Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

Supports and fixed onset

Frozen analytical data and the scalar insertion target › Supports and fixed onset · lines 306–359

X-F-ST@1#eq:sets, X-F-ST@1#eq:separation

Fix

m \geq 0, f \in C_{c}^{\infty} (O; R), g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1]) .

Choose a nonempty relatively compact open plateau $U$ with

U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M) .

The formal group $G_{p}$ is generated through second order by $κ ξ + κ^{2} ζ$ , where $ξ, ζ \in g_{p}$ .

Reserve $K$ for the Killing operator

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} = (L_{ξ} η)_{μν} .

For observation support write

K_{f} = supp f, S = supp g \cap J_{η}^{-} (K_{f}), C = K_{f} \cup S, D_{f} = J_{η}^{-} (K_{f}) \cap M_{T_{*}} .

These sets are compact.

Fix $δ > 0$ so that

K_{f} \cup supp g \cup U \subset {2 δ < t < T_{*}} .

Fix $0 < ε < δ$ , and set

σ (t) = χ (t / ε),

where $χ$ is smooth, zero for $t \leq 0$ , one for $t \geq 1$ , and flat at both endpoints. The width remains positive throughout.

The classical coefficients $h, k$ are arbitrary smooth real symmetric tensors near the closed slab. No equations or spatial decay are imposed. They are not the host metric $G$ .

Displayed formulas in this passage

X-F-ST@1#eq:sets · line 332
$K_{f} = supp f, S = supp g \cap J_{η}^{-} (K_{f}), C = K_{f} \cup S, D_{f} = J_{η}^{-} (K_{f}) \cap M_{T_{*}} .$
X-F-ST@1#eq:separation · line 342
$K_{f} \cup supp g \cup U \subset {2 δ < t < T_{*}} .$
unlabeled · line 309
$m \geq 0, f \in C_{c}^{\infty} (O; R), g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1]) .$
unlabeled · line 316
$U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 326
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} = (L_{ξ} η)_{μν} .$
unlabeled · line 348
$σ (t) = χ (t / ε),$