Manuscript IX-A@1 · IX-A.tex · proposition · 55aa6f6d

Proposition: Sharp adjoint

The sharp adjoint and an intrinsic boundary trace › Linear coordinate functionals · lines 623–670 · proof included or linked

IX-A@1#prop:adjoint, IX-A@1#eq:qa, IX-A@1#eq:boundarymap

Back to manuscript

The functional $J (a)$ has the compact distributional kernel

q_{a} = L^{*} a = C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}),

where $n_{μ} = \partial_{μ} t$ and

B^{α β} (v) = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .

There is no $δ^{'} (t)$ term. Moreover,

supp q_{a} \subset J^{-} (supp a) \cap {t \geq 0}, WF (q_{a}) \subset N^{*} Σ.

Proof. On the future half-space,

L h = G_{R}^{0} (θ C h) .

Transposition of the retarded and advanced inverses gives

J (a) [h] = ⟨ G_{A}^{0} a, θ C h ⟩ = ⟨ C^{*} (θ b), h ⟩ .

All relevant support intersections lie in a compact truncated causal region.

The distributional product rule yields (eq:qa). The operator $C^{*}$ has order one and hence differentiates the step function only once. The coefficients $b$ and $C^{*} b$ are smooth. The only possible singular covectors are therefore conormal to $Σ$ .

Displayed formulas in this passage

IX-A@1#eq:qa · line 626
$q_{a} = L^{*} a = C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}),$
IX-A@1#eq:boundarymap · line 633
$B^{α β} (v) = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .$
unlabeled · line 642
$supp q_{a} \subset J^{-} (supp a) \cap {t \geq 0}, WF (q_{a}) \subset N^{*} Σ.$
unlabeled · line 651
$L h = G_{R}^{0} (θ C h) .$
unlabeled · line 655
$J (a) [h] = ⟨ G_{A}^{0} a, θ C h ⟩ = ⟨ C^{*} (θ b), h ⟩ .$

Proposition: Sharp adjoint

Displayed formulas in this passage

Nearby in this section