Manuscript Em02-E@1 · Em02-E-v1.tex · section · f3795050

Relational coordinate map

Clock slices and harmonic spatial labels › Relational coordinate map · lines 647–675

Em02-E@1#eq:Z, Em02-E@1#eq:alpha

Let $Z_{λ} (s, \cdot)$ be the formal spatial inverse of $Y_{λ} (s, \cdot)$ . Then

Z_{λ} z_{1} = x + λ z_{1} + \frac{λ ^{2}}{2} z_{2}, = - Y_{1}, z_{2} = - Y_{2} + 2 Y_{1}^{j} \partial_{j} Y_{1} .

The relational map is

α_{λ} (s, x) = Ψ_{λ} (s, Z_{λ} (s, x)) .

Consequently its first and second displacement coefficients are

α_{1} = (u_{1}, z_{1}), α_{2} = (u_{2} + 2 z_{1}^{i} \partial_{i} u_{1}, z_{2}) .

These identities follow by composing the Taylor series. They implement (eq:observable) with $X_{Γ_{0}} (s, x) = (ϕ_{0} (s), x)$ .

Displayed formulas in this passage

Em02-E@1#eq:Z · line 651
$Z_{λ} z_{1} = x + λ z_{1} + \frac{λ ^{2}}{2} z_{2}, = - Y_{1}, z_{2} = - Y_{2} + 2 Y_{1}^{j} \partial_{j} Y_{1} .$
Em02-E@1#eq:alpha · line 666
$α_{1} = (u_{1}, z_{1}), α_{2} = (u_{2} + 2 z_{1}^{i} \partial_{i} u_{1}, z_{2}) .$
unlabeled · line 660
$α_{λ} (s, x) = Ψ_{λ} (s, Z_{λ} (s, x)) .$