ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Nonflat anisotropic Kasner exponents

Example

On any nonempty open interval $I \subseteq (0, \infty)$ , the exponents $(p_{1}, p_{2}, p_{3}) = (- 1 / 3, 2 / 3, 2 / 3)$ with $a_{i} = A_{i} (t / t_{0})^{p_{i}}$ , $A_{i} > 0$ lengths in metres and $t_{0} > 0$ a time in seconds, give a nonstatic Ricci-flat diagonal Bianchi I metric. Its directional expansion rates are unequal, the scalar $σ^{2} : = \frac{1}{2} \sum_{i} (H_{i} - H)^{2}$ equals $1 / (3 t^{2})$ , and its curvature component $R_{0101} = - 4 a_{1}^{2} / (9 t^{2})$ is nonzero. Thus the Riemann tensor is nonzero and the metric is not flat.

Example references

Diagonal Bianchi I model

C2 curvature for the diagonal Bianchi I quotient metric

Curvature components of the diagonal Bianchi I metric

Kasner vacuum constraints

Diagonal-source shear equation and isotropic-source decay

Facts & Assumptions

Given: The stated power-law Bianchi I metric with $A_{i} > 0$ , $t_{0} > 0$ , and $t \in I \subseteq (0, \infty)$ , in the inherited ${A C}_{ω}$ context.

[A1]

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

[F1]

The rates, mean expansion, volume and shear scalar are $H_{i} = {\dot{a}}_{i} / a_{i}$ , $H = \frac{1}{3} \sum_{i} H_{i}$ , $V = \prod_{i} a_{i}$ and $σ^{2} = \frac{1}{2} \sum_{i} (H_{i} - H)^{2}$ (Diagonal Bianchi I model).

[F2]

The C2 connection and curvature use the coordinate order $R^{a}_{b c d}$ and the definitions of $Ric$ and $G$ stated on this page (C2 curvature for the diagonal Bianchi I quotient metric).

[F3]

$R_{0 i 0 i} = - a_{i}^{2} ({\dot{H}}_{i} + H_{i}^{2})$ and ${Ric}_{00}, {Ric}_{i i}$ have the explicit component formulas in the curvature lemma (Curvature components of the diagonal Bianchi I metric).

[F4]

For a nonstatic power-law Bianchi I metric, $Ric = 0$ exactly when $\sum_{i} p_{i} = \sum_{i} p_{i}^{2} = 1$ (Kasner vacuum constraints).

[F5]

With isotropic diagonal source, $V (H_{i} - H_{j})$ is constant and $σ^{2} = C V^{- 2}$ , where $C$ has units $m^{6} s^{- 2}$ ; the zero source is isotropic (Diagonal-source shear equation and isotropic-source decay).

Verification

technique · direct finite substitution and one explicit curvature component

1.1givenA1F4algebra

Work under the inherited ${A C}_{ω}$ context [A1]. For $p = (- 1 / 3, 2 / 3, 2 / 3)$ , $\sum_{i} p_{i} = 1$ and $\sum_{i} p_{i}^{2} = 1 / 9 + 4 / 9 + 4 / 9 = 1$ ; none of the exponents is zero, so [F4] gives the Ricci-flat conclusion.

2.1step 1.1F1F2F5algebra

The rates are $H_{1} = - 1 / (3 t)$ and $H_{2} = H_{3} = 2 / (3 t)$ , so $H = 1 / (3 t)$ , $V = A_{1} A_{2} A_{3} (t / t_{0})$ and $σ^{2} = \frac{1}{2} (4 / 9 + 1 / 9 + 1 / 9) t^{- 2} = 1 / (3 t^{2})$ . By Ricci-flatness from step 1.1 and the definition in [F2], $R = 0$ and $G^{i}_{i} = 0$ for every $i$ . Thus the spatial equations in [F5] hold with $Λ = 0$ and isotropic source $Q_{i} = 0$ , giving $V^{2} σ^{2} = (A_{1} A_{2} A_{3})^{2} / (3 t_{0}^{2})$ .

3.1step 1.1F2F3algebra∎

From [F3], $R_{0101} = - a_{1}^{2} ({\dot{H}}_{1} + H_{1}^{2}) = - a_{1}^{2} (1 / (3 t^{2}) + 1 / (9 t^{2})) = - 4 a_{1}^{2} / (9 t^{2}) \neq 0$ throughout $I$ . Thus the metric has nonzero curvature and is not flat, while step 1.1 established Ricci-flatness.

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)