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

Lemma: Signature-correct boundary witness

The compatibility boundary: regular approximation is not admission › A nonzero boundary trace for every nonzero smearing · lines 1175–1217 · proof included or linked

IX-A@1#lem:witness, IX-A@1#eq:witness, IX-A@1#eq:witness-positive

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Let $f \in C_{c}^{\infty} (M_{+}; R)$ be nonzero. Set

ϕ_{f} = \partial_{t} (t f), a_{μ} = f \partial_{μ} ϕ_{f}, b = G_{A}^{0} a .

Then

\int_{R^{3}} b_{0} (0, x) d^{3} x = \int_{M} [\partial_{t} (t f)]^{2} d^{4} x > 0.

In particular, $b ∣_{Σ} \neq = 0$ .

Proof. Let

F_{0} (t) = \int_{R^{3}} a_{0} (t, x) d^{3} x, B_{0} (t) = \int_{R^{3}} b_{0} (t, x) d^{3} x .

On bounded time intervals the supports are uniformly spatially bounded. Since $□_{η} b_{0} = a_{0}$ , spatial integration gives

- B_{0}^{''} (t) = F_{0} (t) .

The advanced solution has zero data at sufficiently late times. Therefore

B_{0} (0) = - \int_{0}^{\infty} t F_{0} (t) d t = - \int_{M} t f \partial_{t} ϕ_{f} d^{4} x .

Put $u = \partial_{t} (t f)$ , so $ϕ_{f} = u$ . Compact integration by parts gives

- \int t f \partial_{t} u = \int u^{2} .

If $u = 0$ , then $t f$ is independent of $t$ . Compact time support forces $t f = 0$ , hence $f = 0$ , a contradiction.

Displayed formulas in this passage

IX-A@1#eq:witness · line 1178
$ϕ_{f} = \partial_{t} (t f), a_{μ} = f \partial_{μ} ϕ_{f}, b = G_{A}^{0} a .$
IX-A@1#eq:witness-positive · line 1184
$\int_{R^{3}} b_{0} (0, x) d^{3} x = \int_{M} [\partial_{t} (t f)]^{2} d^{4} x > 0.$
unlabeled · line 1195
$F_{0} (t) = \int_{R^{3}} a_{0} (t, x) d^{3} x, B_{0} (t) = \int_{R^{3}} b_{0} (t, x) d^{3} x .$
unlabeled · line 1201
$- B_{0}^{''} (t) = F_{0} (t) .$
unlabeled · line 1206
$B_{0} (0) = - \int_{0}^{\infty} t F_{0} (t) d t = - \int_{M} t f \partial_{t} ϕ_{f} d^{4} x .$
unlabeled · line 1212
$- \int t f \partial_{t} u = \int u^{2} .$

Lemma: Signature-correct boundary witness

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