Manuscript VII-H@1 · VII-H.tex · section · d20d2a9f

An exactly evaluated nonzero invariant

An exactly evaluated nonzero invariant · lines 1009–1045

VII-H@1#eq:bump, VII-H@1#eq:bump-moments, VII-H@1#eq:Fstar

Choose $t_{0} > a > 0$ , and let

ρ (u) = {e^{- 1/ (1 - u^{2})}, 0, ∣ u ∣ < 1, ∣ u ∣ \geq 1.

Set

b (t) = \frac{ρ (( t - t _{0} ) / a )}{a \int _{- 1} ^{1} ρ ( u ) ^{2} d u} .

Then

\int b (t)^{2} d t = 1, \int t b (t)^{2} d t = t_{0} .

The second equality uses the evenness of $ρ$ .

Define the real compact test

F_{*} (t, x) = (t^{2} - ∣ x ∣^{2}) b (t) b (∣ x ∣) .

It is smooth because its spatial support avoids $x = 0$ . Its support avoids $z = 0$ , although it crosses the nonzero future null cone.

Displayed formulas in this passage

VII-H@1#eq:bump · line 1021
$b (t) = \frac{ρ (( t - t _{0} ) / a )}{a \int _{- 1} ^{1} ρ ( u ) ^{2} d u} .$
VII-H@1#eq:bump-moments · line 1028
$\int b (t)^{2} d t = 1, \int t b (t)^{2} d t = t_{0} .$
VII-H@1#eq:Fstar · line 1036
$F_{*} (t, x) = (t^{2} - ∣ x ∣^{2}) b (t) b (∣ x ∣) .$
unlabeled · line 1013
$ρ (u) = {e^{- 1/ (1 - u^{2})}, 0, ∣ u ∣ < 1, ∣ u ∣ \geq 1.$