Feynman Path Integral Formulation

154 5 Semiclassical Gravity
essentially non-perturbative, and non-analytic in the external field perturbation E.
Furthermore, a comparison of the exact answer in Eq. (5.61) with the WKB result
of Eq. (5.54) shows that the latter only gives the leading term in an infinite series of
progressively smaller contribution.
The result obtained for electrons and positrons in strong uniform electric fields
are clearly not transferable as is to the gravitational case. For once, there is no notion
of oppositely charged particles in gravity. Thus the naive replacement |eE|→
m/4MG for a particle-antiparticle pair, say close to the horizon of a black hole, does
not seem to make much sense. Yet the strong electric field QED calculation shows
that quantum mechanical tunneling events can take place, and that their effects can
be computed to first order using the WKB approximation. A second lesson from
QED is that in general higher order corrections should be expected.
5.4 Black Hole Particle Emission
Originally quantum gravitational effects for black holes were ignored, since the radius
of curvature outside the black hole is much larger than the Planck length, the
length scale on which one would expect quantum fluctuations of the metric to become
important. If the gravitational field is able to create locally virtual pairs, the
local energy density associated with such a pair would be much smaller than the
energy scale associated with the local curvature. It can be shown though that in the
vicinity of a black hole horizon particle production is possible, due to vacuum fluctuations
and tunneling. The resulting effects add up over time, are therefore macroscopic
and could in principle be observable.
Normally when describing a stationary non-rotating black hole one uses the
Schwarzschild metric in standard form
ds 2 = −
(
1 − 2GM
r
)
dt 2 +
(
1 − 2MG
r
) −1
dr 2 + r 2 dΩ 2 2 , (5.62)
with dΩ2 2 ≡ dθ 2 + sin 2 θdφ 2 . The metric shows a singularity at r = 2MG. Since
none of the curvature invariants are singular on the horizon r = 2MG, one would
expect the singularity to be perhaps an artifact of the coordinate system, here most
suitably describing the viewpoint of an observer stationary at infinity. On the other
hand a freely falling observer is expected to pass initially unscathed through the
black hole horizon, and indeed a singularity-free coordinate system can be found describing
such an observer (Kruskal, 1960). In these new coordinates, defined by the
transformation from the original Schwarzschild coordinates (r,θ,φ,t) → (r ′ ,θ,φ,t ′ )
(
r ′ 2 − t
′2 = T
2 r
) ( r
)
2GM − 1 exp
2GM
2r ′ t ′ ( t
)
r ′2 + t ′ 2 = tanh , (5.63)
2GM