String Theory Demystified

240 String Theory Demystifi ed
of thermodynamics. Black holes have entropy and temperature, and the laws of
thermodynamics have analogs that Hawking and his colleagues dubbed the laws of
black hole mechanics.
One of the most dramatic results of Hawking’s work was the implication that
black holes are associated with information loss. Physically speaking, we can
associate information with pure states in quantum mechanics. In ordinary quantum
physics, it is not possible for a pure quantum state to evolve into a mixed state. This
is related to the unitary nature of time evolution. What Hawking found was that
pure quantum states evolved into mixed states. This is because the character of the
radiation emitted by a black hole is thermal, it’s purely random—so a pure state that
falls into the black hole is emitted as a mixed state. The implication is that perhaps
a quantum theory of gravity would drastically alter quantum theory to allow for
nonunitary evolution. This is bad because nonunitary transformations do not
preserve probabilities. Either black holes destroy quantum mechanics or we have
not included an aspect of the analysis that would maintain the missing information
required to keep pure states evolving into pure states.
However, it is important to realize that the analysis done by Hawking and others
in this context was done using semiclassical methods. That is, a classical spacetime
background with quantum fi elds was studied. Given this fact, the results can’t
necessarily be trusted.
String theory is a fully quantum theory so evolution is unitary. And it turns out
that the application of string theory to black hole physics has produced one of the
theories most dramatic results to date. Using string theory, it is possible to count
the microscopic states of a black hole and compare this to the result obtained using the
laws of black hole mechanics (which state that entropy is proportional to area). It is
found that there is an exact agreement using the two methods. This is a spectacular
result in favor of string theory.
In this chapter we will quickly review the study of black holes in general relativity,
state the laws of black hole mechanics, and then illustrate the entropy calculation
using string theory.
Black Holes in General Relativity
The existence of black holes is predicted by Einstein’s theory of general relativity.
Readers interested in a detailed description of black holes in this context may want
to consult Relativity Demystifi ed.
The Einstein fi eld equations are a set of differential equations which relate the
curvature of space-time to the matter-energy content as follows:
1
Rµν − gµν R+ gµν Λ = 8πG T
(14.1)
4 µν
2