Manuscript V@1 · V.tex · section · e7ddf37d

The declared plane and its gauge intersection

Restricted compact representatives on a two-dimensional TT plane › The declared plane and its gauge intersection · lines 1657–1695

Fix $t_{0} < T$ . A Fourier-gap Schwartz spatial-TT wave satisfies

□ k_{μν} = 0, k_{0 μ} = 0, \partial_{i} k_{ij} = 0, δ^{ij} k_{ij} = 0,

and has Schwartz Cauchy data whose spatial Fourier transforms vanish near $p = 0$ .

Choose independent real waves $e, r$ of this kind and set

V_{fin} = span_{R} {e, r} .

A common gap may be chosen for this finite family. For example, if $a_{ij}$ is nonzero real Fourier-gap Schwartz TT data, then

e_{ij} (t) = cos ((t - t_{0}) ∣ D ∣) a_{ij}, r_{ij} (t) = \frac{sin (( t - t _{0} ) ∣ D ∣ )}{∣ D ∣} a_{ij}

are independent. Such data are obtained by applying the Fourier TT projector

Π_{ij, k l} (p) = \frac{1}{2} (π_{ik} π_{j l} + π_{i l} π_{j k} - π_{ij} π_{k l}), π_{ij} = δ_{ij} - p_{i} p_{j} /∣ p ∣^{2},

to a nonzero compact smooth Fourier tensor supported away from zero, with the usual reality symmetry.

Let

G_{S} = C^{\infty} (R; S (R^{3}; R^{4})) .

Displayed formulas in this passage

unlabeled · line 1660
$□ k_{μν} = 0, k_{0 μ} = 0, \partial_{i} k_{ij} = 0, δ^{ij} k_{ij} = 0,$
unlabeled · line 1670
$V_{fin} = span_{R} {e, r} .$
unlabeled · line 1676
$e_{ij} (t) = cos ((t - t_{0}) ∣ D ∣) a_{ij}, r_{ij} (t) = \frac{sin (( t - t _{0} ) ∣ D ∣ )}{∣ D ∣} a_{ij}$
unlabeled · line 1682
$Π_{ij, k l} (p) = \frac{1}{2} (π_{ik} π_{j l} + π_{i l} π_{j k} - π_{ij} π_{k l}), π_{ij} = δ_{ij} - p_{i} p_{j} /∣ p ∣^{2},$
unlabeled · line 1692
$G_{S} = C^{\infty} (R; S (R^{3}; R^{4})) .$