Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Direct cancellation of the second-order coefficient

Direct cancellation of the second-order coefficient · lines 2114–2183

VII-A@1#eq:Jw, VII-A@1#eq:Jdifference

For a vector field $w$ , define the differential operator

J (w) = (w^{ν} \partial_{ν} w^{μ}) \partial_{μ} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} .

Then

A_{2}^{(g)} = ϕ_{2, g} - X_{w} ϕ_{1, g} - X_{v} ϕ_{0} + J (w) ϕ_{0} .

A direct expansion gives

J (w + ξ) - J (w) = X_{w} X_{ξ} + X_{X_{ξ} w} + \frac{1}{2} X_{ξ}^{2} + \frac{1}{2} X_{X_{ξ} ξ} .

Here $X_{ξ} w$ denotes the vector with components $ξ^{ν} \partial_{ν} w^{μ}$ .

For completeness, the first-derivative part of the left-hand side is

(w^{ν} \partial_{ν} ξ^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ξ^{ν} \partial_{ν} ξ^{μ}) \partial_{μ},

and its second-derivative part is

w^{μ} ξ^{ν} \partial_{μ} \partial_{ν} + \frac{1}{2} ξ^{μ} ξ^{ν} \partial_{μ} \partial_{ν} .

These are exactly the corresponding parts of (eq:Jdifference).

Using (eq:phi1transform)–(eq:vtransform), the difference $A_{2}^{'} - A_{2}$ is

X_{ξ} ϕ_{1, g} + (X_{ζ} + \frac{1}{2} X_{ξ}^{2}) ϕ_{0} - X_{w + ξ} (ϕ_{1, g} + X_{ξ} ϕ_{0}) + X_{w} ϕ_{1, g} - X_{X_{ξ} w + ζ + \frac{1}{2} X_{ξ} ξ} ϕ_{0} + (J (w + ξ) - J (w)) ϕ_{0} .

Terms containing $ϕ_{1, g}$ and $ζ$ cancel immediately. The remaining operator on $ϕ_{0}$ is

- \frac{1}{2} X_{ξ}^{2} - X_{w} X_{ξ} - X_{X_{ξ} w} - \frac{1}{2} X_{X_{ξ} ξ} + J (w + ξ) - J (w),

which vanishes by (eq:Jdifference).

This calculation verifies the second-order relational cancellation without suppressing the quadratic terms in the finite gauge parameter.

Displayed formulas in this passage

VII-A@1#eq:Jw · line 2117
$J (w) = (w^{ν} \partial_{ν} w^{μ}) \partial_{μ} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} .$
VII-A@1#eq:Jdifference · line 2132
$J (w + ξ) - J (w) = X_{w} X_{ξ} + X_{X_{ξ} w} + \frac{1}{2} X_{ξ}^{2} + \frac{1}{2} X_{X_{ξ} ξ} .$
unlabeled · line 2161
$X_{ξ} ϕ_{1, g} + (X_{ζ} + \frac{1}{2} X_{ξ}^{2}) ϕ_{0} - X_{w + ξ} (ϕ_{1, g} + X_{ξ} ϕ_{0}) + X_{w} ϕ_{1, g} - X_{X_{ξ} w + ζ + \frac{1}{2} X_{ξ} ξ} ϕ_{0} + (J (w + ξ) - J (w)) ϕ_{0} .$
unlabeled · line 2125
$A_{2}^{(g)} = ϕ_{2, g} - X_{w} ϕ_{1, g} - X_{v} ϕ_{0} + J (w) ϕ_{0} .$
unlabeled · line 2144
$(w^{ν} \partial_{ν} ξ^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ξ^{ν} \partial_{ν} ξ^{μ}) \partial_{μ},$
unlabeled · line 2152
$w^{μ} ξ^{ν} \partial_{μ} \partial_{ν} + \frac{1}{2} ξ^{μ} ξ^{ν} \partial_{μ} \partial_{ν} .$
unlabeled · line 2173
$- \frac{1}{2} X_{ξ}^{2} - X_{w} X_{ξ} - X_{X_{ξ} w} - \frac{1}{2} X_{X_{ξ} ξ} + J (w + ξ) - J (w),$