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

Theorem: Second-order local descent

Finite-time harmonic coordinates and descent · lines 770–808 · proof included or linked

VI-B@1#thm:descent, VI-B@1#eq:observable-coefficients

The coefficients of (eq:observable) are

O_{f}^{[0]} O_{f}^{[1]} O_{f}^{[2]} ψ_{2} = \int f ϕ_{0} d^{4} x, = \int f (ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0}) d^{4} x, = \int f ψ_{2} d^{4} x, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

Each is invariant under (eq:gauge-coefficients). Consequently they descend to the orbit set $J_{R, b} / G$ of the stated local formal gauge group.

Proof. Taylor expansion using (eq:inverse) gives all displayed terms. No factor $1/2$ is attached to $ϕ_{2}$ , because (eq:expansion) has no factorials.

Lemma lem:coordinate-covariance implies

Y_{γ^{'}} = F^{- 1} \circ Y_{γ} (mod κ^{3}) .

Together with $φ^{'} = φ \circ F$ , this gives

φ^{'} \circ Y_{γ^{'}} = φ \circ F \circ F^{- 1} \circ Y_{γ} = φ \circ Y_{γ} (mod κ^{3}) .

The same $f (X) d^{4} X$ is used on both sides, so equality is coefficientwise. All integrals are finite, and the required coordinate coefficients are defined on compact causal pasts.

Displayed formulas in this passage

VI-B@1#eq:observable-coefficients · line 773
$O_{f}^{[0]} O_{f}^{[1]} O_{f}^{[2]} ψ_{2} = \int f ϕ_{0} d^{4} x, = \int f (ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0}) d^{4} x, = \int f ψ_{2} d^{4} x, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
unlabeled · line 796
$Y_{γ^{'}} = F^{- 1} \circ Y_{γ} (mod κ^{3}) .$
unlabeled · line 800
$φ^{'} \circ Y_{γ^{'}} = φ \circ F \circ F^{- 1} \circ Y_{γ} = φ \circ Y_{γ} (mod κ^{3}) .$

Theorem: Second-order local descent

Displayed formulas in this passage

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