Manuscript XII-F@1 · XII-F.tex · section · 54cb3169

Compact realization of the decisive witness

F3: cancellation on a subclass and a nonzero shell witness › Compact realization of the decisive witness · lines 870–914

XII-F@1#eq:main-parameter, XII-F@1#eq:witness-diamond, XII-F@1#eq:main-fields

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Fix times

ε < t_{a} < t_{b} < t_{c} < t_{d} < T .

Choose a nonzero nonnegative bump $a \in C_{c}^{\infty} ((t_{a}, t_{b}))$ . Choose a nonzero nonnegative observation

f \in C_{c}^{\infty} ((t_{c}, t_{d}) \times B_{r}; [0, \infty)),

where $B_{r}$ is a spatial ball centered at the origin.

Choose a spatial cutoff $ζ$ equal to one on $B_{R}$ , where

R > r + t_{d} - t_{a},

and supported in a larger ball. Set

ξ = a (t) ζ (x) \partial_{t} .

Choose an open relatively compact $U$ containing $supp ξ$ with $U \subset {ε < t < T}$ . Choose $g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1])$ equal to one on a neighbourhood of both $U$ and

J^{-} (K_{f}) \cap J^{+} (S_{ξ}) \cap M_{T} .

These sets are compact and separated from the temporal endpoints, so such a switch exists. This freezes an admissible tuple $(f, g, U, ξ)$ before evaluation. It satisfies (UCP), hence (CP).

Put

m = 0, ψ = t^{2} + (x^{1})^{2}, γ_{μν} = ψ η_{μν}, ϕ = ψ, b^{ν} = D^{ν} γ, c = c ˉ = 0.

Displayed formulas in this passage

XII-F@1#eq:main-parameter · line 889
$ξ = a (t) ζ (x) \partial_{t} .$
XII-F@1#eq:main-fields · line 906
$m = 0, ψ = t^{2} + (x^{1})^{2}, γ_{μν} = ψ η_{μν}, ϕ = ψ, b^{ν} = D^{ν} γ, c = c ˉ = 0.$
unlabeled · line 873
$ε < t_{a} < t_{b} < t_{c} < t_{d} < T .$
unlabeled · line 879
$f \in C_{c}^{\infty} ((t_{c}, t_{d}) \times B_{r}; [0, \infty)),$
unlabeled · line 885
$R > r + t_{d} - t_{a},$
XII-F@1#eq:witness-diamond · line 897
$J^{-} (K_{f}) \cap J^{+} (S_{ξ}) \cap M_{T} .$

Compact realization of the decisive witness

Displayed formulas in this passage

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