Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Direct quadratic cancellation

Direct quadratic cancellation · lines 2376–2415

This calculation makes the quadratic cancellation independent of any notation for an inverse map.

Let $B_{w} = w \cdot \partial w - v$ . Equations (eq:coordinatelaws) imply

B_{w}^{'} = B_{w} + w \cdot \partial ξ + \frac{1}{2} ξ \cdot \partial ξ - ζ .

The order-one difference of $ϕ_{1} - w \cdot \partial ϕ_{0}$ vanishes.

At order two, insert (eq:scalarlaw) and $w^{'} = w + ξ$ into (eq:relationalcoeff). The coefficient of $\partial_{μ} ϕ_{0}$ left after subtracting the unprimed expression is

ζ^{μ} + \frac{1}{2} (ξ \cdot \partial ξ)^{μ} - (w \cdot \partial ξ)^{μ} - (ξ \cdot \partial ξ)^{μ} + (w \cdot \partial ξ)^{μ} + \frac{1}{2} (ξ \cdot \partial ξ)^{μ} - ζ^{μ} = 0.

The terms with derivatives of $ϕ_{1}$ are $X_{ξ} ϕ_{1} - ξ \cdot \partial ϕ_{1} = 0$ . The second-derivative terms in $ϕ_{0}$ are

\frac{1}{2} ξ^{μ} ξ^{ν} - w^{μ} ξ^{ν} - ξ^{μ} ξ^{ν} + w^{μ} ξ^{ν} + \frac{1}{2} ξ^{μ} ξ^{ν},

contracted with the symmetric Hessian; they vanish. This proves the coefficient cancellation using the complete quadratic laws.

Displayed formulas in this passage

unlabeled · line 2384
$B_{w}^{'} = B_{w} + w \cdot \partial ξ + \frac{1}{2} ξ \cdot \partial ξ - ζ .$
unlabeled · line 2397
$ζ^{μ} + \frac{1}{2} (ξ \cdot \partial ξ)^{μ} - (w \cdot \partial ξ)^{μ} - (ξ \cdot \partial ξ)^{μ} + (w \cdot \partial ξ)^{μ} + \frac{1}{2} (ξ \cdot \partial ξ)^{μ} - ζ^{μ} = 0.$
unlabeled · line 2406
$\frac{1}{2} ξ^{μ} ξ^{ν} - w^{μ} ξ^{ν} - ξ^{μ} ξ^{ν} + w^{μ} ξ^{ν} + \frac{1}{2} ξ^{μ} ξ^{ν},$