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 201–223

Hawk01-A@1#eq:metric, Hawk01-A@1#eq:coordinates

For $M > 0$ , consider the exterior $r > 2 M$ with metric

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

Set

r_{*} = r + 2 M lo g (\frac{r}{2 M} - 1), u = t - r_{*}, κ = \frac{1}{4 M}, β_{H} = \frac{2 π}{κ} = 8 π M .

The field in the proved source calculation is a real, free, massless, minimally coupled scalar satisfying $□_{g} ϕ = 0$ .

Displayed formulas in this passage

Hawk01-A@1#eq:metric · line 204
$d s^{2} = - F (r) d t^{2} + F (r)^{- 1} d r^{2} + r^{2} d Ω^{2}, F (r) = 1 - \frac{2 M}{r} .$
Hawk01-A@1#eq:coordinates · line 211
$r_{*} = r + 2 M lo g (\frac{r}{2 M} - 1), u = t - r_{*}, κ = \frac{1}{4 M}, β_{H} = \frac{2 π}{κ} = 8 π M .$