Manuscript Hawk01-A@1 · Hawk01-A.tex · section · 11979653

Geometry, field, and boundary state

The four-dimensional semiclassical source › Geometry, field, and boundary state · lines 238–266

Hawk01-A@1#eq:radial, Hawk01-A@1#eq:greybody

This is a stationary boundary-state prescription of the kind used to isolate the late-time source term Unruh1976,CheungEm03A. Equation (eq:affine) is exact in this prescribed stationary model. It is not asserted to be an exact affine map throughout an arbitrary collapse spacetime. A collapse application requires a separate peeling estimate and transient error control.

After spherical-harmonic decomposition, the radial equation is

[- \frac{d ^{2}}{d r _{*}^{2}} + V_{ℓ} (r)] f_{ℓ ω} (r_{*}) = ω^{2} f_{ℓ ω} (r_{*}), V_{ℓ} (r) = F (r) [\frac{ℓ ( ℓ + 1 )}{r ^{2}} + \frac{2 M}{r ^{3}}] .

Let $t_{ℓ} (ω)$ be the horizon-to-infinity transmission amplitude with unit incident flux, and define

Γ_{ℓ} (ω; M) = ∣ t_{ℓ} (ω) ∣^{2} .

Displayed formulas in this passage

Hawk01-A@1#eq:radial · line 248
$[- \frac{d ^{2}}{d r _{*}^{2}} + V_{ℓ} (r)] f_{ℓ ω} (r_{*}) = ω^{2} f_{ℓ ω} (r_{*}), V_{ℓ} (r) = F (r) [\frac{ℓ ( ℓ + 1 )}{r ^{2}} + \frac{2 M}{r ^{3}}] .$
Hawk01-A@1#eq:greybody · line 262
$Γ_{ℓ} (ω; M) = ∣ t_{ℓ} (ω) ∣^{2} .$

Geometry, field, and boundary state

Displayed formulas in this passage

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