Manuscript XX-E@1 · XX-E.tex · section · cf7a8f40

The complete quadratic lapse

The identical freeze and a necessary smearing correction › The complete quadratic lapse · lines 372–425

XX-E@1#eq:ell, XX-E@1#eq:n, XX-E@1#eq:qloc, XX-E@1#eq:composition

Define

ℓ z = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ .

The complete second coefficient is

n_{1} = ℓ z, n_{2} = ℓ ζ + q (z), q (z) = q_{loc} (z) - 2 \calRz \cdot \nabla ℓ z,

where

q_{loc} (z) = - \frac{t}{4 A} h_{00}^{2} + \frac{t}{A a ^{2}} h_{0 i} h_{0 i} + \frac{t ^{2}}{A ^{2}} h_{00} φ ˙ + \frac{2 t ^{3}}{A ^{3}} φ ˙^{2} + \frac{2 t ^{2}}{A ^{2} a ^{2}} h_{0 i} \partial_{i} φ + \frac{t ^{3}}{A ^{3} a ^{2}} \partial_{i} φ \partial_{i} φ + \frac{t ^{2}}{A ^{2}} φ h ˙_{00} + \frac{t}{A ^{2}} φ h_{00} + \frac{2 t ^{3}}{A ^{3}} φ φ ¨ + \frac{6 t ^{2}}{A ^{3}} φ φ ˙ + \frac{t}{A ^{3}} φ^{2} .

This is the expansion in the supplied XX-D.

For clarity, the underlying composition identity is

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, Y_{1} = \calRz .

The second spatial rod is constructed by its elliptic equation but multiplies $\partial_{i} N_{0} = 0$ in this composition. Thus $Y_{2}$ is absent for a proved reason, not by a tensor or source truncation.

Displayed formulas in this passage

XX-E@1#eq:ell · line 375
$ℓ z = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ .$
XX-E@1#eq:n · line 382
$n_{1} = ℓ z, n_{2} = ℓ ζ + q (z), q (z) = q_{loc} (z) - 2 \calRz \cdot \nabla ℓ z,$
XX-E@1#eq:qloc · line 390
$q_{loc} (z) = - \frac{t}{4 A} h_{00}^{2} + \frac{t}{A a ^{2}} h_{0 i} h_{0 i} + \frac{t ^{2}}{A ^{2}} h_{00} φ ˙ + \frac{2 t ^{3}}{A ^{3}} φ ˙^{2} + \frac{2 t ^{2}}{A ^{2} a ^{2}} h_{0 i} \partial_{i} φ + \frac{t ^{3}}{A ^{3} a ^{2}} \partial_{i} φ \partial_{i} φ + \frac{t ^{2}}{A ^{2}} φ h ˙_{00} + \frac{t}{A ^{2}} φ h_{00} + \frac{2 t ^{3}}{A ^{3}} φ φ ¨ + \frac{6 t ^{2}}{A ^{3}} φ φ ˙ + \frac{t}{A ^{3}} φ^{2} .$
XX-E@1#eq:composition · line 415
$n_{2} = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1}, N_{0} = t / A, Y_{1} = \calRz .$