What is black hole entropy? Black holes and branes in string theory

Entropy describes the degradation of energy to perform work. What is energy? On the high school level, we simply defined energy as the capacity to do work. However, the real question is, What factor, precisely, is doing this work? Why does energy have the capacity to perform work? The HOWs in life are easy, the WHYs are the tough ones. Two important thermodynamic quantities are temperature and entropy. Temperature we all know from our fevers, weather reports and ovens. Entropy however is foreign to everyday life for most people.
Suppose we have a box filled with gas of some type of molecule called M. The temperature of that gas in that box tells us the average kinetic energy of those vibrating molecules of gas. Each molecule as a quantum particle has quantized energy states, and if we understand the quantum theory of those molecules, theorists can count up the available quantum microstates of those molecules and get some number. The entropy is the logarithm of that number.
When it was discovered that black holes can decay by quantum processes, it was also discovered that black holes seem to have the thermodynamic properties of temperature and entropy. The temperature of the black hole is inversely proportional to its mass, so the black hole gets hotter and hotter as it decays.
The entropy of a black hole is one fourth of the area of the event horizon, so the entropy gets smaller and smaller as the black hole decays and the event horizon area becomes smaller and smaller.
But until string theory there was not a clear relation between quantum microstates of a quantum theory and this supposed black hole entropy.

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.
A special type of black hole that is very important in string theory is called a BPS black hole. A BPS black hole has both charge (electric and/or magnetic) and mass, and the mass and the charges satisfy an equality that leads to unbroken supersymmetry in the spacetime near the black hole. This supersymmetry is very important because it results in the disappearance of messy quantum corrections, so that precise answers about the physics near the black hole horizon can be found by simple calculations.
In the previous section we learned that string theories contain objects called p-branes and D-branes. Since a point can be thought of as a zero-brane, a natural generalization of a black hole is a black p-brane. And there are also BPS black p-branes.
But there's also a relationship between black p-branes and D-branes. At large values of the charge, spacetime geometry is a good description of of a black p-brane system. But when the charge is small, 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 number of available quantum states. This answer depends on the charges of the D-branes in the system.
When we go back to the geometrical limit of the equivalent black hole of p-brane system with the same charges and masses, we find that the entropy of the D-brane system matches the entropy as calculated from the black hole or p-brane 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. This is an expanded version of a talk given at \IInd Recontre du Vietnam" held at Ho Chi Minh City in October, 1995. We discuss several aspects of black hole entropy in string theory. We rst explain why the geometric entropy in two dimensional noncritical string theory is nonperturbatively nite. We then explain the philosophy of regarding massive string states as black branes and how the BeckensteinHawking entropy for extremal BPS black holes may be understood as coming from degeneracy of string states. This is then discussed in the context of D-strings in Type IIB superstrings. We then describe
non-BPS excitations of D-strings and their entropy and explore the possibility that their decay describes Hawking radiation. For these D-strings and other D-branes the entropy and temperature are consequences of the physical motion of stuck open strings along the D-brane and this leads to a simple space-time interpretation. Finally we speculate that the horizon may be itself regarded as a D-brane. In this contribution I will discuss some aspects of black hole entropy and how string theory has helped us to understand this rather mysterious quantity. This is based on a talk given at IInd Recontre du Vietnam. However I have added some developments which took place very recently to make the article meaningful at this time. Even before the phenomenon of black hole radiation was discovered, Beckenstein noticed a profound analogy between black holes in classical general relativity and the laws of thermodynamics. In particular he found that the surface gravity at the horizon of a black hole can be regarded as a temperature and the area of the horizon as an entropy. Hawking showed that black holes radiate due to quantum eects with a temperature which is indeed proportional to the surface gravity and this gave a precise formula for the \Beckenstein- Hawking entropy"We now turn to the question of Beckenstein-Hawking entropy. It is an old idea that very massive elementary particles behave as black holes in strong coupling. Recently this idea has proved
to be very fruitful in understanding the Hawking-Beckenstein entropy in string theory. The idea is as follows. It is well known that in string theory there are very massive states and there
are large number of states of a given mass. Could it be that this degeneracy of states is the origin of the Beckenstein-Hawking entropy ?
However at the face of it, the idea seems to run into trouble for usual Schwarzschild black holes. Here the entropy S for mass M is S M2 whereas it is known that in string theory
the degeneracy of states grows as eM . It was argued in , that quantum effects might renormalize the mass suitably. Luckily there are states in string theory whose masses are not renormalized - these are the BPS staurated states. The above idea may be tested for such states which are to be identied
with extremal black holes . Indeed such states do behave like extremal black holes in scattering processes . However, most of these extremal holes have zero horizon area and should lead to a zero entropy ! In it was, however, proposed that the area of the streched horizon rather than the event horizon should be identied with the entropy. Indeed for a class of such BPS saturated extremal holes in heterotic string theory compactied on T 6 it was found that the dependence of the streched horizon area on the parameters sepcifying the solution agrees with the dependence of the degeneracy of string states on these parameters (which are generically charges) Recently this connection has been better understood owing to the realization that a certain
class of solitons in superstring theories can be described in terms of ob jects called D-branes Consider a theory of closed strings, e.g. the Type IIB superstring theory. Now add to this
some open strings whose ends are restricted to move only in p of the spatial dimensions. This means we have imposed Dirichelt boundary conditions on (9p) coordinates. The values ofthese (9p) coordinates then dene a p- dimensional ob ject moving in time - a p-brane. This p brane is a soliton in the closed string theory. The collective coordinates which describe the low energy excitations of this soliton are precisely the lowest mass modes of the open strings
whose ends are stuck on the brane. Closed superstring theories have generally two types of gauge elds coming from the NSNS and R-R sectors on the world sheet. Consider for example the Type IIB theory. The It has also been argued using thermodynamic arguments that the entropy of an extremal black hole could be proportional to the mass ten dimensional gauge elds of this theory consist of (1) NS-NS sector : dilaton , metricgAB and antisymmetric tensor eld B(1)
AB (2) R-R sector : an axion eld , antisymmetric tensor B(2)
AB and a rank four gauge eld with self dual eld strength CABCD. The ob ject which carries charges under the NS-NS gauge eld B(AB is in fact the elementary string itself. At low
energies the elementary string is described by a classical solution of the eective eld theory- the macroscopic string [25]. However there are no states in the perturbative spectrum of the elementary string theory which carry charges of the R-R gauge eld B(2) AB. Remarkably, these missing states carrying R-R charges are D-branes [26]. The eld content shows that these D-branes describe p-branes with odd p for Type IIB and even p for Type IIA. The Type-IIB theory is conjectured to be self-dual under an SL(2; Z) symmetry which transform (; ) and (B(1) AB; B(1)AB) into each other. In fact there are an innite number of classical string solutions labelled by (m; n) where m denotes a quantized NS-NS charge and n a quantized R-R charge . These solutions are in fact low energy descriptions of bound
states of D-branes and elementary strings . In particular the string with (0; 1) charge has a nonzero B(2) (plus metric and dilaton) but zero B(1) - we will call this the D-string. The
duality conjecture then implies that the states of the D-string behave exactly like the states of the elementary string with the coupling replaced by the inverse coupling. Evidence for this has accumulated over the past few months

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Taken on: February 7, 2018

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