Manuscript Em03-A@1 · Em03-A-v1.tex · section · 6d54627d

Geometry and field

Scope, evidence, and the declared model › Geometry and field · lines 165–221

Em03-A@1#eq:metric, Em03-A@1#eq:action, Em03-A@1#eq:tortoise

We use units

c = ℏ = k_{B} = 1, G > 0, M > 0.

The background is the right Schwarzschild exterior

E = R_{t} \times (r_{h}, \infty)_{r} \times S^{2}, r_{h} = 2 GM,

with metric

g = - F (r) d t^{2} + F (r)^{- 1} d r^{2} + r^{2} d Ω^{2}, F (r) = 1 - \frac{r _{h}}{r} .

The Killing vector is $χ = \partial_{t}$ , normalized by $g (χ, χ) \to - 1$ as $r \to \infty$ .

The field is a real, massless, minimally coupled scalar:

S [ϕ] = - \frac{1}{2} \int g^{ab} (\nabla_{a} ϕ) (\nabla_{b} ϕ) d vol_{g}, □_{g} ϕ = 0.

Schwarzschild is Ricci flat, so the massless conformally coupled and minimally coupled wave operators coincide on this background.

Let

x = r + r_{h} lo g (\frac{r}{r _{h}} - 1), u = t - x, v = t + x .

The additive constant in $x$ is immaterial. We use real orthonormal spherical harmonics

Y_{ℓ α}, ℓ \in N \cup {0}, α = 1, \dots, 2 ℓ + 1.

This avoids imposing an independent reality convention on complex $m$ -modes. Every result below is for one fixed $(ℓ, α)$ .

Displayed formulas in this passage

Em03-A@1#eq:metric · line 179
$g = - F (r) d t^{2} + F (r)^{- 1} d r^{2} + r^{2} d Ω^{2}, F (r) = 1 - \frac{r _{h}}{r} .$
Em03-A@1#eq:action · line 190
$S [ϕ] = - \frac{1}{2} \int g^{ab} (\nabla_{a} ϕ) (\nabla_{b} ϕ) d vol_{g}, □_{g} ϕ = 0.$
Em03-A@1#eq:tortoise · line 202
$x = r + r_{h} lo g (\frac{r}{r _{h}} - 1), u = t - x, v = t + x .$
unlabeled · line 168
$c = ℏ = k_{B} = 1, G > 0, M > 0.$
unlabeled · line 174
$E = R_{t} \times (r_{h}, \infty)_{r} \times S^{2}, r_{h} = 2 GM,$
unlabeled · line 212
$Y_{ℓ α}, ℓ \in N \cup {0}, α = 1, \dots, 2 ℓ + 1.$