Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

Exterior curvature certificate and finite cutoffs

Model 3: a specified sourced form and an exterior pairing › Exterior curvature certificate and finite cutoffs · lines 1318–1388

IV-B@1#eq:curvature-current

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Use

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 $ψ$ , set

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,

with all other entries zero. Direct differentiation and integration by parts give

\partial_{μ} J_{ψ}^{μν} = 0, \int J_{ψ}^{μν} h_{μν} d^{4} x = \int ψ R^{(1)}_{1212} (h) d^{4} x .

Choose nonzero real radial $F \in C_{c}^{\infty} (R^{3})$ , supported in $∣ x ∣ \leq r_{0}$ , and scalar data $(F, 0)$ . Their energy is

E = \frac{1}{2} \int (∣\nabla F ∣^{2} + m^{2} F^{2}) d x > 0.

The initial momentum density vanishes. Newton's radial formula and trace reversal give

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

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

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

Smoothness and finite propagation permit a small future neighborhood where the curvature remains positive and the uncut scalar vanishes. Choose a nonzero nonnegative $ψ$ there. Then

d_{3} = \int ψ R^{(1)}_{1212} (R^{NW} T (φ_{*})) d^{4} x > 0.

Let $z_{*}$ be the Schwartz coherent label for $(F, 0)$ . Since $χ_{Λ} z_{*} \to z_{*}$ in Schwartz topology, (eq:coherent3) and response continuity allow a sufficiently large finite $Λ$ with

⟨ B_{Λ} (J_{ψ}) ⟩_{Coh (z_{*})} \geq 3 d_{3} /4.

The weighted Poisson tail and (eq:B3) imply

⟨ B_{Λ, M} (J_{ψ}) ⟩_{Coh (z_{*})} ⟶ ⟨ B_{Λ} (J_{ψ}) ⟩_{Coh (z_{*})} .

For example, writing

T_{M} (ν)^{2} = e^{- ν} n > M \sum (n + 2)^{2} \frac{ν ^{n}}{n !}, ν = ∥ z_{*} ∥^{2},

the expectation error is at most $2 C_{Λ} ν^{2} + 5 ν + 4 T_{M} (ν)$ . Fix a finite $M$ for which the compressed mean is at least $d_{3} /2$ . Its vacuum mean is exactly zero.

Displayed formulas in this passage

IV-B@1#eq:curvature-current · line 1336
$\partial_{μ} J_{ψ}^{μν} = 0, \int J_{ψ}^{μν} h_{μν} d^{4} x = \int ψ R^{(1)}_{1212} (h) d^{4} x .$
unlabeled · line 1321
$R^{(1)}_{1212} (h) = \partial_{1} \partial_{2} h_{12} - \frac{1}{2} \partial_{1}^{2} h_{22} - \frac{1}{2} \partial_{2}^{2} h_{11} .$
unlabeled · line 1328
$J_{ψ}^{11} = - \frac{1}{2} \partial_{2}^{2} ψ, J_{ψ}^{22} = - \frac{1}{2} \partial_{1}^{2} ψ, J_{ψ}^{12} = J_{ψ}^{21} = \frac{1}{2} \partial_{1} \partial_{2} ψ,$
unlabeled · line 1345
$E = \frac{1}{2} \int (∣\nabla F ∣^{2} + m^{2} F^{2}) d x > 0.$
unlabeled · line 1350
$j_{ij} (0, x) = \frac{E}{4 π ∣ x ∣} δ_{ij} (∣ x ∣ > r_{0}) .$
unlabeled · line 1355
$R^{(1)}_{1212} (j) (0, x_{*}) = \frac{E}{4 π r _{*}^{3}} > 0.$
unlabeled · line 1361
$d_{3} = \int ψ R^{(1)}_{1212} (R^{NW} T (φ_{*})) d^{4} x > 0.$
unlabeled · line 1370
$⟨ B_{Λ} (J_{ψ}) ⟩_{Coh (z_{*})} \geq 3 d_{3} /4.$
unlabeled · line 1374
$⟨ B_{Λ, M} (J_{ψ}) ⟩_{Coh (z_{*})} ⟶ ⟨ B_{Λ} (J_{ψ}) ⟩_{Coh (z_{*})} .$
unlabeled · line 1380
$T_{M} (ν)^{2} = e^{- ν} n > M \sum (n + 2)^{2} \frac{ν ^{n}}{n !}, ν = ∥ z_{*} ∥^{2},$

Exterior curvature certificate and finite cutoffs

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