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Flat-torus vacuum data from a constant TT tensor

Physical scope

Exact mathematical example of compact flat-torus vacuum constraint data with nonzero transverse-traceless part; it makes no empirical claim or spacetime-development assertion.

Example

Let $Σ = T_{2 π}^{3}$ have dimensionless periodic coordinates $x^{i}$ and conformal metric $\tilde{h} = L_{*}^{2} \sum_{i = 1}^{3} (d x^{i})^{2}$ , where $L_{*} > 0$ is a length. Let $\tilde{σ}$ have constant covariant components ${\tilde{σ}}_{11} = a_{*}$ , ${\tilde{σ}}_{22} = - a_{*}$ and all other components zero, where $a_{*} > 0$ is a length. Fix a nonzero constant $τ$ with units $m^{- 1}$ . Then $∣ \tilde{σ} ∣_{\tilde{h}}^{2} = A : = 2 a_{*}^{2} / L_{*}^{4} > 0$ and $\tilde{R} = 0$ . The unique positive constant solution $ϕ_{0} = (3 A / (2 τ^{2}))^{1 / 12}$ gives $h = ϕ_{0}^{4} \tilde{h}$ and $K = ϕ_{0}^{- 2} \tilde{σ} + (τ / 3) h$ , which satisfy the vacuum constraint equations. The tracefree part of $K$ is nonzero.

Example references

Vacuum conformal constraint data

Conformal reduction of the vacuum constraint equations

Exact positive constant solution of the CMC constraint equation

Christoffel formula for the levi civita connection

Dual connection

Product connection on tensor and hom bundles

Curvature is a type (1,3) tensor

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s

Facts & Assumptions

Given: The flat torus, $L_{*} > 0$ , $a_{*} > 0$ , and a constant $τ \neq 0$ with the units and coordinate convention in the Example. Assume ${A C}_{ω}$ for the torus supplier and the inherited embedded Lorentzian constraint interpretation.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ), Vacuum conformal constraint data: ${A C}_{ω}$ is the axiom that every countable family of nonempty sets has a choice function; the torus supplier assumes it for its quotient convention, and the conformal-data definition requires it for an embedded Lorentzian interpretation.

[F1]

Vacuum conformal constraint data: $\tilde{D}$ is the Levi-Civita connection of $\tilde{h}$ ; TT means ${\tilde{h}}^{i j} {\tilde{σ}}_{i j} = 0$ and ${\tilde{D}}^{i} {\tilde{σ}}_{i j} = 0$ ; and the ansatz is $h = ϕ^{4} \tilde{h}$ , $K = ϕ^{- 2} \tilde{σ} + (τ / 3) h$ .

[F2]

Conformal reduction of the vacuum constraint equations: the momentum constraint vanishes for this TT, constant-trace ansatz, and the Hamiltonian constraint is equivalent to the displayed Lichnerowicz equation.

[F3]

Exact positive constant solution of the CMC constraint equation: when $\tilde{R} = 0$ and $∣ \tilde{σ} ∣_{\tilde{h}}^{2} = A > 0$ is constant, the unique positive constant solution for $τ \neq 0$ is $(3 A / (2 τ^{2}))^{1 / 12}$ .

[F4]

Christoffel formula for the levi civita connection: in a chart, $Γ^{k}_{i j} = \frac{1}{2} {\tilde{h}}^{k ℓ} (\partial_{i} {\tilde{h}}_{j ℓ} + \partial_{j} {\tilde{h}}_{i ℓ} - \partial_{ℓ} {\tilde{h}}_{i j})$ .

[F5]

Dual connection, Product connection on tensor and hom bundles: the covariant derivative on covariant tensor fields obeys the dual and product Leibniz rules.

[F6]

Curvature is a type (1,3) tensor: curvature is tensorial, so its components in a coordinate basis determine it pointwise.

[F7]

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s: $T_{2 π}^{3} = (R / 2 π Z)^{3}$ has the quotient/product topology; smooth maps are those with smooth periodic lifts to $R^{3}$ . The angular expressions here are local coordinates and invariant differentials under translations by $2 π Z^{3}$ .

Verification

1.1givenA1F1F4F5F6F7algebra

The displayed constant metric and tensor components are invariant under the quotient translations of [F7], so they define smooth tensor fields on the torus. The metric is positive definite because $L_{*} > 0$ . The metric components are constant, so [F4] makes the Christoffel symbols of $\tilde{h}$ vanish in the periodic coordinates. Thus ${\tilde{D}}_{\partial_{i}} \partial_{j} = 0$ and $[\partial_{i}, \partial_{j}] = 0$ ; the curvature definition in [F1] gives $\tilde{R} (\partial_{i}, \partial_{j}) \partial_{k} = 0$ for all coordinate indices. By tensoriality [F6], the curvature and its scalar trace vanish, so $\tilde{R} = 0$ . The dual and tensor-factor rules in [F5] give $({\tilde{D}}_{i} \tilde{σ})_{k ℓ} = \partial_{i} {\tilde{σ}}_{k ℓ} - {\tilde{Γ}}^{m}_{i k} {\tilde{σ}}_{m ℓ} - {\tilde{Γ}}^{m}_{i ℓ} {\tilde{σ}}_{k m} = 0$ , since the components are constant and the Christoffel symbols vanish. Hence ${\tilde{D}}^{i} {\tilde{σ}}_{i j} = 0$ , and its trace is $L_{*}^{- 2} (a_{*} - a_{*} + 0) = 0$ . Its squared norm is $L_{*}^{- 4} (a_{*}^{2} + a_{*}^{2}) = 2 a_{*}^{2} / L_{*}^{4} = A > 0$ ; therefore the input tensor is TT with the stated constant norm.

2.1A1F1F2F3step 1.1algebra∎

By [F3], the positive constant $ϕ_{0} = (3 A / (2 τ^{2}))^{1 / 12}$ solves the scalar constraint equation. The reduction in [F2] then gives both vacuum constraints for the constructed $(h, K)$ . From [F1] and step 1.1, ${tr}_{h} K = τ$ , so its tracefree part is $K - \frac{1}{3} ({tr}_{h} K) h = ϕ_{0}^{- 2} \tilde{σ} \neq 0$ , because $a_{*} > 0$ and $ϕ_{0} > 0$ . Varying $L_{*}, a_{*}, τ$ over the stated ranges gives the claimed parameterized family.

Flat-torus vacuum data from a constant TT tensor

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources