Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a

Geometry and smoothness

Setting, source input, and signs › Geometry and smoothness · lines 295–334

XVI-D@1#eq:switch

Let

M = R^{4}, M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T,

with

η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ.

Smoothness on the closed slab means smoothness up to its time boundaries; all Cauchy traces used below are therefore defined. No spatial boundedness, decay, or finite total-energy condition is imposed on smooth solutions.

Spacetime indices are raised and lowered with $η$ . Spatial Euclidean contractions use $δ_{ij}$ . The frequency parameter is $λ$ . Repeated spacetime derivative indices are denoted by $α, μ, ν$ , not by $λ$ .

Fix a real smooth switch

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

Thus $σ$ is flat at $0$ , its positive-order derivatives vanish at $ε$ , and $supp σ^{'} \subset [0, ε]$ .

Displayed formulas in this passage

XVI-D@1#eq:switch · line 323
$σ (t) = σ_{ε} (t) = χ (t / ε), χ \in C^{\infty} (R; R), χ (s) = 0 (s \leq 0), χ (s) = 1 (s \geq 1) .$
unlabeled · line 298
$M = R^{4}, M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T,$
unlabeled · line 306
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ.$