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

Lemma: Partial-wave reduction

Geometry and exact exterior scattering › The radial equation · lines 461–512 · proof included or linked

Em03-A@1#lem:radial, Em03-A@1#eq:radialwave, Em03-A@1#eq:potential, Em03-A@1#eq:potentialL1

Writing

ϕ (t, r, Ω) = \frac{1}{r} ℓ, α \sum ψ_{ℓ α} (t, x) Y_{ℓ α} (Ω)

reduces $□_{g} ϕ = 0$ to

(- \partial_{t}^{2} + \partial_{x}^{2} - V_{ℓ} (x)) ψ_{ℓ α} = 0,

where

V_{ℓ} (x) = F (r (x)) (\frac{ℓ ( ℓ + 1 )}{r ( x ) ^{2}} + \frac{r _{h}}{r ( x ) ^{3}}) .

The potential is nonnegative, smooth, and integrable:

\int_{R} V_{ℓ} (x) d x = \frac{ℓ ( ℓ + 1 )}{r _{h}} + \frac{1}{2 r _{h}} .

Proof. In Schwarzschild coordinates,

□_{g} ϕ = - F^{- 1} \partial_{t}^{2} ϕ + r^{- 2} \partial_{r} (r^{2} F \partial_{r} ϕ) + r^{- 2} Δ_{S^{2}} ϕ .

Insert $r^{- 1} ψ Y_{ℓ α}$ , use $Δ_{S^{2}} Y_{ℓ α} = - ℓ (ℓ + 1) Y_{ℓ α}$ , and use $\partial_{x} = F \partial_{r}$ . Multiplication by $r F$ gives (eq:radialwave).

All terms in (eq:potential) are nonnegative. Since $d x = F^{- 1} d r$ ,

\int_{R} V_{ℓ} (x) d x = \int_{r_{h}}^{\infty} (\frac{ℓ ( ℓ + 1 )}{r ^{2}} + \frac{r _{h}}{r ^{3}}) d r,

which evaluates to (eq:potentialL1).

Displayed formulas in this passage

Em03-A@1#eq:radialwave · line 470
$(- \partial_{t}^{2} + \partial_{x}^{2} - V_{ℓ} (x)) ψ_{ℓ α} = 0,$
Em03-A@1#eq:potential · line 476
$V_{ℓ} (x) = F (r (x)) (\frac{ℓ ( ℓ + 1 )}{r ( x ) ^{2}} + \frac{r _{h}}{r ( x ) ^{3}}) .$
Em03-A@1#eq:potentialL1 · line 484
$\int_{R} V_{ℓ} (x) d x = \frac{ℓ ( ℓ + 1 )}{r _{h}} + \frac{1}{2 r _{h}} .$
unlabeled · line 464
$ϕ (t, r, Ω) = \frac{1}{r} ℓ, α \sum ψ_{ℓ α} (t, x) Y_{ℓ α} (Ω)$
unlabeled · line 493
$□_{g} ϕ = - F^{- 1} \partial_{t}^{2} ϕ + r^{- 2} \partial_{r} (r^{2} F \partial_{r} ϕ) + r^{- 2} Δ_{S^{2}} ϕ .$
unlabeled · line 506
$\int_{R} V_{ℓ} (x) d x = \int_{r_{h}}^{\infty} (\frac{ℓ ( ℓ + 1 )}{r ^{2}} + \frac{r _{h}}{r ^{3}}) d r,$