Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Second classical rod and the retained lapse term

Newton inversion and uniform spatial-tail estimates › Second classical rod and the retained lapse term · lines 1161–1189

XX-C@1#eq:lapse

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For classical coefficient fields with common compact spatial support on $J$ , the second rod equation is

Y_{2}^{a} = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .

Here $B_{1}, B_{2}$ are the induced-metric coefficients. Both $B_{2}$ and $B_{1} \partial Y_{1}$ are compact in space. Theorem thm:dipole therefore gives the same inverse-square estimate for $Y_{2}$ .

The complete quadratic lapse retains

n_{2} = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1}, N_{0} = t / A .

The absence of $Y_{2}$ follows from $\partial_{i} N_{0} = 0$ , not from dropping its construction.

The tail theorem controls the classical rod term on this domain. It does not define its quantum Wick contraction.

Displayed formulas in this passage

XX-C@1#eq:lapse · line 1175
$n_{2} = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1}, N_{0} = t / A .$
unlabeled · line 1165
$Y_{2}^{a} = - G_{sp} \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$