Manuscript Hawk02-A@1 · Hawk02-A.tex · section · 7fbae745

Geometry, field, state, and normalization

Stage 1: the fixed-background comparison object › Geometry, field, state, and normalization · lines 337–371

Hawk02-A@1#eq:schwarzschild, Hawk02-A@1#eq:radial

The comparison object is a massless, minimally coupled real scalar on the Schwarzschild exterior,

d s^{2} = - F (r) d t^{2} + F (r)^{- 1} d r^{2} + r^{2} d Ω^{2}, F (r) = 1 - \frac{2 GM}{r},

with $t$ normalized at asymptotic infinity. The state is Unruh-type: no incoming occupation from past null infinity and the specified affine-vacuum horizon boundary condition.

For this comparison,

κ (M) = \frac{1}{4 GM}, β_{H} (M) = 8 π GM .

The scalar radial problem is

(\frac{d ^{2}}{d r _{*}^{2}} + ω^{2} - V_{ℓ} (r)) χ = 0, V_{ℓ} (r) = F (r) (\frac{ℓ ( ℓ + 1 )}{r ^{2}} + \frac{2 GM}{r ^{3}}),

with $d r_{*} / d r = F^{- 1}$ . Its transmission probability is $Γ_{ℓ} (ω; M)$ .

The fixed-background Hawking comparison and emission-rate calculations are supplied by Hawking and Page Hawking1975,Page1976. The particular boundary-current and packet conventions used in the corpus are recorded in Em03-A CheungEm03A ({thm:main}).

Displayed formulas in this passage

Hawk02-A@1#eq:schwarzschild · line 341
$d s^{2} = - F (r) d t^{2} + F (r)^{- 1} d r^{2} + r^{2} d Ω^{2}, F (r) = 1 - \frac{2 GM}{r},$
Hawk02-A@1#eq:radial · line 356
$(\frac{d ^{2}}{d r _{*}^{2}} + ω^{2} - V_{ℓ} (r)) χ = 0, V_{ℓ} (r) = F (r) (\frac{ℓ ( ℓ + 1 )}{r ^{2}} + \frac{2 GM}{r ^{3}}),$
unlabeled · line 351
$κ (M) = \frac{1}{4 GM}, β_{H} (M) = 8 π GM .$

Geometry, field, state, and normalization

Displayed formulas in this passage

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