Manuscript XIX-A@1 · XIX-A.tex · section · e30d8062

A detecting compact perturbation

Second refusal: original rods are not modified rods › A detecting compact perturbation · lines 813–839

XIX-A@1#eq:scalar-metric, XIX-A@1#eq:modified-clock

Consider the scalar metric subclass

q_{e} = - (1 + 2 e A) d t^{2} + a (t)^{2} (1 + 2 eD) δ_{ij} d x^{i} d x^{j} (mod e^{3}), T_{e} = T_{0} (t),

where $A, D$ are spatially compact smooth functions.

For the original rods, the clock slices are exactly $t = constant$ . For the modified rods, it suffices to use the first clock jet

Θ_{e} = T_{0} (t) + e \frac{v ( t )}{H ( t )} D (t, x) + O (e^{2}) .

Define $Z_{e}^{i}$ to be harmonic on the $Θ_{e}$ -slices with the same asymptotic Cartesian boundary condition.

Equation (eq:modified-clock) has the first jet of the modified slicing used in BFHPR. No choice of its second-order coefficient is needed for the quadratic distinction proved below. In both relational observables, the {time coordinate remains $T_{e}$ }; only the spatial rods are replaced.

Displayed formulas in this passage

XIX-A@1#eq:scalar-metric · line 816
$q_{e} = - (1 + 2 e A) d t^{2} + a (t)^{2} (1 + 2 eD) δ_{ij} d x^{i} d x^{j} (mod e^{3}), T_{e} = T_{0} (t),$
XIX-A@1#eq:modified-clock · line 827
$Θ_{e} = T_{0} (t) + e \frac{v ( t )}{H ( t )} D (t, x) + O (e^{2}) .$

A detecting compact perturbation

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