Manuscript XX-F@1 · XX-F.tex · section · de0bd24f

Background, clock, and output density

Frozen geometric and free-field data › Background, clock, and output density · lines 248–309

XX-F@1#eq:action, XX-F@1#eq:background, XX-F@1#eq:jet, XX-F@1#eq:density

Use signature $(-, +, +, +)$ and

S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x .

The notation means the local action density with compact variations; no finite total background action is asserted.

Fix $M > 0$ , $p > 1$ , $t_{*} > 0$ , and

A ϕ ˉ (t) V (ϕ) = 2 p M, = ϕ_{*} + A lo g (t / t_{*}), = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} e^{- 2 (ϕ - ϕ_{*}) / A} . a (t) g ˉ = (t / t_{*})^{p}, = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, H (t) = p / t,

Work on $I \times R^{3}$ , where $I = (t_{-}, t_{+})$ and $0 < t_{-} < t_{+} < \infty$ . The identities

ϕ ˉ ˙ = A / t, - 2 M^{2} H ˙ = ϕ ˉ ˙^{2}, 3 M^{2} H^{2} = \frac{1}{2} ϕ ˉ ˙^{2} + V (ϕ ˉ)

and $A^{2} = 2 p M^{2}$ verify the background equations.

The factorial jet is

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

The clock label and raw matter-scalar lapse remain

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

The positive classical timelike branch is used.

All observable smearings below use the fixed label density $d t d^{3} x$ . For comparison with XX-D's background-volume convention,

f_{D} = a^{- 3} f_{A}, \int f_{A} n d t d^{3} x = \int f_{D} n d V_{g ˉ} .

This identification changes test coordinates, not the geometric prescription.

Displayed formulas in this passage

XX-F@1#eq:action · line 251
$S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x .$
XX-F@1#eq:background · line 262
$A ϕ ˉ (t) V (ϕ) = 2 p M, = ϕ_{*} + A lo g (t / t_{*}), = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} e^{- 2 (ϕ - ϕ_{*}) / A} . a (t) g ˉ = (t / t_{*})^{p}, = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, H (t) = p / t,$
XX-F@1#eq:jet · line 285
$g_{s} = g ˉ + s h + \frac{s ^{2}}{2} k, ϕ_{s} = ϕ ˉ + s φ + \frac{s ^{2}}{2} ψ, z = (h, φ), ζ = (k, ψ) .$
XX-F@1#eq:density · line 301
$f_{D} = a^{- 3} f_{A}, \int f_{A} n d t d^{3} x = \int f_{D} n d V_{g ˉ} .$
unlabeled · line 277
$ϕ ˉ ˙ = A / t, - 2 M^{2} H ˙ = ϕ ˉ ˙^{2}, 3 M^{2} H^{2} = \frac{1}{2} ϕ ˉ ˙^{2} + V (ϕ ˉ)$
unlabeled · line 292
$T_{s} = t_{*} e^{(ϕ_{s} - ϕ_{*}) / A}, N_{s} = (- g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s}))^{- 1/2} .$