Kiefer C. Quantum gravity

206 QUANTIZATION OF BLACK HOLES
The temperature (7.11) refers to an observer at infinite distance from the hole.
For finite distance one has to modify this expression by a redshift factor. If the
observer is static, he is in a state of acceleration, and therefore the temperature
contains both the Hawking and the Unruh effect. Close to the horizon, only
the Unruh effect remains, and a freely falling observer does not experience any
temperature at all.
For the total luminosity of the black hole, one finds in the Schwarzschild case,
L = − dM dt
= 1
2π
∞∑
(2l +1)
l=0
∫ ∞
0
Γ ωl
dωω
e 2πκ−1 ω
− 1 . (7.12)
To obtain the expression for the Kerr–Newman black hole, one has to replace
2πκ −1 ω with 2πκ −1 (ω −mΩ H −qΦ), where m is the azimuthal quantum number
of the spherical harmonics. The spectrum thus involves a chemical potential. The
term Γ ωl —called ‘greybody factor’ because it encodes a deviation from the blackbody
spectrum—takes into account the fact that some of the particle modes are
back-scattered into the black hole by means of space–time curvature.
For the special case of the Schwarzschild metric, where κ =(4GM) −1 , (7.11)
becomes the expression (1.33). One can estimate the life-time of such a black
hole by making the plausible assumption that the decrease in mass is equal to
the energy radiated to infinity. This corresponds to a heuristic implementation
of the back reaction of Hawking radiation on the black hole. (Without this back
reaction, energy would not be conserved.) Using Stefan–Boltzmann’s law, one
gets
dM
dt
which when integrated yields
( ) 4 1
∝−ATBH 4 ∝−M 2 × = − 1
M M 2 ,
t(M) ∝ (M 3 0 − M 3 ) ≈ M 3 0 . (7.13)
Here M 0 is the initial mass. It has been assumed that after the evaporation
M ≪ M 0 . Very roughly, the life-time of a black hole is thus given by
( ) 3 ( ) 3 M0
τ BH ≈ t P ≈ 10 65 M0
years . (7.14)
m P M ⊙
Hawking used the semiclassical approximation in which the non-gravitational
fields are quantum but the gravitational field is treated as an external classical
field; cf. Section 5.4. This approximation is expected to break down when the
black-hole mass approaches the Planck mass, that is, after a time given by (7.14).
Only a full theory of quantum gravity can describe the final stage of black-hole
evaporation. This would then necessarily include the full back-reaction effect of
the Hawking radiation on the black hole.
The original derivation of the Hawking effect deals with non-local ‘particle’
modes (Hawking 1975). Alternatively, one can base the analysis solely on the