Manuscript En02-E@2 · En02-E-v2.tex · section · c2769986

Specified stationary radiation model

Specified stationary radiation model · lines 256–291

En02-E@2#eq:radial

Fix $M_{*} > 0$ , $r_{h} = 2 G M_{*}$ , and one real angular harmonic of degree $ℓ$ . The exterior metric and tortoise coordinate are

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

The Killing field $\partial_{t}$ is normalized at infinity. For the massless minimally coupled scalar, the radial equation is

q^{''} (x) + (ω^{2} - V_{ℓ} (x)) q (x) = 0, V_{ℓ} (x) = F (r (x)) (\frac{ℓ ( ℓ + 1 )}{r ( x ) ^{2}} + \frac{r _{h}}{r ( x ) ^{3}}) .

Surface gravity and inverse horizon temperature are

κ = \frac{1}{2} F^{'} (r_{h}) = \frac{1}{2 r _{h}}, β = \frac{2 π}{κ} = 8 π G M_{*} .

The state is the quasifree product prescription with the affine vacuum on the complete past horizon and the Killing-frequency vacuum on past null infinity. On the exterior half-horizon,

u = t - x, U = - κ^{- 1} e^{- κ u} .

The bulk-to-boundary construction and the Hadamard property on the stated Schwarzschild domain are external inputs DMP. They are not a finite-core construction.

Displayed formulas in this passage

En02-E@2#eq:radial · line 270
$q^{''} (x) + (ω^{2} - V_{ℓ} (x)) q (x) = 0, V_{ℓ} (x) = F (r (x)) (\frac{ℓ ( ℓ + 1 )}{r ( x ) ^{2}} + \frac{r _{h}}{r ( x ) ^{3}}) .$
unlabeled · line 261
$g = - F (r) d t^{2} + F (r)^{- 1} d r^{2} + r^{2} d Ω^{2}, F (r) = 1 - \frac{r _{h}}{r}, x = r + r_{h} lo g (r / r_{h} - 1) .$
unlabeled · line 277
$κ = \frac{1}{2} F^{'} (r_{h}) = \frac{1}{2 r _{h}}, β = \frac{2 π}{κ} = 8 π G M_{*} .$
unlabeled · line 285
$u = t - x, U = - κ^{- 1} e^{- κ u} .$