Manuscript II-B@1 · II-B-v1.tex · proposition · bfd2eb3c

Proposition: Classical second-order composition

Optional classical lapse-composition interface · lines 2746–2775 · proof included or linked

II-B@1#prop:lapse

Its first two coefficients are

n_{1} = N_{1} + N ˙_{0} u_{1},

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} .

The second spatial coefficient $a_{2}^{i}$ does not occur.

Proof. Taylor expansion gives

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

Use the stated temporal dependence of $N_{0}$ and substitute the coefficients of $a_{1}, a_{2}$ . The spatial terms are

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

Also $\partial_{i} N_{0} = 0$ , so $a_{2}^{i}$ drops out.

Displayed formulas in this passage

unlabeled · line 2749
$n_{1} = N_{1} + N ˙_{0} u_{1},$
unlabeled · line 2752
$n_{2} = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
unlabeled · line 2762
$n_{2} = N_{2} + 2 a_{1}^{μ} \partial_{μ} N_{1} + a_{2}^{μ} \partial_{μ} N_{0} + a_{1}^{μ} a_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .$
unlabeled · line 2769
$- 2 Y_{1}^{i} \partial_{i} N_{1} - 2 N ˙_{0} Y_{1}^{i} \partial_{i} u_{1} = - 2 Y_{1}^{i} \partial_{i} n_{1} .$