Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Action and background

Identical frozen background, operator, and rods › Action and background · lines 251–304

XX-C@1#eq:action, XX-C@1#eq:background

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, M > 0.

Only compactly supported variations of the local action density are required.

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

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

The spacetime is

M = I \times R^{3}, I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.

No endpoint boundary condition is imposed on this open slab.

Writing $H = a ˙ / a = p / t$ and $v_{μ} = \partial_{μ} ϕ ˉ$ , the background equations are

3 M^{2} H^{2} = \frac{1}{2} ϕ ˉ ˙^{2} + V (ϕ ˉ), - 2 M^{2} H ˙ = ϕ ˉ ˙^{2}, □_{g ˉ} ϕ ˉ - V^{'} (ϕ ˉ) = 0.

They follow directly from $A^{2} = 2 p M^{2}$ .

Each $t = constant$ surface is Cauchy. Indeed, along a causal curve

∣ d x / d t ∣ \leq a (t)^{- 1},

so a causal curve with a time endpoint inside $I$ has a finite spatial endpoint and is extendible. The background is therefore globally hyperbolic.

Displayed formulas in this passage

XX-C@1#eq:action · line 254
$S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x, M > 0.$
XX-C@1#eq:background · line 267
$A g ˉ = 2 p M, = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, a (t) V (ϕ) = (t / t_{*})^{p}, = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} e^{- 2 (ϕ - ϕ_{*}) / A} . ϕ ˉ (t) = ϕ_{*} + A lo g (t / t_{*}),$
unlabeled · line 280
$M = I \times R^{3}, I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.$
unlabeled · line 289
$3 M^{2} H^{2} = \frac{1}{2} ϕ ˉ ˙^{2} + V (ϕ ˉ), - 2 M^{2} H ˙ = ϕ ˉ ˙^{2}, □_{g ˉ} ϕ ˉ - V^{'} (ϕ ˉ) = 0.$
unlabeled · line 298
$∣ d x / d t ∣ \leq a (t)^{- 1},$