Manuscript XIX-A@1 · XIX-A.tex · definition · e30d8062

Definition: Finite-time harmonic prescription

The finite-time harmonic relational scalar › An explicit comparison prescription · lines 307–330

XIX-A@1#def:harmonic, XIX-A@1#eq:harmonic, XIX-A@1#eq:harmonic-data

The formal coordinate map

H_{q_{e}}^{μ} = x^{μ} + e W_{1}^{μ} + \frac{e ^{2}}{2} W_{2}^{μ}

is defined by

\partial_{α} (a_{e}^{α β} \partial_{β} H_{q_{e}}^{μ}) = 0 (mod e^{3}),

and the ordinary coordinate-time Cauchy conditions

W_{j}^{μ} (t_{0}, x) = 0, \partial_{t} W_{j}^{μ} (t_{0}, x) = 0, j = 1, 2.

A metric-dependent unit-normal derivative is not substituted for $\partial_{t}$ in these conditions.

Displayed formulas in this passage

XIX-A@1#eq:harmonic · line 314
$\partial_{α} (a_{e}^{α β} \partial_{β} H_{q_{e}}^{μ}) = 0 (mod e^{3}),$
XIX-A@1#eq:harmonic-data · line 322
$W_{j}^{μ} (t_{0}, x) = 0, \partial_{t} W_{j}^{μ} (t_{0}, x) = 0, j = 1, 2.$
unlabeled · line 310
$H_{q_{e}}^{μ} = x^{μ} + e W_{1}^{μ} + \frac{e ^{2}}{2} W_{2}^{μ}$