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

Two harmonic prescriptions

Classification on the spacelike-compact solution spaces › Two harmonic prescriptions · lines 937–977

V@1#eq:residualsurvives

For $h \in D_{ret}$ , set

w_{R} (h) = G_{1, R} D (h), s_{ret} h = h - K w_{R} (h) .

For every spacelike-compact smooth $h$ , let $w_{0} (h)$ solve

□ w_{0} (h) = D (h), w_{0} (h) ∣_{0} = w ˙_{0} (h) ∣_{0} = 0,

and set

s_{0} h = h - K w_{0} (h) .

Finite propagation gives smooth spacelike-compact compensators. Both prescriptions satisfy $D (s h) = 0$ , and both preserve the vacuum equation.

Define

B_{R} (h) = B_{raw} (s_{ret} h), h \in S_{ret}, B_{0} (h) = B_{raw} (s_{0} h), h \in S .

For compact $ξ$ , $s_{ret} (K ξ) = 0$ . For a residual wave $□ ξ = 0$ , however,

s_{ret} (K ξ) = s_{0} (K ξ) = K ξ .

More generally,

s_{0} (K ξ) = K H_{w} (Data_{0} ξ),

where $H_{w}$ is componentwise massless Cauchy evolution. Thus these are harmonic prescriptions, not sections of the complete gauge quotient.

Displayed formulas in this passage

V@1#eq:residualsurvives · line 965
$s_{ret} (K ξ) = s_{0} (K ξ) = K ξ .$
unlabeled · line 940
$w_{R} (h) = G_{1, R} D (h), s_{ret} h = h - K w_{R} (h) .$
unlabeled · line 945
$□ w_{0} (h) = D (h), w_{0} (h) ∣_{0} = w ˙_{0} (h) ∣_{0} = 0,$
unlabeled · line 950
$s_{0} h = h - K w_{0} (h) .$
unlabeled · line 958
$B_{R} (h) = B_{raw} (s_{ret} h), h \in S_{ret}, B_{0} (h) = B_{raw} (s_{0} h), h \in S .$
unlabeled · line 970
$s_{0} (K ξ) = K H_{w} (Data_{0} ξ),$