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

Proposition: Scalar expansion

The finite-time harmonic relational scalar › The relational scalar is not the coordinate map · lines 405–429 · proof included or linked

XIX-A@1#prop:scalar-expansion, XIX-A@1#eq:scalar-second

The coefficients of (eq:rel-scalar) are

R_{H}_{0} R_{H}_{1} R_{H}_{2} = T_{0}, = θ_{1} - W_{1}^{μ} \partial_{μ} T_{0}, = θ_{2} - 2 W_{1}^{μ} \partial_{μ} θ_{1} + (2 W_{1}^{ν} \partial_{ν} W_{1}^{μ} - W_{2}^{μ}) \partial_{μ} T_{0} + W_{1}^{μ} W_{1}^{ν} \partial_{μ} \partial_{ν} T_{0} .

Proof. Solving $H_{q_{e}} \circ H_{q_{e}}^{- 1} = id$ coefficientwise gives

H_{q_{e}}^{- 1} (x) = x - e W_{1} (x) + \frac{e ^{2}}{2} (2 W_{1}^{ν} \partial_{ν} W_{1} - W_{2}) (x) .

Taylor expansion of $T_{e}$ at this inverse map gives the formula.

Displayed formulas in this passage

XIX-A@1#eq:scalar-second · line 408
$R_{H}_{0} R_{H}_{1} R_{H}_{2} = T_{0}, = θ_{1} - W_{1}^{μ} \partial_{μ} T_{0}, = θ_{2} - 2 W_{1}^{μ} \partial_{μ} θ_{1} + (2 W_{1}^{ν} \partial_{ν} W_{1}^{μ} - W_{2}^{μ}) \partial_{μ} T_{0} + W_{1}^{μ} W_{1}^{ν} \partial_{μ} \partial_{ν} T_{0} .$
unlabeled · line 422
$H_{q_{e}}^{- 1} (x) = x - e W_{1} (x) + \frac{e ^{2}}{2} (2 W_{1}^{ν} \partial_{ν} W_{1} - W_{2}) (x) .$