds, and $p_{i} \in R$ are dimensionless exponents. These $p_{i}$ are called Kasner exponents for this power-law branch; define their sums $P : = \sum_{i = 1}^{3} p_{i}$ and $Q : = \sum_{i = 1}^{3} p_{i}^{2}$ . For a nonstatic datum, meaning that not all $p_{i}$ vanish, the geometric vacuum equation $Ric = 0$ in the $C^{2}$ curvature definition of this page holds exactly when $P = Q = 1$ . If all $p_{i} = 0$ , then the full curvature tensor vanishes, so the metric is flat in the curvature sense. These are exact statements about the specified finite-dimensional metrics; no singularity or generic cosmological conclusion follows.

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, the power-law scale factors in the Statement, and the mathematical vacuum condition $Ric = 0$ where invoked. The inherited foundational context assumes ${A C}_{ω}$ only.

[A1]

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

[F1]

The scale factors are positive $C^{2}$ lengths on $I \subseteq (0, \infty)$ , and the definitions of $H_{i}$ , $H$ and $V$ are those of the diagonal Bianchi I datum (Diagonal Bianchi I model).

[F2]

The local Ricci components are ${Ric}_{00} = - \sum_{i} ({\dot{H}}_{i} + H_{i}^{2})$ and ${Ric}_{i i} = a_{i}^{2} ({\dot{H}}_{i} + H_{i} H_{Σ}) / c^{2}$ , with zero off-diagonal components; the only potentially nonzero curvature components are the two displayed families (Curvature components of the diagonal Bianchi I metric).

Proof

technique · direct substitution into the local curvature components

1.1givenA1F1F2algebra

Work in the inherited ${A C}_{ω}$ context [A1]; the calculation itself uses only finite sums and differentiation. Put $P = \sum_{i} p_{i}$ and $Q = \sum_{i} p_{i}^{2}$ . The power laws give $H_{i} = p_{i} / t$ , ${\dot{H}}_{i} = - p_{i} / t^{2}$ , and $H_{Σ} = P / t$ , so [F2] yields ${Ric}_{00} = (P - Q) / t^{2}$ and ${Ric}_{i i} = a_{i}^{2} p_{i} (P - 1) / (c^{2} t^{2})$ .

2.1step 1.1algebra

Suppose not all $p_{i}$ vanish and $Ric = 0$ . Choose an index $k$ with $p_{k} \neq 0$ . Since $a_{k}, c, t$ are positive, the equation ${Ric}_{k k} = 0$ from step 1.1 implies $P = 1$ . The equation ${Ric}_{00} = 0$ then implies $Q = P = 1$ .

2.2step 1.1F2algebra

Conversely, if $P = Q = 1$ , step 1.1 makes every diagonal Ricci component zero, and [F2] makes every off-diagonal component zero; hence $Ric = 0$ .

3.1givenF1F2algebra∎

If $p_{1} = p_{2} = p_{3} = 0$ , then every $H_{i}$ and ${\dot{H}}_{i}$ is zero. Both curvature-component families in [F2] therefore vanish, and every other component is zero by that lemma, so the full curvature tensor vanishes.

Depends on

Dependency tree · 2 levels

Sources

Rishabh Jha, Introduction to Hamiltonian Formulation of General Relativity and Homogeneous Cosmologies, SciPost Physics Lecture Notes 73 (2023), arXiv:2204.03537v2, Section 4.3.1, equations (159)–(172), printed pp. 36–38 (PDF pp. 37–39)