Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

A joint wavefront upper bound

The full insertion kernel and the characteristic-cone test › A joint wavefront upper bound · lines 434–471

VII-C@1#eq:Lret, VII-C@1#eq:GammaB

We order the mixed kernel variables as $(x, z)$ , with $x$ the scalar slot and $z$ the graviton slot. There is a compact tensor-valued distribution $b_{f} (x, z)$ such that

B_{f} [h, ϕ] = ⟨ b_{f}, ϕ \otimes h ⟩ .

The compact-kernel assertion follows either from the explicit construction below or from continuity of the bilinear functional (eq:B) and Proposition prop:reconstruction.

Let $E_{ret}$ denote the retarded fundamental solution of $□$ . For $x^{0} > 0$ ,

(L h) (x) = \int E_{ret} (x - z) θ (z^{0}) (C h) (z) d^{4} z .

The identity is local in $x$ , so harmless compact cutoffs can be inserted in the integration variables.

For covectors in the fixed Cartesian coordinates, define

Λ_{diag} Λ_{\partial} Λ_{B} = {(ξ, ζ) \neq = 0 : ζ = - ξ}, = {(ξ, ζ) \neq = 0 : η^{- 1} (ξ, ξ) = 0, ξ + ζ \in R d t}, = Λ_{diag} \cup Λ_{\partial}, Γ_{B} = (M \times M) \times Λ_{B} .

The case $ξ = 0$ is included in $Λ_{\partial}$ .

Displayed formulas in this passage

VII-C@1#eq:Lret · line 448
$(L h) (x) = \int E_{ret} (x - z) θ (z^{0}) (C h) (z) d^{4} z .$
VII-C@1#eq:GammaB · line 458
$Λ_{diag} Λ_{\partial} Λ_{B} = {(ξ, ζ) \neq = 0 : ζ = - ξ}, = {(ξ, ζ) \neq = 0 : η^{- 1} (ξ, ξ) = 0, ξ + ζ \in R d t}, = Λ_{diag} \cup Λ_{\partial}, Γ_{B} = (M \times M) \times Λ_{B} .$
unlabeled · line 439
$B_{f} [h, ϕ] = ⟨ b_{f}, ϕ \otimes h ⟩ .$

A joint wavefront upper bound

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