Manuscript XIX-B@1 · XIX-B.tex · section · b5670709

VI's coordinate equation

The common slab and the native prescriptions › VI's coordinate equation · lines 223–254

XIX-B@1#eq:VI-data, XIX-B@1#eq:VI-first

The registered VI coefficient convention is

γ_{κ} = η_{-} + κh + κ^{2} j, X_{γ}^{μ} = x^{μ} + κ w^{μ} + κ^{2} v^{μ} (mod κ^{3}),

without factorials at order two. Its coordinate conditions are

□_{γ_{κ}} X_{γ}^{μ} = 0 (mod κ^{3}), X_{γ}^{μ} ∣_{t = 0} = x^{μ} ∣_{t = 0}, \partial_{t} X_{γ}^{μ} ∣_{t = 0} = δ^{μ}_{0} .

Thus $w, v, \partial_{t} w, \partial_{t} v$ vanish at $t = 0$ . The first equation is

□_{-} w^{μ} = D_{-} (h)^{μ} .

These are ordinary coordinate-time data, not metric-dependent unit-normal data. VI's coordinate proposition is stated for every smooth $h, j$ , whereas its descended solution-jet observable has the additional equation-space restriction CheungVI (:coordinates, :DK, :descent).

Displayed formulas in this passage

XIX-B@1#eq:VI-data · line 233
$□_{γ_{κ}} X_{γ}^{μ} = 0 (mod κ^{3}), X_{γ}^{μ} ∣_{t = 0} = x^{μ} ∣_{t = 0}, \partial_{t} X_{γ}^{μ} ∣_{t = 0} = δ^{μ}_{0} .$
XIX-B@1#eq:VI-first · line 242
$□_{-} w^{μ} = D_{-} (h)^{μ} .$
unlabeled · line 226
$γ_{κ} = η_{-} + κh + κ^{2} j, X_{γ}^{μ} = x^{μ} + κ w^{μ} + κ^{2} v^{μ} (mod κ^{3}),$