Chapter 10
Black holes in string theory
10.1 Introduction
Besides 'predicting' general relativity as the effective low-energy (classical)
theory of gravitation, string theory also provides a (perturbative) quantum theory
of gravity. Since it is the only such theory known, one might hope that string
theory would offer insights into the quantum aspects of gravitation that are not
available elsewhere. This is indeed the case.
In 1971, Hawking [I] showed that area of the event horizon of a black
hole must increase with time. This prompted the observation that the area
of the event horizon is analogous to the entropy of a thermodynamic system.
It was subsequently shown that quantum mechanics requires that black holes
besides absorbing radiation must also emit it and that black holes are indeed
thermodynamic systems. Specifically, it was shown by Bekenstein [2] and
Hawking [3] that the entropy is proportional to the area of the event horizon.
Our experience with statistical mechanics as the microscopic theory underlying
thermodynamics leads us to expect that this entropy is associated with the
number of microstates of the (black-hole) system. It is precisely this aspect
that is illuminated by string theory. As we shall see, for certain black-hole
solutions of string theory, the number of microstates can be calculated, and the
resulting multiplicity reproduces precisely the Bekenstein-Hawking formula for
the entropy.
In the next section, we review the definition of the black-hole event horizon
and outline the proof that the area of its two-dimensional section cannot decrease.
In section 10.3, we show why quantum mechanics requires black holes to have a
temperature that is determined by their 'surface gravity' and entropy proportional
to the area of the (two-dimensional section of the) event horizon. The number of
perturbative microstates in string theory is evaluated in section 10.4 and shown to
be quite inadequate to explain the derived entropy of black holes. The special
class of black holes for which string theory is able to provide a microscopic
explanation of their entropy are (certain) 'extreme' black holes, which have both
DOl: 10.1201/9780367806637-10
275
Black holes in string theory
10.1 Introduction
Besides 'predicting' general relativity as the effective low-energy (classical)
theory of gravitation, string theory also provides a (perturbative) quantum theory
of gravity. Since it is the only such theory known, one might hope that string
theory would offer insights into the quantum aspects of gravitation that are not
available elsewhere. This is indeed the case.
In 1971, Hawking [I] showed that area of the event horizon of a black
hole must increase with time. This prompted the observation that the area
of the event horizon is analogous to the entropy of a thermodynamic system.
It was subsequently shown that quantum mechanics requires that black holes
besides absorbing radiation must also emit it and that black holes are indeed
thermodynamic systems. Specifically, it was shown by Bekenstein [2] and
Hawking [3] that the entropy is proportional to the area of the event horizon.
Our experience with statistical mechanics as the microscopic theory underlying
thermodynamics leads us to expect that this entropy is associated with the
number of microstates of the (black-hole) system. It is precisely this aspect
that is illuminated by string theory. As we shall see, for certain black-hole
solutions of string theory, the number of microstates can be calculated, and the
resulting multiplicity reproduces precisely the Bekenstein-Hawking formula for
the entropy.
In the next section, we review the definition of the black-hole event horizon
and outline the proof that the area of its two-dimensional section cannot decrease.
In section 10.3, we show why quantum mechanics requires black holes to have a
temperature that is determined by their 'surface gravity' and entropy proportional
to the area of the (two-dimensional section of the) event horizon. The number of
perturbative microstates in string theory is evaluated in section 10.4 and shown to
be quite inadequate to explain the derived entropy of black holes. The special
class of black holes for which string theory is able to provide a microscopic
explanation of their entropy are (certain) 'extreme' black holes, which have both
DOl: 10.1201/9780367806637-10
275