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

Finite-time harmonic coordinates and descent

Finite-time harmonic coordinates and descent · lines 746–769

VI-B@1#eq:inverse, VI-B@1#eq:observable

The formal inverse coordinate map is

Y_{γ}^{μ} (X) = X^{μ} - κ w^{μ} (X) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (X) (mod κ^{3}) .

Substitution in $X_{γ} (Y_{γ} (X))$ verifies this identity directly.

For real $f \in C_{c}^{\infty} (D_{R, b}^{\circ})$ , interpreted as a test in label coordinates, define

O_{f} [γ, φ] = \int f (X) φ (Y_{γ} (X)) d^{4} X (mod κ^{3}) .

The measure is the fixed label density. It is not a fluctuating spacetime volume element.

Displayed formulas in this passage

VI-B@1#eq:inverse · line 749
$Y_{γ}^{μ} (X) = X^{μ} - κ w^{μ} (X) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (X) (mod κ^{3}) .$
VI-B@1#eq:observable · line 761
$O_{f} [γ, φ] = \int f (X) φ (Y_{γ} (X)) d^{4} X (mod κ^{3}) .$