Manuscript XIX-A@1 · XIX-A.tex · proposition · e30d8062

Proposition: Rod dependence of the quadratic relational lapse

The scalar-clock lapse and the original spatial rods › A self-contained second-order composition identity · lines 621–668 · proof included or linked

XIX-A@1#prop:lapse-composition, XIX-A@1#eq:n1, XIX-A@1#eq:n2

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The coefficients of $N_{E} = N \circ α$ 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} .

In particular, $Y_{2}$ does not occur in $n_{2}$ .

Proof. Let

α_{e} = x + e α_{1} + \frac{e ^{2}}{2} α_{2} .

The equations $T_{e} \circ α_{e} = T_{0}$ and $Y_{e}^{i} \circ α_{e} = x^{i}$ imply

α_{1}^{0} = u_{1}, α_{1}^{i} = z_{1}^{i} = - Y_{1}^{i} .

At second order in the clock equation,

v α_{2}^{0} + v ˙ u_{1}^{2} + 2 u_{1} θ ˙_{1} + 2 z_{1}^{i} \partial_{i} θ_{1} + θ_{2} = 0.

Since $\partial_{i} u_{1} = - \partial_{i} θ_{1} / v$ , this gives

α_{2}^{0} = u_{2} + 2 z_{1}^{i} \partial_{i} u_{1} .

Taylor expansion yields

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} .

Because $N_{0}$ depends only on $t$ , this becomes

N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} + 2 z_{1}^{i} \partial_{i} (N_{1} + N ˙_{0} u_{1}) .

Substitute $z_{1} = - Y_{1}$ . The second spatial displacement multiplies $\partial_{i} N_{0} = 0$ , so $Y_{2}$ is absent. The first-order formula follows from the same Taylor expansion.

Displayed formulas in this passage

XIX-A@1#eq:n1 · line 624
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
XIX-A@1#eq:n2 · line 624
$n_{1} n_{2} = N_{1} + N ˙_{0} u_{1}, = N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} - 2 Y_{1}^{i} \partial_{i} n_{1} .$
unlabeled · line 636
$α_{e} = x + e α_{1} + \frac{e ^{2}}{2} α_{2} .$
unlabeled · line 641
$α_{1}^{0} = u_{1}, α_{1}^{i} = z_{1}^{i} = - Y_{1}^{i} .$
unlabeled · line 645
$v α_{2}^{0} + v ˙ u_{1}^{2} + 2 u_{1} θ ˙_{1} + 2 z_{1}^{i} \partial_{i} θ_{1} + θ_{2} = 0.$
unlabeled · line 650
$α_{2}^{0} = u_{2} + 2 z_{1}^{i} \partial_{i} u_{1} .$
unlabeled · line 654
$n_{2} = N_{2} + 2 α_{1}^{μ} \partial_{μ} N_{1} + α_{2}^{μ} \partial_{μ} N_{0} + α_{1}^{μ} α_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .$
unlabeled · line 661
$N_{2} + 2 u_{1} N ˙_{1} + N ˙_{0} u_{2} + N ¨_{0} u_{1}^{2} + 2 z_{1}^{i} \partial_{i} (N_{1} + N ˙_{0} u_{1}) .$

Proposition: Rod dependence of the quadratic relational lapse

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