Manuscript X-G@1 · X-G.tex · proposition · 01566e14

Proposition: Evaluated free-solution defect

Source switching and an exact-upgrade obstruction › A pure-gauge background in the ramp · lines 1194–1244 · proof included or linked

X-G@1#prop:Dsigned, X-G@1#eq:Dnonzero

For the background (eq:rampgauge),

D_{w} (u, v) = \frac{5 M _{χ}}{8} \int_{0}^{ε} σ_{ε}^{''} ψ d t = \frac{5 M _{χ}}{8} \int_{0}^{ε} ζ (σ_{ε}^{''})^{2} d t > 0.

Proof. By (eq:commutator), on a free input, $Q_{w} u = - P_{gr} L_{ξ} u$ .

Directly,

(L_{ξ} u)_{00} = 2 ψ^{'} χ_{0}, (L_{ξ} u)_{0 i} = ψ \partial_{i} χ_{0}, (L_{ξ} u)_{ij} = 0,

and

(L_{ξ} v)_{00} = (ψ + 2 t ψ^{'}) χ_{0}, (L_{ξ} v)_{0 i} = t ψ \partial_{i} χ_{0}, (L_{ξ} v)_{ij} = 0.

For a tensor with no spatial–spatial components,

(P_{gr} a)^{00} = - \frac{1}{8} □ a_{00} .

It follows that

(Q_{w} u)^{00} = \frac{1}{8} □ (2 ψ^{'} χ_{0}), (Q_{w} v)^{00} = \frac{1}{8} □ ((ψ + 2 t ψ^{'}) χ_{0}) .

After spatial integration, the Laplacian terms vanish. Since

(ψ + 2 t ψ^{'})^{''} - t (2 ψ^{'})^{''} = 5 ψ^{''},

we obtain

D_{w} (u, v) = \frac{5 M _{χ}}{8} \int σ_{ε} ψ^{''} d t .

Twice integrating by parts gives (eq:Dnonzero).

Displayed formulas in this passage

X-G@1#eq:Dnonzero · line 1197
$D_{w} (u, v) = \frac{5 M _{χ}}{8} \int_{0}^{ε} σ_{ε}^{''} ψ d t = \frac{5 M _{χ}}{8} \int_{0}^{ε} ζ (σ_{ε}^{''})^{2} d t > 0.$
unlabeled · line 1212
$(L_{ξ} u)_{00} = 2 ψ^{'} χ_{0}, (L_{ξ} u)_{0 i} = ψ \partial_{i} χ_{0}, (L_{ξ} u)_{ij} = 0,$
unlabeled · line 1218
$(L_{ξ} v)_{00} = (ψ + 2 t ψ^{'}) χ_{0}, (L_{ξ} v)_{0 i} = t ψ \partial_{i} χ_{0}, (L_{ξ} v)_{ij} = 0.$
unlabeled · line 1224
$(P_{gr} a)^{00} = - \frac{1}{8} □ a_{00} .$
unlabeled · line 1228
$(Q_{w} u)^{00} = \frac{1}{8} □ (2 ψ^{'} χ_{0}), (Q_{w} v)^{00} = \frac{1}{8} □ ((ψ + 2 t ψ^{'}) χ_{0}) .$
unlabeled · line 1235
$(ψ + 2 t ψ^{'})^{''} - t (2 ψ^{'})^{''} = 5 ψ^{''},$
unlabeled · line 1239
$D_{w} (u, v) = \frac{5 M _{χ}}{8} \int σ_{ε} ψ^{''} d t .$