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

Theorem: Complete classical original-rod lapse

Complete raw and relational lapse through second order › The complete inverse relational map · lines 688–746 · proof included or linked

XX-A@1#thm:lapse, XX-A@1#eq:n1, XX-A@1#eq:n2, XX-A@1#eq:n1-explicit

The frozen relational lapse has the expansion

n_{s} = N_{0} + s n_{1} + \frac{s ^{2}}{2} n_{2} (mod s^{3}),

where

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

All terms are supplied by (eq:clock-coeff)–(eq:Y2) and (eq:W0)–(eq:N2). The second spatial rod $Y_{2}$ is constructed but does not enter $n_{2}$ .

For this background,

n_{1} = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ, N ¨_{0} = 0.

Proof. Composition with a formal inverse gives

n_{1} = N_{1} + α_{1}^{μ} \partial_{μ} N_{0},

n_{2} = N_{2} + 2 α_{1}^{μ} \partial_{μ} N_{1} + α_{2}^{μ} \partial_{μ} N_{0} + α_{1}^{μ} α_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .

The background $N_{0}$ depends only on time. Substitution of (eq:alpha1)–(eq:alpha2) gives

N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} N_{1} - 2 N ˙_{0} Y_{1}^{i} \partial_{i} u_{1},

which is (eq:n2). The coefficient $α_{2}^{i}$ , containing $Y_{2}$ , multiplies $\partial_{i} N_{0} = 0$ .

Since $v_{0} = A / t$ , $g ˉ^{00} = - 1$ , and $h^{00} = h_{00}$ ,

W_{1} = \frac{A ^{2}}{t ^{2}} h_{00} + \frac{2 A}{t} φ ˙ .

Equations (eq:N1), (eq:time-inverse), and $N ˙_{0} = 1/ A$ give (eq:n1-explicit).

Displayed formulas in this passage

XX-A@1#eq:n1 · line 695
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
XX-A@1#eq:n2 · line 695
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
XX-A@1#eq:n1-explicit · line 709
$n_{1} = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ, N ¨_{0} = 0.$
unlabeled · line 691
$n_{s} = N_{0} + s n_{1} + \frac{s ^{2}}{2} n_{2} (mod s^{3}),$
unlabeled · line 721
$n_{1} = N_{1} + α_{1}^{μ} \partial_{μ} N_{0},$
unlabeled · line 724
$n_{2} = N_{2} + 2 α_{1}^{μ} \partial_{μ} N_{1} + α_{2}^{μ} \partial_{μ} N_{0} + α_{1}^{μ} α_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .$
unlabeled · line 732
$N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} N_{1} - 2 N ˙_{0} Y_{1}^{i} \partial_{i} u_{1},$
unlabeled · line 741
$W_{1} = \frac{A ^{2}}{t ^{2}} h_{00} + \frac{2 A}{t} φ ˙ .$