Feynman Path Integral Formulation

158 5 Semiclassical Gravity
∫ E
(
ImS = Im 2πi · 12 dE ′ 4G(M − E ′ )=4π GME 1 − E )
. (5.73)
0
2M
Note the sign change due to the fact that z i > z f . This last result, essential in getting
the right sign for the tunneling amplitude, can be visualized by noting that the outgoing
particle starts at r = 2MG− ε (which is just barely inside the initial location
of the horizon) and then traverses the contraction horizon (through the classically
forbidden region) to materialize at r = 2G(M − E ′ )+ε, which is just outside of the
final location for the horizon!
The quantum-mechanical tunneling rate is then given, in the WKB approximation,
by
{ (
Γ ∼ exp(−2ImS) =exp −8π GME 1 − E )}
. (5.74)
2M
When the quadratic correction in E is neglected (small energy change in the black
hole mass), one obtains the Boltzmann weight for a particle with energy E and
inverse Hawking temperature T −1 = 8πGM.
It is tempting to pursue for a while the thermodynamic analogy. Since one can
associate with the black hole a temperature T = 1/8πMG, one can also define an
entropy for it, using the thermodynamic relation dE = TdS, here with dE = dM.
This gives
S = 8πG
∫ M
M ′ dM ′ = 4πGM 2 ,
M 0
(5.75)
assuming the integration constant is zero (“a zero mass black hole has zero entropy”).
Note then that the expression in the exponent of Eq. (5.74) is precisely
the change in the Hawking-Beckenstein (Beckenstein, 1973; 1974; Hawking, 1976;
t’Hooft, 1985) black hole entropy,
ΔS = 4πGM 2 − 4πG(M − E) 2 = 8πG(ME − 1 2 E2 ) . (5.76)
One largely unresolved puzzle in the context of the semiclassical picture is: what
microstates are being counted when one assigns to the black hole an entropy S =
k logN.
5.5 Method of In and Out Vacua
The original derivation by Hawking of black hole radiance relies on a slightly different
set of arguments (Hawking, 1975). Here we will only outline the main steps
of the argument. One start by considering a massless real scalar field with wave
equation
g μν ∇ μ ∇ ν φ(x) =0 , (5.77)
where the covariant derivative ∇ μ is defined for an asymptotically flat spacetime,
describing initially the gravitationally collapsed object that gave origin to the black
hole. The quantum operator φ can be expanded