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

The unsmeared scalar lapse

Complete classical lapse expansion › The unsmeared scalar lapse · lines 680–704

Em02-E@1#eq:lapse

Define

W_{λ} = - g_{λ}^{μν} \partial_{μ} ϕ_{λ} \partial_{ν} ϕ_{λ}, N_{ϕ, λ} = W_{λ}^{- 1/2} .

The timelike-clock branch has $W_{0} = v^{2} > 0$ .

In this section, raising a spacetime index on $h, k$ uses $g_{0}$ , not the Euclidean spatial metric. Set

v_{μ} = \partial_{μ} ϕ_{0}, p_{μ} = \partial_{μ} φ, w_{μ} = \partial_{μ} ψ .

Write $W_{λ} = W_{0} + λ W_{1} + λ^{2} W_{2} /2$ .

Displayed formulas in this passage

Em02-E@1#eq:lapse · line 683
$W_{λ} = - g_{λ}^{μν} \partial_{μ} ϕ_{λ} \partial_{ν} ϕ_{λ}, N_{ϕ, λ} = W_{λ}^{- 1/2} .$
unlabeled · line 697
$v_{μ} = \partial_{μ} ϕ_{0}, p_{μ} = \partial_{μ} φ, w_{μ} = \partial_{μ} ψ .$