Manuscript VII-B@1 · VII-B.tex · proposition · be632835

Proposition: Exact adjoint and boundary coefficient

Exact sharp-Cauchy adjoint · lines 680–729 · proof included or linked

VII-B@1#prop:adjoint, VII-B@1#eq:q-adjoint, VII-B@1#eq:boundary-map, VII-B@1#eq:q-support

Let $n_{μ} = \partial_{μ} t$ . The compactly supported distributional adjoint $q_{a} = L^{*} a$ is

q_{a}^{α β} B^{α β} (v) = C^{*} (θ b)^{α β} = θ (C^{*} b)^{α β} + δ (t) B^{α β} (b ∣_{Σ_{0}}), = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .

Furthermore,

supp q_{a} \subset J^{-} (supp a) \cap M_{+} .

There is no $δ^{'} (t)$ term in this adjoint.

Proof. For smooth $h$ , the retarded representation and formal adjointness give

ℓ_{a} [h] = ⟨ a, E_{R} (θ C h)⟩ = ⟨ E_{A} a, θ C h ⟩ = ⟨ C^{*} (θ b), h ⟩ .

The pairings involve only the compact causal set $J^{-} (supp a) \cap M_{+}$ . Hence the calculation is valid without a global compact-support assumption on $h$ .

Applying the product rule in distributions and using

\partial_{μ} θ = n_{μ} δ (t)

gives (eq:q-adjoint). Since $C^{*}$ has differential order one, it differentiates the step function at most once. This proves the absence of a delta derivative. Differentiation does not enlarge support, giving (eq:q-support).

Displayed formulas in this passage

VII-B@1#eq:q-adjoint · line 684
$q_{a}^{α β} B^{α β} (v) = C^{*} (θ b)^{α β} = θ (C^{*} b)^{α β} + δ (t) B^{α β} (b ∣_{Σ_{0}}), = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .$
VII-B@1#eq:boundary-map · line 684
$q_{a}^{α β} B^{α β} (v) = C^{*} (θ b)^{α β} = θ (C^{*} b)^{α β} + δ (t) B^{α β} (b ∣_{Σ_{0}}), = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .$
VII-B@1#eq:q-support · line 698
$supp q_{a} \subset J^{-} (supp a) \cap M_{+} .$
unlabeled · line 708
$ℓ_{a} [h] = ⟨ a, E_{R} (θ C h)⟩ = ⟨ E_{A} a, θ C h ⟩ = ⟨ C^{*} (θ b), h ⟩ .$
unlabeled · line 722
$\partial_{μ} θ = n_{μ} δ (t)$

Proposition: Exact adjoint and boundary coefficient

Displayed formulas in this passage

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