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#### 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 that system

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 system?
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 very different.
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 (in units 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.

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