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Curvature components of the diagonal Bianchi I metric

Statement

For a diagonal Bianchi I datum, write $H_{Σ} : = H_{1} + H_{2} + H_{3} = 3 H$ . In every local quotient chart, the only nonzero Christoffel symbols are $Γ^{0}_{i i} = a_{i}^{2} H_{i} / c^{2}$ and $Γ^{i}_{0 i} = Γ^{i}_{i 0} = H_{i}$ . The possibly nonzero lowered curvature components are exactly those obtained from the following values by the displayed sign reversals:

$R_{0 i 0 i} = R_{i 0 i 0} = - a_{i}^{2} ({\dot{H}}_{i} + H_{i}^{2}), R_{0 i i 0} = R_{i 00 i} = a_{i}^{2} ({\dot{H}}_{i} + H_{i}^{2}),$

$R_{i j i j} = R_{j i j i} = \frac{a_{i}^{2} a_{j}^{2}}{c^{2}} H_{i} H_{j}, R_{i j j i} = R_{j i i j} = - \frac{a_{i}^{2} a_{j}^{2}}{c^{2}} H_{i} H_{j} (1 \leq i < j \leq 3) .$

The Ricci components are

${Ric}_{00} = - \sum_{i = 1}^{3} ({\dot{H}}_{i} + H_{i}^{2}), {Ric}_{i i} = \frac{a_{i}^{2}}{c^{2}} ({\dot{H}}_{i} + H_{i} H_{Σ}),$

and every off-diagonal Ricci component is zero.

Statement references

Diagonal Bianchi I model

C2 curvature for the diagonal Bianchi I quotient metric

Facts & Assumptions

Given: A diagonal Bianchi I datum on a nonempty open interval $I \subseteq R$ , its positive $C^{2}$ scale factors and metric, and the inherited ${A C}_{ω}$ context.

[A1]

The quotient and product construction for this datum carries the axiom of countable choice ${A C}_{ω}$ and no stronger choice principle (Diagonal Bianchi I model).

[F1]

In a local quotient chart the metric has entries $g_{00} = - c^{2}$ , $g_{i i} = a_{i}^{2}$ , and $g_{0 i} = 0$ for $i = 1, 2, 3$ ; each $a_{i}$ is positive and $C^{2}$ , and all spatial derivatives of the metric entries vanish (Diagonal Bianchi I model).

[F2]

The connection, curvature, Ricci tensor and their sign/index conventions are given by the coordinate formulas in the local $C^{2}$ definition (C2 curvature for the diagonal Bianchi I quotient metric).

Proof

technique · direct coordinate calculation

1.1givenA1F1F2algebra

Work under the inherited ${A C}_{ω}$ context [A1]; no choice is used in the following finite coordinate calculation. The inverse metric has $g^{00} = - c^{- 2}$ , $g^{i i} = a_{i}^{- 2}$ and zero off-diagonal entries by [F1]. Substitution in [F2] gives $Γ^{0}_{i i} = \frac{1}{2} (- c^{- 2}) (- 2 a_{i} {\dot{a}}_{i}) = a_{i}^{2} H_{i} / c^{2}$ and $Γ^{i}_{0 i} = Γ^{i}_{i 0} = \frac{1}{2} a_{i}^{- 2} (2 a_{i} {\dot{a}}_{i}) = H_{i}$ . Every other coefficient is zero because each differentiated spatial metric entry is zero and the metric matrix is diagonal.

2.1step 1.1F2algebra

The curvature formula [F2] and step 1.1 give $R^{0}_{i 0 i} = (a_{i}^{2} / c^{2}) ({\dot{H}}_{i} + H_{i}^{2})$ and $R^{i}_{0 i 0} = - ({\dot{H}}_{i} + H_{i}^{2})$ ; lowering the first index gives $R_{0 i 0 i} = R_{i 0 i 0} = - a_{i}^{2} ({\dot{H}}_{i} + H_{i}^{2})$ . Reversing the last two direction indices in the same formula gives $R_{0 i i 0} = R_{i 00 i} = + a_{i}^{2} ({\dot{H}}_{i} + H_{i}^{2})$ . For $i \neq j$ , its only nonzero product in $R^{i}_{j i j}$ is $Γ^{i}_{i 0} Γ^{0}_{j j} = a_{j}^{2} H_{i} H_{j} / c^{2}$ , so $R_{i j i j} = a_{i}^{2} a_{j}^{2} H_{i} H_{j} / c^{2}$ ; direct substitution with the indices reversed gives $R_{j i j i}$ equal to this and $R_{i j j i} = R_{j i i j}$ equal to its negative. Every remaining index pattern contains either a spatial derivative of a metric/connection coefficient or a zero coefficient from step 1.1, so it vanishes.

3.1step 2.1F1F2algebra∎

Contracting the components from step 2.1 with the inverse metric gives ${Ric}_{00} = \sum_{i} R^{i}_{0 i 0} = - \sum_{i} ({\dot{H}}_{i} + H_{i}^{2})$ . For each $i$ , ${Ric}_{i i} = g^{00} R_{0 i 0 i} + \sum_{j \neq i} g^{j j} R_{j i j i} = \frac{a_{i}^{2}}{c^{2}} ({\dot{H}}_{i} + H_{i}^{2}) + \frac{a_{i}^{2}}{c^{2}} \sum_{j \neq i} H_{i} H_{j} = \frac{a_{i}^{2}}{c^{2}} ({\dot{H}}_{i} + H_{i} H_{Σ})$ . The index patterns in step 2.1 give zero for every off-diagonal contraction. These are the displayed Ricci components.

Curvature components of the diagonal Bianchi I metric

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources