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A locally flat Kasner coordinate branch

Example

For lengths $A_{i} > 0$ in metres and a time $t_{0} > 0$ in seconds, set $I = (0, \infty)$ and take the exponents $(p_{1}, p_{2}, p_{3}) = (1, 0, 0)$ , so $a_{1} = A_{1} (t / t_{0})$ and $a_{2} = A_{2}$ , $a_{3} = A_{3}$ . They satisfy $\sum_{i} p_{i} = 1$ and $\sum_{i} p_{i}^{2} = 1$ , and the corresponding diagonal Bianchi I metric is Ricci-flat with identically zero Riemann tensor on $I$ . Define the open coordinate domain $D : = {(T, X, z_{2}, z_{3}) \in R^{4} : T > ∣ X ∣}$ and its Minkowski metric $η : = - d T^{2} + d X^{2} + d z_{2}^{2} + d z_{3}^{2}$ , with all four coordinates measured in metres. In any local quotient chart with lifted dimensionless angular coordinates $x^{1}, x^{2}, x^{3}$ , let $α : = A_{1} / (c t_{0})$ and define

$Φ (t, x^{1}, x^{2}, x^{3}) : = (c t \cosh (α x^{1}), c t \sinh (α x^{1}), A_{2} x^{2}, A_{3} x^{3}) .$

This map is a local isometry from the chart domain, with the Bianchi I metric, onto an open subset of $(D, η)$ . Its scale factor $a_{1}$ tends to zero as $t ↓ 0$ , but that excluded coordinate boundary alone does not establish curvature blowup or a global extension on the torus quotient.

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 on $I = (0, \infty)$ with $A_{i} > 0$ lengths in metres and $t_{0} > 0$ seconds, a local lift of the periodic angular coordinates, and 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]

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

[F2]

The local C2 curvature convention defines the Riemann tensor from the displayed connection coefficients and uses the stated sign/index order (C2 curvature for the diagonal Bianchi I quotient metric).

[F3]

The curvature components are $R_{0 i 0 i} = - a_{i}^{2} ({\dot{H}}_{i} + H_{i}^{2})$ and $R_{i j i j} = a_{i}^{2} a_{j}^{2} H_{i} H_{j} / c^{2}$ ; all other independent index patterns vanish (Curvature components of the diagonal Bianchi I metric).

[F4]

In a nonstatic power-law branch, Ricci-flatness is equivalent to the two Kasner sums; the all-zero exponent case is separately flat (Kasner vacuum constraints).

[F5]

For isotropic diagonal source, each $V (H_{i} - H_{j})$ is constant and $V^{2} σ^{2}$ is constant with units $m^{6} s^{- 2}$ (Diagonal-source shear equation and isotropic-source decay).

Verification

technique · direct component substitution followed by an explicit local coordinate change

1.1givenA1F4algebra

Work under the inherited ${A C}_{ω}$ context [A1]. The exponents have $\sum_{i} p_{i} = 1$ and $\sum_{i} p_{i}^{2} = 1$ , so [F4] gives the Ricci-flat conclusion for this nonstatic branch.

2.1step 1.1F1F2F3F5algebra

The directional rates are $H_{1} = 1 / t$ and $H_{2} = H_{3} = 0$ . Thus every ${\dot{H}}_{i} + H_{i}^{2}$ is zero and every product $H_{i} H_{j}$ with $i \neq j$ is zero; [F3] makes every independent Riemann component vanish, hence the full curvature tensor vanishes on $I$ . Also $H = 1 / (3 t)$ , $V = A_{1} A_{2} A_{3} (t / t_{0})$ and $σ^{2} = 1 / (3 t^{2})$ . Since step 1.1 and [F2] give $R = 0$ and $G^{i}_{i} = 0$ , the equations in [F5] hold with $Λ = 0$ and $Q_{i} = 0$ , yielding $V^{2} σ^{2} = (A_{1} A_{2} A_{3})^{2} / (3 t_{0}^{2})$ .

3.1step 2.1F2algebra∎

On the lifted chart, put $r = c t > 0$ and $θ = α x^{1}$ . Then $(T, X) = (r \cosh θ, r \sinh θ)$ satisfies $T > ∣ X ∣$ , and the inverse in its image is $t = \sqrt{T^{2} - X^{2}} / c$ , $x^{1} = artanh (X / T) / α$ , $x^{2} = z_{2} / A_{2}$ , and $x^{3} = z_{3} / A_{3}$ . Thus $Φ$ is a diffeomorphism onto an open subset of $D$ . The identities $- d T^{2} + d X^{2} = - d r^{2} + r^{2} d θ^{2} = - c^{2} d t^{2} + a_{1} (t)^{2} (d x^{1})^{2}$ and $(d z_{2})^{2} + (d z_{3})^{2} = a_{2}^{2} (d x^{2})^{2} + a_{3}^{2} (d x^{3})^{2}$ give $Φ^{*} η = g$ . The local lift does not define a global coordinate transformation of the periodic torus. Since all curvature components already vanish for every $t > 0$ , the collapse of $a_{1}$ at the excluded endpoint $t = 0$ alone proves no curvature blowup.

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)