Manuscript VI-B@1 · VI-B-v1.tex · section · 33798723

Finite-time harmonic coordinates and descent

Finite-time harmonic coordinates and descent · lines 678–695

VI-B@1#eq:coordinate-data

Define four scalar coordinate functions, indexed by an external label $μ = 0, 1, 2, 3$ :

X_{γ}^{μ} = x^{μ} + κ w^{μ} + κ^{2} v^{μ} (mod κ^{3}) .

Require

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

The label $μ$ does not make $X_{γ}^{μ}$ the components of a spacetime vector.

Displayed formulas in this passage

VI-B@1#eq:coordinate-data · line 687
$□_{γ} X_{γ}^{μ} = 0 (mod κ^{3}), X_{γ}^{μ} ∣_{0} = x^{μ} ∣_{0}, \partial_{t} X_{γ}^{μ} ∣_{0} = δ^{μ}_{0} .$
unlabeled · line 683
$X_{γ}^{μ} = x^{μ} + κ w^{μ} + κ^{2} v^{μ} (mod κ^{3}) .$