What is black hole entropy?
Two important thermodynamic quantities are temperature
and entropy. Temperature is a familiar
quantity measured by direct personal experience. Entropy however is
a more mysterious quantity. It was discovered in an roundabout way,
when scientists noticed that in certain experiments with gases, there
was a constant ratio between the heat exchanged in the process, and
the temperature at which the process occurred. So entropy was discovered
by observing processes in which it was conserved.
But what is entropy, really? That answer only came
after the macroscopic thermodynamic properties of gases and fluids were
understood in terms of the quantum statistical behavior of their microscopic
constituents. Temperature was revealed to be calculable from the average
kinetic energy of a system of identical particles, and entropy was understood
in terms of the number of quantum states available to the particles
in that system.
If we have some system with some energy E, number
of particles N, being kept in a volume V, then the entropy
is proportional to the logarithm of the density of quantum states of
For an ideal gas, this quantity can be calculated
from basic quantum principles to be
The Bekenstein-Hawking entropy of a black hole
is one fourth of the area of the event horizon (in units
where Planck's constant=GN=1). This black hole entropy behaves
just like the entropy of a thermodynamic system should behave. But what
theory will allow us to derive macroscopic black hole entropy using
the density of states of some underlying microscopic quantum statistical
Until string theory, there was no clear idea
how this task could be accomplished. String theory has provided at least
a partial answer to this question in terms of D-branes.
Black holes and branes in string theory
A black hole is an object that is described
by a spacetime geometry that is a solution to the Einstein equation.
In string theory at large distance scales, solutions to the Einstein
equation are only modified by very small corrections. But it has been
discovered through string duality relations that spacetime
geometry is not a fundamental concept in string theory, and at
small distance scales or when the forces are very strong, there is an
alternate description of the same physical system that appears to be
Bearing that in mind, let's start with the simplest
charged black p-brane solution known, which is a charged black hole
in four spacetime dimensions, described by the metric
If the charge and mass are equal in magnitude
where c=GN=1) then we have an extreme black hole, with area
4pQ2, and therefore
with entropy pQ2.
This extreme black hole is a special object because when M=Q, a condition
for unbroken supersymmetry is satisfied that is called the BPS
condition. This BPS condition results in the cancellation
of quantum corrections to the effective action for string theory,
so that precise answers can be found by simple calculations at lowest
order in perturbation theory.
The above black hole can be thought of as a
zero-brane. In the previous section we learned that string theories
contain objects called p-branes and
D-branes. A natural generalization of
a black hole is a black p-brane. And
there are also BPS black p-branes.
Unfortunately, the string theory solution to
the black hole entropy problem cannot be easily illustrated for the
simple charged black hole above. The simplest example that can be calculated
features a system of a one-brane (i.e. a string) with charge Q1
lying parallel to a five-brane with charge Q5, with momentum
p5 in the finite fifth dimension which is proportional to
an integer n5.
The spacetime metric for this system is very
complicated and won't be reproduced here, but from the area of the extreme
object, one can derive the entropy
This is the macroscopic thermodynamic result.
Now how does string theory connect this to a microscopic density of
quantum states? We have to look to the relationship between black p-branes
and D-branes. This p-brane system has charges that match an equivalent
D-brane system. The critical parameter that interpolates between the
geometric limit and the D-brane description is the string coupling g
times the D-brane charge Q. At large values of gQ, spacetime geometry
is a good description of of a black p-brane system. But when gQ is much
smaller than one, the system can be described
by a bunch of weakly interacting D-branes.
In this weakly coupled D-brane limit, with the
BPS condition satisfied, it is possible to calculate the density of
available quantum states. This answer depends on the charges of the
D-branes in the system as follows
The entropy is just the logarithm of the density
of states, so from this we can see that the entropy of the microscopic
D-brane system matches the entropy as calculated from the macroscopic
event horizon area.
This was a fantastic result for string theory.
But can we now say that D-branes provide the fundamental quantum microstates
of a black hole that underlie black hole thermodynamics? The D-brane
calculation is only easily performed for the supersymmetric BPS black
objects. Most black holes in the Universe probably have very little
if any electric or magnetic charge, and are very far from being BPS
objects. It's still a challenge to compute the black hole entropy for
such an object using D-branes.