Manuscript VI@1 · VI.tex · theorem · b5b60afb

Theorem: Second-order classical descent

Finite-time harmonic coordinates and second-order descent › The observable · lines 1069–1120 · proof included or linked

VI@1#thm:descent, VI@1#eq:O0-jet, VI@1#eq:O1-jet, VI@1#eq:O2-jet, VI@1#eq:psi2

On the jet space $J_{T}$ , the functional (eq:observable) has coefficients

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

Each coefficient is invariant under the formal gauge group (eq:formal-pullback). Hence $O_{f}$ descends to the orbit space of $J_{T}$ .

Proof. Taylor expansion of $ϕ (Y_{γ} (X))$ , using (eq:jets) and (eq:inverse-coordinate), gives (eq:O0-jet)–(eq:psi2).

For invariance, equation (eq:coordinate-covariance) says $X_{γ^{'}} = X_{γ} \circ F$ . Therefore

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

Since $ϕ^{'} = F^{*} ϕ = ϕ \circ F$ ,

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

Smearing with the same fixed function of the relational labels proves coefficientwise invariance.

All coefficients are integrals of smooth functions against compact $f$ . The coordinate coefficients at points of $supp f$ depend on finite-time wave problems, so the definition is meaningful on the declared smooth domain.

Displayed formulas in this passage

VI@1#eq:O0-jet · line 1073
$O_{f}^{[0]} O_{f}^{[1]} O_{f}^{[2]} ψ_{2} = \int f ϕ_{0} d^{4} x, = \int f ψ_{1} d^{4} x, = \int f ψ_{2} d^{4} x, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} . ψ_{1} = ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0},$
VI@1#eq:O1-jet · line 1073
$O_{f}^{[0]} O_{f}^{[1]} O_{f}^{[2]} ψ_{2} = \int f ϕ_{0} d^{4} x, = \int f ψ_{1} d^{4} x, = \int f ψ_{2} d^{4} x, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} . ψ_{1} = ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0},$
VI@1#eq:O2-jet · line 1073
$O_{f}^{[0]} O_{f}^{[1]} O_{f}^{[2]} ψ_{2} = \int f ϕ_{0} d^{4} x, = \int f ψ_{1} d^{4} x, = \int f ψ_{2} d^{4} x, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} . ψ_{1} = ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0},$
VI@1#eq:psi2 · line 1073
$O_{f}^{[0]} O_{f}^{[1]} O_{f}^{[2]} ψ_{2} = \int f ϕ_{0} d^{4} x, = \int f ψ_{1} d^{4} x, = \int f ψ_{2} d^{4} x, = ϕ_{2} - w^{μ} \partial_{μ} ϕ_{1} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} . ψ_{1} = ϕ_{1} - w^{μ} \partial_{μ} ϕ_{0},$
unlabeled · line 1101
$Y_{γ^{'}} = F^{- 1} \circ Y_{γ} mod κ^{3} .$
unlabeled · line 1105
$ϕ^{'} \circ Y_{γ^{'}} = ϕ \circ F \circ F^{- 1} \circ Y_{γ} = ϕ \circ Y_{γ} mod κ^{3} .$

Theorem: Second-order classical descent

Displayed formulas in this passage

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