Manuscript IV@1 · IV.tex · section · 351cb2d7

A nondegenerate plane and its admitted section

Model 1: the fixed vacuum TT plane › A nondegenerate plane and its admitted section · lines 726–763

IV@1#eq:TT

For $p \neq = 0$ , define

P_{ij} (p) = δ_{ij} - \frac{p _{i} p _{j}}{∣ p ∣ ^{2}},

Π_{ij, l m} (p) = \frac{1}{2} (P_{i l} P_{j m} + P_{im} P_{j l} - P_{ij} P_{l m}) .

It is the orthogonal projector onto symmetric transverse traceless spatial tensors.

Let $p_{*} = (0, 0, 1)$ , $H = diag (1, - 1, 0)$ , and choose a nonnegative even smooth bump $χ$ supported sufficiently near $p_{*}$ and $- p_{*}$ , away from zero. Put

A_{ij} (p) = χ (p) (Π (p) H)_{ij} .

Choose $χ$ nonzero where $Π (p) H \neq = 0$ . Then $A$ is real Schwartz and TT, and

H^{ij} A_{ij} (0) = (2 π)^{- 3/2} \int χ (p) ∣Π (p) H ∣^{2} d p > 0.

In particular $A (0) \neq = 0$ .

Let $D_{0} = - Δ$ , $M_{A} = \int A_{ij} A_{ij} d x$ , and set

e_{ij} (t) = cos ((t - t_{0}) D_{0}) A_{ij}, r_{ij} (t) = \frac{1}{M _{A}} \frac{sin (( t - t _{0} ) D _{0} )}{D _{0}} A_{ij}, e_{0 μ} = r_{0 μ} = 0.

These waves are Schwartz on finite time slabs.

Displayed formulas in this passage

IV@1#eq:TT · line 732
$Π_{ij, l m} (p) = \frac{1}{2} (P_{i l} P_{j m} + P_{im} P_{j l} - P_{ij} P_{l m}) .$
unlabeled · line 729
$P_{ij} (p) = δ_{ij} - \frac{p _{i} p _{j}}{∣ p ∣ ^{2}},$
unlabeled · line 744
$A_{ij} (p) = χ (p) (Π (p) H)_{ij} .$
unlabeled · line 749
$H^{ij} A_{ij} (0) = (2 π)^{- 3/2} \int χ (p) ∣Π (p) H ∣^{2} d p > 0.$
unlabeled · line 756
$e_{ij} (t) = cos ((t - t_{0}) D_{0}) A_{ij}, r_{ij} (t) = \frac{1}{M _{A}} \frac{sin (( t - t _{0} ) D _{0} )}{D _{0}} A_{ij}, e_{0 μ} = r_{0 μ} = 0.$

A nondegenerate plane and its admitted section

Displayed formulas in this passage

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