Manuscript XVI-B@1 · XVI-B.tex · section · 6a0abffd

Geometry and the switch

Setting, notation, and explicit inputs › Geometry and the switch · lines 294–336

XVI-B@1#eq:switch, XVI-B@1#eq:flat

Work on ambient Minkowski space

M = R^{4}, y = (t, x), η = diag (1, - 1, - 1, - 1),

and, where a slab is needed, set

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .

Greek indices range over $0, 1, 2, 3$ and spatial indices over $1, 2, 3$ . Spacetime indices are raised and lowered with $η$ . Spatial Euclidean contractions use $δ_{ij}$ .

Let

σ_{ε} (t) = χ (t / ε), χ \in C^{\infty} (R; R), χ (s) = {0, 1, s \leq 0, s \geq 1.

Write $σ = σ_{ε}$ when no confusion is possible. Smoothness and the constant plateaus imply

σ^{(j)} (0) = σ^{(j)} (ε) = 0 (j \geq 1), supp σ^{'} \subset [0, ε] .

No estimate uniform as $ε ↓ 0$ is claimed.

Displayed formulas in this passage

XVI-B@1#eq:switch · line 314
$σ_{ε} (t) = χ (t / ε), χ \in C^{\infty} (R; R), χ (s) = {0, 1, s \leq 0, s \geq 1.$
XVI-B@1#eq:flat · line 328
$σ^{(j)} (0) = σ^{(j)} (ε) = 0 (j \geq 1), supp σ^{'} \subset [0, ε] .$
unlabeled · line 297
$M = R^{4}, y = (t, x), η = diag (1, - 1, - 1, - 1),$
unlabeled · line 303
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .$