Manuscript I-B@1 · I-B-v1.tex · theorem · 5357688f

Theorem: Complete classical composition through second order

Optional original-rod lapse attachment › The second-order composition formula · lines 1790–1832 · proof included or linked

I-B@1#thm:lapse-composition, I-B@1#eq:n1, I-B@1#eq:n2

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Under these hypotheses,

n_{ε} = N_{0} + ε n_{1} + \frac{ε ^{2}}{2} n_{2} (mod ε^{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} .

The dot is differentiation in the background coordinate used in this expansion. The coefficient $Y_{2}$ does not enter $n_{2}$ .

Proof. Taylor composition gives

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

Since $N_{0}$ depends only on time, insertion of Lemma lem:inverse-jets yields

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

The final two terms are $- 2 Y_{1}^{i} \partial_{i} (N_{1} + N ˙_{0} u_{1})$ , giving (eq:n2). The coefficient containing $Y_{2}$ multiplies $\partial_{i} N_{0} = 0$ .

Displayed formulas in this passage

I-B@1#eq:n1 · line 1799
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
I-B@1#eq:n2 · line 1799
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
unlabeled · line 1813
$n_{1} n_{2} = N_{1} + α_{1}^{μ} \partial_{μ} N_{0}, = N_{2} + 2 α_{1}^{μ} \partial_{μ} N_{1} + α_{2}^{μ} \partial_{μ} N_{0} + α_{1}^{μ} α_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .$
unlabeled · line 1793
$n_{ε} = N_{0} + ε n_{1} + \frac{ε ^{2}}{2} n_{2} (mod ε^{3}),$
unlabeled · line 1822
$n_{2} = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} N_{1} - 2 N ˙_{0} Y_{1}^{i} \partial_{i} u_{1} .$

Theorem: Complete classical composition through second order

Displayed formulas in this passage

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