String Theory and M-Theory

11.2 Black-hole thermodynamics 565
The AdS/CFT conjecture, described in Chapter 12, certainly would appear
to contradict this reasoning, since the AdS space in which black holes
can form is dual to a unitary conformal field theory. Thus string solutions,
at least ones that are asymptotically AdS, probably provide counterexamples
to Hawking’s claim. That said, it should be admitted that it is an
extremely subtle matter to explain in detail where Hawking’s argument for
information loss breaks down. This question has been discussed extensively
in the literature, but it is not yet completely settled.
EXERCISE 11.4
Show that the temperature of a D = 4 Reissner–Nordström black hole is
(MG4)
T =
2 − Q2G4 .
2πr 2 +
What happens to this temperature in the extremal limit?
SOLUTION
Using the same reasoning as in Section 11.2, we set r = r+(1+ρ2 ) and expand
the Euclideanized metric of the Reissner–Nordström black hole about ρ = 0
to get
ds 2 = 4r3 +
r+ − r−
dρ 2 + ρ 2
(r+ − r−)dτ
2r 2 +
2
The value of β that follows from this expression is
which leads to a temperature
T = r+ − r−
4πr 2 +
β = 4πr2 +
,
r+ − r−
(MG4)
=
2 − Q2G4 .
2πr 2 +
+ r+ − r−
dΩ
4r+
2
In the extremal limit, M √ G4 = |Q|, this gives a vanishing temperature
T = 0. ✷
EXERCISE 11.5
Estimate the Schwarzschild radius, temperature, and entropy of a one solar
mass Schwarzschild black hole. Estimate its lifetime due to the emission of
Hawking radiation. The sun has a mass of M = 2.0 × 10 33 g.
.