Thermodynamics of Black Hole in (N+3)-dimensions from Euclidean N-brane Theory

In this article we consider an N-brane description of an (N+3)-dimensional black hole horizon. First of all, we start by reviewing a previous work where a string theory is used as describing the dynamics of the event horizon of a four dimensional black hole. Then we consider a particle model defined on one dimensional Euclidean line in a three dimensional black hole as a target spacetime metric. By solving the field equations we find a “world line instanton” which connects the past event horizon with the future one. This solution gives us the exact value of the Hawking temperature and to leading order the Bekenstein-Hawking formula of black hole entropy. We also show that this formalism is extensible to an arbitrary spacetime dimension. Finally we make a comment of one-loop quantum correction to the black hole entropy . † E-mail addess: [email protected] The Bekenstein-Hawking formula of the black hole entropy, S = 4 kc Gh̄AH 1,2 is not only so beautiful but also very mysterious for us at present. This formula contains the four fundamental constants of physics, those are, the Boltzman constant k, the Newton one G, the Planck one h̄ and the light velocity c so that it suggests a deep triangle relation among thermodynamics, general relativity and quantum mechanics. Moreover, this formula relates the entropy of a black hole to the area of a event horizon, therefore also implies a connection to geometry. Thus, although the above formula was originally derived in terms of the semi-classical approach, many theoretical physicists never doubt its validity up to some quantum corrections even when we have a quantum theory of gravitation in the future. However, the underlying physical basis by which 1 4 kc Gh̄AH arises as the black hole entropy remains unclear. It is tempted to regard this black entropy as the logarithm of the number of microscopic states compatible with the observed macroscopic state from the viewpoint of the ordinary statistical physics. Then, a crux of an understanding is what those microscopic states are. It might be true that the underlying law explaining the Bekenstein-Hawking formula might presumably not be fully understood until we construct a theory of quantum gravity. But there certainly exists an opposite attitude toward it. Namely this mystery might give us a clue of constructing a theory of quantum gravity. At this point one expects that a quantum black hole plays a similar role as the hydrogen atom at the advent of the quantum mechanics. In a previous paper 3 , we have considered a stringy description of a black hole horizon in four spacetime dimensions where we have described the event horizon of a black hole in terms of the world sheet swept by a string in the Rindler background. It was shown that a nonlinear sigma model action leads to both the Hawking temperature of a black hole and the well-known Bekenstein-Hawking formula of the black hole entropy within the lowest order of approximation. Furthermore we have derived a covariant operator algebra on the event horizon which is a natural 2 generalization to the ’tHooft one 4,5 . Thus it is natural to ask whether this stringy approach to the black hole thermodynamics can be extended to an arbitrary spacetime dimension or is a peculiar thing only in four dimensions. We will see later that we can in fact construct a more general formalism where the event horizon of a black hole in (N+3)-spacetime dimensions is described by a Euclidean N-brane. We start with a brief review of the previous work on a string, i.e., 1-brane, approach to the four dimensional black hole 3 . As an effective action describing the dynamical properties of the black hole horizon, let us consider the Polyakov action of a bosonic string in a curved target spacetime which is given by