Feynman Path Integral Formulation

156 5 Semiclassical Gravity
conservation: the mass of the black hole M needs to be decreased suitably when the
virtual particle is emitted, thus leading to a non-zero real tunneling amplitude, which
can then be shown to agree with the original Hawking calculation.
The computation is most easily carried out in Painlevé coordinates for a static,
non-rotating black hole. The Painlevé line element (Painlevé, 1921; Gullstrand,
1922) reads
(
ds 2 = − 1 − 2MG ) √
2MG
dt 2 + 2 dt dr + dr 2 + r 2 dΩ2 2 . (5.66)
r
r
The corresponding metric describes the same physics, but has several attractive features
when compared to the Schwarzschild metric: none of the metric (or inverse
metric) components diverge on the horizon r = 2MG; furthermore it still covers the
inside and outside of the black hole, and constant time slices simply correspond to
flat Euclidean space. One can show that the Painlevé time is related to the original
Schwarzschild time t s by
t = t s + 2 √ √ √
r − 2MG
2MGr + 2MGln √ √ . (5.67)
r + 2MG
From dτ/dt = 1 it follows that in these coordinates the time t is linearly related to
proper time, τ = t + c, for a radially infalling observer.
In this metric the radial null geodesics have a rather simple form,
dr
dt
√
2MG
= ±1 − , (5.68)
r
where the choice of signs depends on whether the rays go towards infinity (+),
or away from it (−). One can view the above geodesic equation as arising from
a classical mechanics effective potential V ef f (r) = √ 2MG/r − GM/r, shownin
Fig. (5.1), with a total energy fixed at 1/2. Note that the maximum of this function
is precisely at r = 2GM, and that the peak is at the total energy value 1/2, which
seems to make the two classical turning points coincide with the peak.
The fact that the coordinate system is stationary and non-singular allows one to
define what is meant by a vacuum: a state whose quantum fields will annihilate
modes which carry negative frequency with respect to the Painlevé time t. Butit
is important to note that modifications arise when the particle’s self-gravitation is
taken into account, and which are crucial in obtaining the correct result. For a nonrotating
self-gravitating shell of energy E (visualized as an s-wave state) one can
show (Kraus and Wilczek, 1995) that the shell moves on a geodesic still described
by the line element in Eq. (5.66), but with mass M → M + E, whereas if the total
mass is fixed and the black hole mass is allowed to vary, then the shell moves on a
geodesic with mass M → M − E.
In order to compute a tunneling amplitude one would like to use the semiclassical
or WKB approximation, which assumes point particles. One might worry
that a point particle description might not be adequate since the wavelengths