Gary T. Horowitz, UC Santa Barbara

gary@cosmic.physics.ucsb.edu

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.

Sun Sep 1 16:45:26 EDT 1996