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

Perturbations, clock, and lapse

The complete background and classical prescription › Perturbations, clock, and lapse · lines 298–347

XX-B@1#eq:jet, XX-B@1#eq:clock, XX-B@1#eq:raw-lapse, XX-B@1#eq:clock-jet, XX-B@1#eq:theta

Retain the factorial convention

g_{s} = g ˉ + s h + \frac{s ^{2}}{2} k, ϕ_{s} = ϕ ˉ + s φ + \frac{s ^{2}}{2} ψ (mod s^{3}) .

The classical coefficient fields are smooth and have a common compact spatial support on each compact time interval used in the observable calculation. They need not solve the perturbed Euler equations.

The field-dependent cosmic label is

T_{s} = t_{*} exp (\frac{ϕ _{s} - ϕ _{*}}{A}) .

The raw matter-scalar lapse, on the positive branch, is

N_{s} = (- g_{s}^{μν} \partial_{μ} ϕ_{s} \partial_{ν} ϕ_{s})^{- 1/2} .

It is not silently replaced by the lapse calibrated to $T_{s}$ . The two normalizations satisfy

N_{s} = (T_{s} / A) (- g_{s}^{- 1} (d T_{s}, d T_{s}))^{- 1/2} .

Let $t = θ_{s} (τ, x)$ invert the clock at fixed spatial coordinate. Then

T_{s} θ_{s} = t + sχ + \frac{s ^{2}}{2} ζ, = τ + s u_{1} + \frac{s ^{2}}{2} u_{2}, χ u_{1} = \frac{t}{A} φ, = - χ, ζ u_{2} = \frac{t}{A} ψ + \frac{t}{A ^{2}} φ^{2}, = 2 χ χ ˙ - ζ .

All inverse coefficients are evaluated at $t = τ$ .

Displayed formulas in this passage

XX-B@1#eq:jet · line 301
$g_{s} = g ˉ + s h + \frac{s ^{2}}{2} k, ϕ_{s} = ϕ ˉ + s φ + \frac{s ^{2}}{2} ψ (mod s^{3}) .$
XX-B@1#eq:clock · line 313
$T_{s} = t_{*} exp (\frac{ϕ _{s} - ϕ _{*}}{A}) .$
XX-B@1#eq:raw-lapse · line 318
$N_{s} = (- g_{s}^{μν} \partial_{μ} ϕ_{s} \partial_{ν} ϕ_{s})^{- 1/2} .$
XX-B@1#eq:clock-jet · line 332
$T_{s} θ_{s} = t + sχ + \frac{s ^{2}}{2} ζ, = τ + s u_{1} + \frac{s ^{2}}{2} u_{2}, χ u_{1} = \frac{t}{A} φ, = - χ, ζ u_{2} = \frac{t}{A} ψ + \frac{t}{A ^{2}} φ^{2}, = 2 χ χ ˙ - ζ .$
XX-B@1#eq:theta · line 332
$T_{s} θ_{s} = t + sχ + \frac{s ^{2}}{2} ζ, = τ + s u_{1} + \frac{s ^{2}}{2} u_{2}, χ u_{1} = \frac{t}{A} φ, = - χ, ζ u_{2} = \frac{t}{A} ψ + \frac{t}{A ^{2}} φ^{2}, = 2 χ χ ˙ - ζ .$
unlabeled · line 326
$N_{s} = (T_{s} / A) (- g_{s}^{- 1} (d T_{s}, d T_{s}))^{- 1/2} .$