Manuscript XX-A@1 · XX-A.tex · proposition · 3a586110

Proposition: Original and lapse-weighted rods are detected

Classical gauge identity and rod-detection tests › A compact witness distinguishing modified rods · lines 889–930 · proof included or linked

XX-A@1#prop:rod-detection

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There is a compact metric perturbation for which the quadratic relational lapses formed with (eq:rod-exact) and (eq:modified-rods) differ.

Proof. Take a real $w \in C_{c}^{\infty} (J \times R^{3})$ , not identically zero, and choose

h_{00} = - 2 w, h_{0 i} = h_{ij} = 0, φ = k = ψ = 0.

Then $T = t$ , $γ_{s} = a^{2} δ$ , and the original rods are $R_{s}^{a} = x^{a}$ through every displayed order. The raw lapse is

N_{s} = N_{0} 1 + 2 s w, N_{1} = N_{0} w .

The first coefficient of (eq:modified-rods) gives

Y_{1}^{a} = - G \partial_{a} w .

The composition formula therefore gives

n_{2} - n_{2} = 2 N_{0} a \sum (G \partial_{a} w) \partial_{a} w .

At each time, integration by parts and $Δ G w = w$ yield

a \sum \int_{R^{3}} (G \partial_{a} w) \partial_{a} w d^{3} x = - \int_{R^{3}} (G w) Δ w d^{3} x = - \int_{R^{3}} w^{2} d^{3} x .

The difference is supported where $\nabla w$ is supported. Choose a compact spatial smearing equal to one there, and a nonnegative time smearing detecting a time where $w \neq = 0$ . The smeared difference is strictly negative.

Displayed formulas in this passage

unlabeled · line 899
$h_{00} = - 2 w, h_{0 i} = h_{ij} = 0, φ = k = ψ = 0.$
unlabeled · line 906
$N_{s} = N_{0} 1 + 2 s w, N_{1} = N_{0} w .$
unlabeled · line 911
$Y_{1}^{a} = - G \partial_{a} w .$
unlabeled · line 915
$n_{2} - n_{2} = 2 N_{0} a \sum (G \partial_{a} w) \partial_{a} w .$
unlabeled · line 921
$a \sum \int_{R^{3}} (G \partial_{a} w) \partial_{a} w d^{3} x = - \int_{R^{3}} (G w) Δ w d^{3} x = - \int_{R^{3}} w^{2} d^{3} x .$

Proposition: Original and lapse-weighted rods are detected

Displayed formulas in this passage

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