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

Lemma: Full-tensor lapse coefficients

Complete classical lapse expansion › The unsmeared scalar lapse · lines 705–744 · proof included or linked

Em02-E@1#lem:bare-lapse, Em02-E@1#eq:W1, Em02-E@1#eq:W2, Em02-E@1#eq:N

The coefficients are

W_{0} W_{1} W_{2} = v^{2}, = h^{μν} v_{μ} v_{ν} - 2 g_{0}^{μν} v_{μ} p_{ν}, = (k^{μν} - 2 h^{μα} g_{0, α β} h^{β ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} p_{ν} - 2 g_{0}^{μν} p_{μ} p_{ν} - 2 g_{0}^{μν} v_{μ} w_{ν} .

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

N_{0} = v^{- 1}, N_{1} = - \frac{W _{1}}{2 v ^{3}}, N_{2} = \frac{3 W _{1} ^{2}}{4 v ^{5}} - \frac{W _{2}}{2 v ^{3}} .

Proof. The inverse metric expansion is

g_{λ}^{μν} = g_{0}^{μν} - λ h^{μν} + \frac{λ ^{2}}{2} (2 h^{μα} g_{0, α β} h^{β ν} - k^{μν}) .

Multiply it by the two scalar-gradient expansions and retain the minus sign in (eq:lapse). This gives (eq:W1)–(eq:W2). The first two derivatives of $W^{- 1/2}$ give (eq:N).

Displayed formulas in this passage

Em02-E@1#eq:W1 · line 708
$W_{0} W_{1} W_{2} = v^{2}, = h^{μν} v_{μ} v_{ν} - 2 g_{0}^{μν} v_{μ} p_{ν}, = (k^{μν} - 2 h^{μα} g_{0, α β} h^{β ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} p_{ν} - 2 g_{0}^{μν} p_{μ} p_{ν} - 2 g_{0}^{μν} v_{μ} w_{ν} .$
Em02-E@1#eq:W2 · line 708
$W_{0} W_{1} W_{2} = v^{2}, = h^{μν} v_{μ} v_{ν} - 2 g_{0}^{μν} v_{μ} p_{ν}, = (k^{μν} - 2 h^{μα} g_{0, α β} h^{β ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} p_{ν} - 2 g_{0}^{μν} p_{μ} p_{ν} - 2 g_{0}^{μν} v_{μ} w_{ν} .$
Em02-E@1#eq:N · line 722
$N_{0} = v^{- 1}, N_{1} = - \frac{W _{1}}{2 v ^{3}}, N_{2} = \frac{3 W _{1} ^{2}}{4 v ^{5}} - \frac{W _{2}}{2 v ^{3}} .$
unlabeled · line 732
$g_{λ}^{μν} = g_{0}^{μν} - λ h^{μν} + \frac{λ ^{2}}{2} (2 h^{μα} g_{0, α β} h^{β ν} - k^{μν}) .$

Lemma: Full-tensor lapse coefficients

Displayed formulas in this passage

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