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

Action and on-shell background

The complete background and classical prescription › Action and on-shell background · lines 221–277

XX-B@1#eq:action, XX-B@1#eq:background, XX-B@1#eq:metric, XX-B@1#eq:potential

Use signature $(-, +, +, +)$ , a reduced Planck mass $M > 0$ , 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 .

On noncompact slices, this notation specifies the local action density and its compactly supported variations. No finite total background action is assumed.

Fix $p > 1$ , $t_{*} > 0$ , $ϕ_{*} \in R$ , and

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

The spacetime is

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

There are no timelike or spacelike boundary conditions at the missing endpoints $t_{\pm}$ . The interval is open.

Write

E_{μν} = M^{2} G_{μν} - T_{μν}, E_{ϕ} = □_{g} ϕ - V^{'} (ϕ) .

The background solves $E_{μν} = E_{ϕ} = 0$ :

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

The scalar equation follows from $A^{2} = 2 p M^{2}$ . Also

- g ˉ^{- 1} (d ϕ ˉ, d ϕ ˉ) = A^{2} / t^{2} > 0.

Displayed formulas in this passage

XX-B@1#eq:action · line 224
$S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x .$
XX-B@1#eq:background · line 237
$A g ˉ V (ϕ) = 2 p M, = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} exp (- \frac{2 ( ϕ - ϕ _{*} )}{A}) . a (t) = (t / t_{*})^{p}, ϕ ˉ (t) = ϕ_{*} + A lo g (t / t_{*}),$
XX-B@1#eq:metric · line 237
$A g ˉ V (ϕ) = 2 p M, = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} exp (- \frac{2 ( ϕ - ϕ _{*} )}{A}) . a (t) = (t / t_{*})^{p}, ϕ ˉ (t) = ϕ_{*} + A lo g (t / t_{*}),$
XX-B@1#eq:potential · line 237
$A g ˉ V (ϕ) = 2 p M, = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} exp (- \frac{2 ( ϕ - ϕ _{*} )}{A}) . a (t) = (t / t_{*})^{p}, ϕ ˉ (t) = ϕ_{*} + A lo g (t / t_{*}),$
unlabeled · line 251
$M = I \times R^{3}, I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.$
unlabeled · line 259
$E_{μν} = M^{2} G_{μν} - T_{μν}, E_{ϕ} = □_{g} ϕ - V^{'} (ϕ) .$
unlabeled · line 265
$H = \frac{p}{t}, ϕ ˉ ˙ = \frac{A}{t}, 3 M^{2} H^{2} = \frac{1}{2} ϕ ˉ ˙^{2} + V (ϕ ˉ), - 2 M^{2} H ˙ = ϕ ˉ ˙^{2} .$
unlabeled · line 274
$- g ˉ^{- 1} (d ϕ ˉ, d ϕ ˉ) = A^{2} / t^{2} > 0.$

Nearby in this section

lemma: Lemma: Global hyperbolicity of the open slab