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

The second-order composition formula

Optional original-rod lapse attachment › The second-order composition formula · lines 1728–1754

Work in background-adapted coordinates for which the background label map is the identity. Suppose the raw lapse has factorial jets

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

At fixed clock label, let the forward spatial rod map be

R_{ε} (τ, x) = x + ε Y_{1} (τ, x) + \frac{ε ^{2}}{2} Y_{2} (τ, x) (mod ε^{3}) .

Let the inverse time map at fixed spatial argument be

θ_{ε} (τ, x) = τ + ε u_{1} (τ, x) + \frac{ε ^{2}}{2} u_{2} (τ, x) (mod ε^{3}) .

The full inverse first inverts $R_{ε}$ , then substitutes the resulting argument into $θ_{ε}$ .

Displayed formulas in this passage

unlabeled · line 1732
$N_{ε} = N_{0} + ε N_{1} + \frac{ε ^{2}}{2} N_{2} (mod ε^{3}), N_{0} = N_{0} (t) .$
unlabeled · line 1739
$R_{ε} (τ, x) = x + ε Y_{1} (τ, x) + \frac{ε ^{2}}{2} Y_{2} (τ, x) (mod ε^{3}) .$
unlabeled · line 1746
$θ_{ε} (τ, x) = τ + ε u_{1} (τ, x) + \frac{ε ^{2}}{2} u_{2} (τ, x) (mod ε^{3}) .$