Manuscript XX-D@1 · XX-D.tex · theorem · c83a526d

Theorem: Complete relational second jet

Complete classical coefficient through second order › Relational composition and absence of Y 2 · lines 616–651 · proof included or linked

XX-D@1#thm:jet, XX-D@1#eq:composition

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The relational lapse

N_{s} = N_{s} \circ α_{s} = N_{0} + s n_{1} + \frac{s ^{2}}{2} n_{2}

has coefficients

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.

Proof. Taylor expansion 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} .

Because $N_{0}$ is spatially homogeneous, $α_{2}^{i} \partial_{i} N_{0} = 0$ . Substitute the first spatial displacement and the displayed temporal second displacement. The spatial terms combine to $- 2 Y_{1}^{i} \partial_{i} (N_{1} + N ˙_{0} u_{1})$ . This proves (eq:composition) and the stated absence. No quantum substitution was used.

Displayed formulas in this passage

XX-D@1#eq:composition · 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 619
$N_{s} = N_{s} \circ α_{s} = N_{0} + s n_{1} + \frac{s ^{2}}{2} n_{2}$
unlabeled · line 637
$n_{2} = N_{2} + 2 α_{1}^{μ} \partial_{μ} N_{1} + α_{2}^{μ} \partial_{μ} N_{0} + α_{1}^{μ} α_{1}^{ν} \partial_{μ} \partial_{ν} N_{0} .$

Theorem: Complete relational second jet

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