Manuscript IV@1 · IV.tex · section · 351cb2d7

A compact exterior curvature current

Model 3: scalar stress form and exterior record › A compact exterior curvature current · lines 3018–3094

IV@1#eq:Jpsi, IV@1#eq:curvpair, IV@1#eq:d3, IV@1#eq:dLambda

Use the convention

R^{(1)}_{1212} (h) = \partial_{1} \partial_{2} h_{12} - \frac{1}{2} \partial_{1}^{2} h_{22} - \frac{1}{2} \partial_{2}^{2} h_{11} .

For real compact $ψ$ , define the only nonzero entries of $J_{ψ}$ by

J_{ψ}^{11} = - \frac{1}{2} \partial_{2}^{2} ψ, J_{ψ}^{22} = - \frac{1}{2} \partial_{1}^{2} ψ, J_{ψ}^{12} = J_{ψ}^{21} = \frac{1}{2} \partial_{1} \partial_{2} ψ .

Direct differentiation gives $\partial_{μ} J_{ψ}^{μν} = 0$ , and integration by parts gives

\int J_{ψ}^{μν} h_{μν} d^{4} x = \int ψ R^{(1)}_{1212} (h) d^{4} x .

Thus this current annihilates all smooth pure gauges.

Choose nonzero real radial $F \in C_{c}^{\infty} (R^{3})$ , supported in $∣ x ∣ \leq r_{0}$ , and take

ϕ_{*} (0) = F, ϕ ˙_{*} (0) = 0.

Then

ρ = \frac{1}{2} (∣\nabla F ∣^{2} + m^{2} F^{2}), J_{i} = T_{0 i} (0) = 0, E = \int ρ > 0.

Both response prescriptions consequently have identical initial data for this particular source. Newton's radial formula gives, for $∣ x ∣ > r_{0}$ ,

j_{ij} (0, x) = \frac{E}{4 π ∣ x ∣} δ_{ij} .

At $x_{*} = (0, 0, r_{*})$ , $r_{*} > r_{0}$ ,

R^{(1)}_{1212} (j) (0, x_{*}) = \frac{E}{4 π r _{*}^{3}} > 0.

The scalar support at time $t$ is contained in $∣ x ∣ \leq r_{0} + ∣ t ∣$ . Smoothness therefore permits a small future neighborhood of $(0, x_{*})$ on which the curvature remains positive and the scalar vanishes. Choose a nonzero nonnegative $ψ$ supported there. Then

d = \int ψ R^{(1)}_{1212} (R^{NW} T (ϕ_{*})) d^{4} x > 0, supp ψ \cap supp ϕ_{*} = \emptyset .

The coherent label is $z_{*} (p) = ω_{p} /2 F (p)$ , which is Schwartz and produces the stated data. By (eq:coherent3), one may fix a sufficiently large finite $Λ$ such that

d_{Λ} = ⟨ Coh (z_{*}), B_{Λ} (J_{ψ}) Coh (z_{*}) ⟩ \geq d /2 > 0.

A momentum cutoff destroys exact compact spatial support. Thus support separation in (eq:d3) is a classical uncut statement; (eq:dLambda) transfers its compact pairing, not its pointwise support property.

Figure fig:exterior-curvature-witness shows the uncut exterior certificate and the more limited transfer to a finite momentum cutoff.

Displayed formulas in this passage

IV@1#eq:Jpsi · line 3029
$J_{ψ}^{11} = - \frac{1}{2} \partial_{2}^{2} ψ, J_{ψ}^{22} = - \frac{1}{2} \partial_{1}^{2} ψ, J_{ψ}^{12} = J_{ψ}^{21} = \frac{1}{2} \partial_{1} \partial_{2} ψ .$
IV@1#eq:d3 · line 3070
$d = \int ψ R^{(1)}_{1212} (R^{NW} T (ϕ_{*})) d^{4} x > 0, supp ψ \cap supp ϕ_{*} = \emptyset .$
IV@1#eq:dLambda · line 3081
$d_{Λ} = ⟨ Coh (z_{*}), B_{Λ} (J_{ψ}) Coh (z_{*}) ⟩ \geq d /2 > 0.$
unlabeled · line 3021
$R^{(1)}_{1212} (h) = \partial_{1} \partial_{2} h_{12} - \frac{1}{2} \partial_{1}^{2} h_{22} - \frac{1}{2} \partial_{2}^{2} h_{11} .$
IV@1#eq:curvpair · line 3037
$\int J_{ψ}^{μν} h_{μν} d^{4} x = \int ψ R^{(1)}_{1212} (h) d^{4} x .$
unlabeled · line 3046
$ϕ_{*} (0) = F, ϕ ˙_{*} (0) = 0.$
unlabeled · line 3050
$ρ = \frac{1}{2} (∣\nabla F ∣^{2} + m^{2} F^{2}), J_{i} = T_{0 i} (0) = 0, E = \int ρ > 0.$
unlabeled · line 3057
$j_{ij} (0, x) = \frac{E}{4 π ∣ x ∣} δ_{ij} .$
unlabeled · line 3061
$R^{(1)}_{1212} (j) (0, x_{*}) = \frac{E}{4 π r _{*}^{3}} > 0.$

Nearby in this section

figure: Exterior curvature witness