Manuscript Em01-A@1 · Em01-A.tex · lemma · 880b67af

Lemma: Wave-shell equality

A common classical setting › The analytic Green package and its signs · lines 449–485 · proof included or linked

Em01-A@1#lem:wave-equality

If $c, c_{sm}, r$ are compact tensor distributions and

c - c_{sm} = □ r,

then

⟨ c, h ⟩ = ⟨ c_{sm}, h ⟩ (h \in S_{□}) .

If $c_{sm}$ is smooth, then

E_{-} T c = E_{-} T c_{sm}

is smooth and spacelike compact.

Proof. Formal self-adjointness and compact support give

⟨ c - c_{sm}, h ⟩ = ⟨ r, □ h ⟩ = 0 (h \in S_{□}) .

Commutation with $T$ and the Green identities give

E_{-} T c = E_{-} T c_{sm} + E_{-} □ T r = E_{-} T c_{sm} .

The last expression is smooth by Assumption ass:green. Its support is contained in the causal hull of a compact set, hence is spacelike compact.

Displayed formulas in this passage

unlabeled · line 453
$c - c_{sm} = □ r,$
unlabeled · line 457
$⟨ c, h ⟩ = ⟨ c_{sm}, h ⟩ (h \in S_{□}) .$
unlabeled · line 462
$E_{-} T c = E_{-} T c_{sm}$
unlabeled · line 470
$⟨ c - c_{sm}, h ⟩ = ⟨ r, □ h ⟩ = 0 (h \in S_{□}) .$
unlabeled · line 476
$E_{-} T c = E_{-} T c_{sm} + E_{-} □ T r = E_{-} T c_{sm} .$