7.4 Black Holes and Thermodynamics

Back

Since T = ~_

2_ is the black hole temperature, we can now rewrite the 1st law

of black hole mechanics as

dM = TdSBH+HdJ+_HdQ; (H;_H intensive, J;Q extensive)(7.69)

where

SBH =

A

4~

(7.70)

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is the black hole (or Beckenstein-Hawking) entropy.

Clearly, black hole evaporation via Hawking radiation will cause SBH to

decrease in violation of the 2nd law of black hole mechanics (derived on the

assumption of classical physics). But the entropy is

S = SBH + Sext (7.71)

where Sext is the entropy of matter in exterior spacetime. But because

the Hawking radiation is thermal, Sext increases with the result that S is a

non-decreasing function of time. This suggests:

Generalized 2nd Law of Thermodynamics

S = SBH +Sext is always a non-decreasing function of time (in any process).

This was _rst suggested by Beckenstein (without knowledge of the precise

form of SBH) on the grounds that the entropy in the exterior spacetime could

be decreased by throwing matter into a black hole. This would violate the

2nd law of thermodynamics unless the black hole is assigned an entropy.

7.4.1 The Information Problem

Taking Hawking radiation into account, a black hole that forms from gravitational

collapse will eventually evaporate, after which the spacetime has

no event horizon. This is depicted by the following CP diagram:

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_nal point of

evaporation of

black hole

=􀀀

H+

i+

i0

=+

_1

_2

_1 is a Cauchy surface for this spacetime, but _2 is not because its past

domain of dependence D􀀀 (_2) does not include the black hole region. Information

from _1 can propagate into the black hole region instead of to _2.

Thus it appears that information is `lost' into the black hole. This would

imply a non-unitary evolution from _1 to _2, and hence put QFT in curved

spacetime in conict with a basic principle of Q.M. However, from the point

of view of a static external observer, nothing actually passes through H+, so

maybe the information is not really lost. A complete calculation including

all back-reaction e_ects might resolve the issue, but even this is controversial

since some authors claim that the resolution requires an understanding of

the Planck scale physics. The point is that whereas QFT in curved spacetime

predicts Tloc ! 1 on the horizon of a black hole, this should not be

believed when kT reaches the Planck energy (~c=G)1=2 c2 because i) Quantum

Gravity e_ects cannot then be ignored and ii) this temperature is then

of the order maximum (Hagedorn) temperature in string theory.

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