0902.1601 (K. Ghosh)

A few comments on Brick-wall model and the Entropy of a scalar field in
Schwarzschild black hole background    [PDF]

K. Ghosh

In this article we will find the expressions of the entropy of a minimally coupled scalar field in the Schwarzschild and in the Reissner-Nordstrom black hole backgrounds using the brick wall model of t' Hooft. We will use the semiclassical WKB approximation and find the possible states using the WKB quantization rule which is similar to the semiclassical Bohr-Sommerfeld quantization rule. In the original article the WKB quantization rule was used to the modes which are globally stationary. In a previous article we found that the WKB quantization rule together with a proper counting of the states lead to a new expression for the matter field entropy which is not proportional to the area of the horizon. The expression of the entropy is logarithmically divergent in the brick wall cut-off parameter in contrary to an inverse power divergence obtained earlier. In this article we will first show that similar situation remains to be valid for the non-extreme Reissner-Nordstrom black hole background. We will consider the entropy for a thin shell of matter field of a given thickness surrounding the black hole horizon. The thickness is chosen to be large compared to the Planck length. When expressed in terms of a covariant cut-off parameter, the entropy of a thin shell of matter field surrounding the horizon in the Schwarzschild black hole background is given by an expression proportional to the area of the black hole horizon. The leading order divergent term in the cut-off parameters remains to be logarithmically divergent. Similar situation also remains valid for the non-extreme Reissner-Nordstrom black hole.

View original: http://arxiv.org/abs/0902.1601