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

Optional classical lapse-composition interface

Optional classical lapse-composition interface · lines 2721–2745

This appendix is a calculus statement for a separately supplied classical geometry. It is not used in the conical release.

Suppose a scalar lapse prescription and a coordinate map are given with expansions

N_{ε} = N_{0} + ε N_{1} + \frac{ε ^{2}}{2} N_{2} + O (ε^{3}),

Y_{ε}^{μ} (x) = x^{μ} + ε a_{1}^{μ} (x) + \frac{ε ^{2}}{2} a_{2}^{μ} (x) + O (ε^{3}) .

Assume $N_{0} = N_{0} (x^{0})$ , and write

a_{1}^{0} = u_{1}, a_{1}^{i} = - Y_{1}^{i}, a_{2}^{0} = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1} .

Define the relationally composed scalar $n_{ε} = N_{ε} \circ Y_{ε}$ .

Displayed formulas in this passage

unlabeled · line 2729
$N_{ε} = N_{0} + ε N_{1} + \frac{ε ^{2}}{2} N_{2} + O (ε^{3}),$
unlabeled · line 2733
$Y_{ε}^{μ} (x) = x^{μ} + ε a_{1}^{μ} (x) + \frac{ε ^{2}}{2} a_{2}^{μ} (x) + O (ε^{3}) .$
unlabeled · line 2739
$a_{1}^{0} = u_{1}, a_{1}^{i} = - Y_{1}^{i}, a_{2}^{0} = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1} .$