Feynman Path Integral Formulation

5.4 Black Hole Particle Emission 157
involved could be comparable to the black hole size, but this is not so due to the
blueshift of frequencies in the vicinity of the horizon. In the following it will therefore
be assumed that such particles can be described by point-like objects.
In the WKB approximation, the imaginary part of the amplitude for an s-wave
outgoing positive energy particle which crosses the horizon outward from r in to r out
is given by
∫ rout
∫ rout
∫ pr
ImS = Im p r dr = Im dp ′ r dr , (5.69)
r in
r in 0
where the actual emission rate is the square of the amplitude, Γ ∼ exp(−2ImS).
Using Hamilton’s equation for the classical trajectory ˙q = ∂H/∂ p, here in the form
dp r =
( ) dr −1
dH , (5.70)
dt
with H = M −E and thus dH = −dE, and inserting the radial geodesic dr/dt given
by Eq. (5.68), one obtains
∫ rout
∫ E
ImS = −Im
r in 0
1 −
drdE ′
√
2G(M −E ′ )
r
. (5.71)
The r-integral can now be done by residues, first by transforming to the variable
z = √ r, and then by adding a Feynman iε to the energy, which slightly displaces the
pole to the upper half-plane,
∫ E ∫ zout
ImS = −Im dE ′
0 z in
2z 2 dz
z − √ 2G(M − E ′ + iε) . (5.72)
After closing the contour in the upper half plane and keeping only the imaginary
part of the amplitude (the real part contributes an irrelevant phase) one has
Fig. 5.1 The effective potential
V ef f (r), obtained from the
geodesic equation in Painlevé
coordinates (here shown for
G = M = 1). The maximum
occurs on the horizon
r = 2MG.