Manuscript XX-B@1 · XX-B.tex · theorem · 56d708a5

Theorem: Retained classical coefficient

Complete lapse coefficient and the classical principal-symbol check › Raw lapse coefficients · lines 1534–1567 · proof included or linked

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

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The complete original-rod lapse coefficients are

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} .

For the frozen background,

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

The constructed $Y_{2}$ does not enter $n_{2}$ .

Proof. Taylor composition gives

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 $N_{0} = t / A$ has no spatial derivative. Substitution of (eq:alpha) gives (eq:n2). The spatial part of $α_{2}$ , containing $Y_{2}$ , multiplies $\partial_{i} N_{0} = 0$ . The first-order explicit expression follows from (eq:clock-jet) and (eq:Nlapse).

Displayed formulas in this passage

XX-B@1#eq:n1 · line 1537
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
XX-B@1#eq:n2 · line 1537
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
unlabeled · line 1545
$n_{1} = - \frac{t}{2 A} h_{00} - \frac{t ^{2}}{A ^{2}} φ ˙ - \frac{t}{A ^{2}} φ, N ¨_{0} = 0.$
unlabeled · line 1555
$n_{2} = N_{2} + 2 α_{1}^{μ} \partial_{μ} N_{1} + α_{2}^{μ} \partial_{μ} N_{0} + α_{1}^{μ} α_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .$

Theorem: Retained classical coefficient

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