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A flat-torus zero mode defeats uniqueness and coercivity

Physical scope

Exact mathematical boundary example for the vacuum conformal constraint equation on a flat torus; no observation, global development, or general uniqueness claim is made.

Example

Let $Σ = T_{2 π}^{3}$ with flat conformal metric $\tilde{h} = L_{*}^{2} \sum_{i} (d x^{i})^{2}$ , set $\tilde{σ} = 0$ and $τ = 0$ , and fix $Λ = 0$ . Every positive constant $ϕ = d$ solves the Lichnerowicz equation, so the same free data admit infinitely many positive constant conformal factors. At each such $d$ , the linearized scalar operator is $A_{0} v = - 8 \tilde{Δ} v$ . Define coercivity here to mean that $\int v A_{0} v d μ \geq c_{0} ∥ v ∥_{H^{1}}^{2}$ for some $c_{0} > 0$ and every smooth real $v$ , using the normalized torus measure and the Sobolev norm of the Fourier supplier. The operator is not coercive.

Example references

Vacuum conformal constraint data

Conformal reduction of the vacuum constraint equations

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

Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms

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

Facts & Assumptions

Given: The flat three-torus with $L_{*} > 0$ , zero TT tensor and zero mean curvature parameter, together with the normalized measure and integer Sobolev spaces of the torus supplier. Assume ${A C}_{ω}$ for the Fourier Sobolev suppliers and inherited embedded Lorentzian interpretation.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ), Vacuum conformal constraint data, The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s, Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms: ${A C}_{ω}$ is the axiom that every countable family of nonempty sets has a choice function; the Fourier Sobolev suppliers state it explicitly, and the local GR definition carries it only for embedded interpretation.

[F1]

Vacuum conformal constraint data: the Lichnerowicz equation uses $\tilde{R}$ , $\tilde{σ}$ , $τ$ and $\tilde{Δ} = {div}_{\tilde{h}} {grad}_{\tilde{h}}$ .

[F2]

Conformal reduction of the vacuum constraint equations: the vacuum constraint equations are equivalent to the scalar equation once the automatic momentum constraint is used.

[F3]

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s: the torus has characters $χ_{k} = e^{i k \cdot x}$ , normalized measure $μ$ , and derivative Sobolev spaces $H^{s}$ .

[F4]

Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms: Fourier coefficients differentiate by multiplication by $i k$ , and the Fourier-weight norm is equivalent to the derivative Sobolev norm.

Verification

1.1A1F1F2algebra

Here $\tilde{R} = 0$ , $∣ \tilde{σ} ∣^{2} = 0$ and $τ = 0$ , so the Lichnerowicz equation of [F2] is $- 8 \tilde{Δ} ϕ = 0$ . For any positive constant $d$ , $\tilde{Δ} d = 0$ , and therefore $ϕ = d$ solves it. Distinct positive constants give distinct solutions of the scalar equation for these same free data; the corresponding constraint pairs have $h = d^{4} \tilde{h}$ and $K = 0$ by [F1] and satisfy both constraints by [F2].

2.1A1F3F4step 1.1algebra∎

Differentiating the scalar expression at $ϕ = d$ with respect to a smooth direction $v$ gives $A_{0} v = - 8 \tilde{Δ} v$ . Take the nonzero constant function $v = 1$ . Its coordinate derivatives vanish, so $\tilde{Δ} v = 0$ and $\int v A_{0} v d μ = 0$ . By [F3], the normalized measure has total mass one and the derivative Sobolev norm includes the $L^{2}$ norm, so $∥ 1 ∥_{H^{1}}^{2} = \int 1 d μ = 1 > 0$ . No positive $c_{0}$ can satisfy the stated coercivity bound. Equivalently, the Fourier multiplier at $k = 0$ is zero; [F4] identifies this constant as the zero mode.

A flat-torus zero mode defeats uniqueness and coercivity

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources