Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Raw lapse coefficients

Complete lapse coefficient and the classical principal-symbol check › Raw lapse coefficients · lines 1488–1533

XX-B@1#eq:Wlapse, XX-B@1#eq:Nlapse, XX-B@1#eq:alpha

With

W_{s} = - g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s}) = W_{0} + s W_{1} + \frac{s ^{2}}{2} W_{2},

the complete coefficients are

W_{0} W_{1} W_{2} = A^{2} / t^{2}, = h^{μν} v_{μ} v_{ν} - 2 g ˉ^{μν} v_{μ} \partial_{ν} φ, = (k^{μν} - 2 h^{μ}_{ρ} h^{ρ ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} \partial_{ν} φ - 2 g ˉ^{μν} \partial_{μ} φ \partial_{ν} φ - 2 g ˉ^{μν} v_{μ} \partial_{ν} ψ .

For $N_{s} = N_{0} + s N_{1} + s^{2} N_{2} /2$ ,

N_{0} = \frac{t}{A}, N_{1} = - \frac{1}{2} W_{0}^{- 3/2} W_{1}, N_{2} = \frac{3}{4} W_{0}^{- 5/2} W_{1}^{2} - \frac{1}{2} W_{0}^{- 3/2} W_{2} .

The forward coordinate map is

C_{s} (t, x) = (T_{s} (t, x), R_{s} (T_{s} (t, x), x)) .

Its inverse $α_{s} = id + s α_{1} + s^{2} α_{2} /2$ has

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

Define $n_{s} = N_{s} \circ α_{s}$ .

Displayed formulas in this passage

XX-B@1#eq:Wlapse · line 1496
$W_{0} W_{1} W_{2} = A^{2} / t^{2}, = h^{μν} v_{μ} v_{ν} - 2 g ˉ^{μν} v_{μ} \partial_{ν} φ, = (k^{μν} - 2 h^{μ}_{ρ} h^{ρ ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} \partial_{ν} φ - 2 g ˉ^{μν} \partial_{μ} φ \partial_{ν} φ - 2 g ˉ^{μν} v_{μ} \partial_{ν} ψ .$
XX-B@1#eq:Nlapse · line 1510
$N_{0} = \frac{t}{A}, N_{1} = - \frac{1}{2} W_{0}^{- 3/2} W_{1}, N_{2} = \frac{3}{4} W_{0}^{- 5/2} W_{1}^{2} - \frac{1}{2} W_{0}^{- 3/2} W_{2} .$
XX-B@1#eq:alpha · line 1525
$α_{1}^{0} α_{2}^{0} = u_{1}, = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1}, α_{1}^{i} α_{2}^{i} = - Y_{1}^{i}, = 2 Y_{1}^{j} \partial_{j} Y_{1}^{i} - Y_{2}^{i} .$
unlabeled · line 1491
$W_{s} = - g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s}) = W_{0} + s W_{1} + \frac{s ^{2}}{2} W_{2},$
unlabeled · line 1518
$C_{s} (t, x) = (T_{s} (t, x), R_{s} (T_{s} (t, x), x)) .$