Black hole entropy from a single long string

In this article we derive the Bekenstein-Hawking formula of black hole entropy from a single long string. We consider a open string in the Rindler metric which can be obtained in the large mass limit from the Schwarzschild black hole metric. By solving the field equations we find a nontrivial solution with the exact value of the Hawking temperature. We see that this solution gives us the Bekenstein-Hawking formula of black hole entropy to leading order of approximation. This string has effectively a rescaled string tension having a relation with a redshift factor. It is also pointed out that this formalism is extensible to other black holes in an arbitrary spacetime dimension. The present work might lead us to a surprising idea that the recent picture which the black hole entropy arises from D-brane excitations has the root of the picture that the black hole entropy is stocked in a single long string with a rescaled string tension. PACS number(s): 11.25.-w, 04.70.Dy † E-mail address: [email protected] The Bekenstein-Hawking formula of the black hole entropy, S = 1 4 kc Gh̄AH [1,2] contains the four fundamental constants of physics, which 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 AH of a event horizon, therefore also implies a connection to geometry. Thus, although the above formula was originally derived in terms of the semiclassical approach, it is expected that its validity might be taken over apart from quantum corrections even when we have a quantum theory of gravitation in the future. However, for more than twenty years the underlying physical basis by which 1 4 kc Gh̄AH arises as the black hole entropy remained 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. Recently this situation has been completely changed by a remarkable work by Strominger and Vafa [3]. They showed through an application of D-brane method [4] that the Bekenstein-Hawking entropy of an extremal five-dimensional black hole precisely equals the number of BPS saturated states in string theory with the given charges. Since then a lot of related works have appeared [5]. However, it seems to be unsatisfying at least for the present author that there is no direct way from string theory of counting the string states to match the Bekenstein-Hawking entropy since string theory is believed to be a unified theory including a theory of quantum gravity. In other words, there should be a physical principle from string theory which explains why the number of string states is equal to that of statiscal states associated with the Bekenstein-Hawking entropy formula in various black hole models discovered recently [5]. In this respect, there has more recently appeared one interesting idea that black hole entropy might be