Update on Black Hole Microstates in String Theory

Gary T. Horowitz, UC Santa Barbara

Last January, Strominger and Vafa (hep-th/9601029) showed that the Bekenstein-Hawking entropy of a static five dimensional extreme black hole was precisely reproduced by counting states in string theory with the same mass and charge (for macroscopic black holes). This touched off an explosion of interest and in the next few months, this agreement was shown to hold for near extremal as well as extremal, four and five dimensional black holes, including rotation. I wrote a review of these developments in April (gr-qc/9604051). What I would like to do here is summarize some of the progress since then.

Perhaps the most important new development is a calculation by Das and Mathur (hep-th/9606185) showing that the rate of Hawking radiation from a near extremal black hole agrees with the string theory prediction based on interactions between the microstates. The fact that the spectrum is thermal with the same temperature as the black hole is not a surprise, given that it was already known that the entropy as a function of energy was the same in the two systems. However, the fact that the overall coefficient agrees is highly nontrivial and quite remarkable. This result has implications for the black hole information puzzle. Recall that in string theory, there is a length scale set by the string tension. Newton's constant is related to this length and the string coupling g by (in four dimensions). At weak coupling, , an extreme black hole is described by a flat space configuration of objects known as D-branes. A near extremal black hole is described by an `excited state' of D-branes. In this description, there is no analog of the event horizon and the emission from excited D-branes is manifestly unitary. The apparent thermal nature of the radiation arises from the large number of degress of freedom, just like an ordinary hot object. At strong coupling, the gravitational field becomes stronger and one obtains a near extremal black hole. The fact that the rate of Hawking evaporation from this black hole agrees with the string calculation is further evidence that radiation from near extremal black holes is also unitary.

In another development, there has been a great increase in the class of solutions for which the Bekenstein-Hawking entropy has been shown to agree with the counting of string states. Previously, it was shown that for black holes depending on a finite number of parameters (including mass, charges and angular momentum) the entropy as a function of these parameters was reproduced by counting states of D-branes at weak string coupling. Recently with Don Marolf, we extended this to the case where the solution depends on arbitrary functions (hep-th/9605224, hep-th/9606113).

One does not usually expect a solution with an event horizon to depend on arbitrary functions, since the `no-hair' theorems show that stationary black holes are characterized by only a few parameters. If one tries to add a wave to the spacetime, it either falls down the hole, or radiates to infinity. However it turns out that extremal black strings, i.e. one dimensional extended objects with an event horizon, are different (Larsen and Wilczek hep-th/9511064). They can support traveling waves of arbitrary profile. These waves affect the horizon area and the distribution of momentum along the black string. By counting states of D-branes with the same momentum distribution as the black string, one finds perfect agreement with the Bekenstein Hawking entropy for all wave profiles (hep-th/9605224, hep-th/9606113).

An outstanding open question is to extend these results to black holes which are far from extremality. There are indications that we are getting close to taking this important next step.

Jorge Pullin
Sun Sep 1 16:45:26 EDT 1996