ar X iv : h ep - t h / 96 07 00 1 v 3 2 4 M ar 1 99 7 INJE – TP – 96 – 4 Propagating waves in an extremal black string

We investigate the black string in the context of the string theories. It is shown that the graviton is the only propagating mode in the (2+1)– dimensional extremal black string background. Both the dilation and axion turn out to be non-propagating modes. 1 Recently there has been much progress in understanding the microscopic origin for the black hole entropy. This was possible by using a new description of solitonic states in string theory [1–6]. For the simplest five-dimensional extremal black hole, Strominger and Vafa [1] counted the number of degeneracy corresponding to BPS-saturated states in the string theory for given charge. And they showed that for large charge, the number of states increases as e A/4 , where A is the area of the horizon for extremal black hole. That is, the statistical interpretation for the Bekenstein-Hawking entropy was made possible in five dimensions. On the other hand, the five-dimensional black hole is just a six-dimesional black string which winds around a compact internal circle [7]. The microstates of five-dimensional extremal black hole arise from the fields moving around a circle in the internal dimensions. In order to understand this situation, it is useful to take this internal direction as a space-time direction explicitly. This is a black string solution in six dimensions. As far as we know, full understanding of the black p-brane including the black string (p = 1) was not completed, compared with the black holes. This is because the black p-brane solutions are obtained in higher dimensions(D > 4) [8–10]. On the other hand, Horowitz and Tseytlin [11,12] have done much work on finding traveling wave solutions of string theories in various dimensions. They related a limit of their F-model to the extremal black string in three dimensions. The three-dimensional black string is used as a toy model for investigating higher dimensional black strings [13]. Unlike the higher dimensional cases, the exact conformal field theory is known for two and three dimensions. Thus the particle contents can also be determined. A simple extension of Witten's construction for a gauged WZW model yields the three-dimensional charged black string [14,15]. This solution is characterized by three parameters: M (mass), Q (axion charge per unit length), and k (a constant related to the asymptotic value of the derivative of the dilaton). For 0 < |Q| < M, the black string is similar to the Reissner-Nordström black hole in four …