Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

Prescribed background and equations

The supported triangular Cauchy quotient › Prescribed background and equations · lines 1059–1089

III-B@1#eq:plateau, III-B@1#eq:C1

Choose $0 < ρ < r_{*}$ , a real cutoff $χ_{0} \in C_{c}^{\infty} (B_{r_{*}})$ equal to one on $B_{ρ}$ , and $A_{*} > 0$ . Let

P Φ = 0, Φ (0, x) = A_{*} χ_{0} (x), \partial_{s} Φ (0, x) = 0.

Finite propagation gives

Φ (s, x) = A_{*} cos (m s) when ∣ x ∣ + ∣ s ∣ < ρ .

The $C_{1}$ model is

□ h = 0, D h = 0, P φ = - L_{h} Φ.

Its metric equation is vacuum linearized gravity. The scalar background stress is not inserted into that equation.

This is not the full Einstein–Klein–Gordon linearization about a nonlinear on-shell background. The latter has different equations and compatibility conditions Hack.

Displayed formulas in this passage

III-B@1#eq:plateau · line 1071
$Φ (s, x) = A_{*} cos (m s) when ∣ x ∣ + ∣ s ∣ < ρ .$
III-B@1#eq:C1 · line 1078
$□ h = 0, D h = 0, P φ = - L_{h} Φ.$
unlabeled · line 1065
$P Φ = 0, Φ (0, x) = A_{*} χ_{0} (x), \partial_{s} Φ (0, x) = 0.$