Manuscript III@2 · III.tex · proposition · 82dad3e5

Proposition: The reduced correction need not be free

The coupled first-order jet and retarded harmonic section › The complete quotient functional · lines 1371–1395 · proof included or linked

III@2#prop:nonfree

There are slice data with $P ψ_{1} \neq = 0$ .

Proof. Let

k_{11} = cos (t - x^{3}), k_{22} = - cos (t - x^{3}),

and all other components vanish. This tensor is trace free, divergence free, and a componentwise wave, so it is harmonic vacuum data.

Use the jet lemma to choose $ϕ_{0}$ whose value and gradient vanish at the origin, with Hessian

\partial_{1}^{2} ϕ_{0} = 1, \partial_{2}^{2} ϕ_{0} = - 1

and all other components zero there. Then $L_{k} ϕ_{0} (0) = - 2$ . Every corresponding correction satisfies $P ψ_{1} (0) = 2$ .

Displayed formulas in this passage

unlabeled · line 1378
$k_{11} = cos (t - x^{3}), k_{22} = - cos (t - x^{3}),$
unlabeled · line 1388
$\partial_{1}^{2} ϕ_{0} = 1, \partial_{2}^{2} ϕ_{0} = - 1$