Manuscript XX-E@1 · XX-E.tex · section · cf7a8f40

Action, background, and rods

The identical freeze and a necessary smearing correction › Action, background, and rods · lines 266–371

XX-E@1#eq:action, XX-E@1#eq:background, XX-E@1#eq:jets, XX-E@1#eq:clocklapse, XX-E@1#eq:rods, XX-E@1#eq:u, XX-E@1#eq:C, XX-E@1#eq:R

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Use signature $(-, +, +, +)$ and the local action density

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 .

Only compact variations of this density are needed.

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

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

Work on $M = I \times R^{3}$ , with

I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.

Substitution of $A^{2} = 2 p M^{2}$ verifies the Einstein–scalar background equations. No limit toward $t = 0$ is taken.

Retain

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 and raw lapse are

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

on the positive timelike branch.

On the actual $T_{s} = t$ slice, let $γ_{s}$ be the induced metric. The original rods solve

\partial_{i} (det γ_{s} γ_{s}^{ij} \partial_{j} R_{s}^{a}) = 0, R_{s}^{a} - x^{a} ⟶ 0 (∣ x ∣ \to \infty) .

No lapse weight is inserted.

The inverse clock displacement is

u_{1} = - \frac{t}{A} φ, u_{2} = - \frac{t}{A} ψ + \frac{2 t ^{2}}{A ^{2}} φ φ ˙ + \frac{t}{A ^{2}} φ^{2} .

In particular, the first induced metric is

(γ_{1})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij} .

Its original rod-source map is

(\calCz)^{ia} = \frac{1}{2 a ^{2}} δ^{ia} δ^{j k} h_{j k} - a^{- 2} h_{ia} - \frac{p}{A} δ^{ia} φ .

Use the unchanged spatial Newton inverse

(G_{sp} b) (x) (\calRz)^{a} P_{i} (r) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, = - G_{sp} \partial_{i} (\calCz)^{ia} = \int P_{i} (x - y) (\calCz)^{ia} (t, y) d^{3} y, = - \frac{r _{i}}{4 π ∣ r ∣ ^{3}} . Δ G_{sp} b = b,

It is not replaced by a retarded spacetime inverse.

Displayed formulas in this passage

XX-E@1#eq:action · line 269
$S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x .$
XX-E@1#eq:background · line 280
$A ϕ ˉ (t) g ˉ = 2 p M, = ϕ_{*} + A lo g (t / t_{*}), = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, a (t) v (t) V (ϕ) = (t / t_{*})^{p}, = A / t, = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} e^{- 2 (ϕ - ϕ_{*}) / A} . H (t) = p / t,$
XX-E@1#eq:jets · line 306
$g_{s} = g ˉ + s h + \frac{s ^{2}}{2} k, ϕ_{s} = ϕ ˉ + s φ + \frac{s ^{2}}{2} ψ, z = (h, φ), ζ = (k, ψ) .$
XX-E@1#eq:clocklapse · line 313
$T_{s} = t_{*} e^{(ϕ_{s} - ϕ_{*}) / A}, N_{s} = (- g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s}))^{- 1/2}$
XX-E@1#eq:rods · line 323
$\partial_{i} (det γ_{s} γ_{s}^{ij} \partial_{j} R_{s}^{a}) = 0, R_{s}^{a} - x^{a} ⟶ 0 (∣ x ∣ \to \infty) .$
XX-E@1#eq:u · line 334
$u_{1} = - \frac{t}{A} φ, u_{2} = - \frac{t}{A} ψ + \frac{2 t ^{2}}{A ^{2}} φ φ ˙ + \frac{t}{A ^{2}} φ^{2} .$
XX-E@1#eq:C · line 346
$(\calCz)^{ia} = \frac{1}{2 a ^{2}} δ^{ia} δ^{j k} h_{j k} - a^{- 2} h_{ia} - \frac{p}{A} δ^{ia} φ .$
XX-E@1#eq:R · line 354
$(G_{sp} b) (x) (\calRz)^{a} P_{i} (r) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d^{3} y, = - G_{sp} \partial_{i} (\calCz)^{ia} = \int P_{i} (x - y) (\calCz)^{ia} (t, y) d^{3} y, = - \frac{r _{i}}{4 π ∣ r ∣ ^{3}} . Δ G_{sp} b = b,$
unlabeled · line 298
$I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.$
unlabeled · line 342
$(γ_{1})_{ij} = h_{ij} + 2 a^{2} H u_{1} δ_{ij} .$