Near horizon geometry, Brick wall model and the Entropy of a scalar field in the Reissner-Nordstrom black hole backgrounds

In this article we will find the entropy of a scalar field in the Reissner-Nordstrom black hole backgrounds using the brick wall model of t' Hooft. We will use the semi-classical WKB approximation. We will consider the modes which are globally stationary so that the WKB quantization rule used in the brick wall model remains valid. In the Schwarzschild black hole this consideration had led to a new expression of the entropy different from the conventional expression which is inversely divergent in the brick wall cut-off parameter and in terms of a proper distance cut-off parameter, is proportional to the area of the event horizon. The new expression of the scalar field entropy obtained in this article is logarithmically divergent in the brick wall cut-off parameter and is not proportional to the area of the black hole event horizon. For the extremal Reissner-Nordstrom black hole background the entropy of the scalar field is again divergent in the brick wall cut-off parameter and vanishes if the temperature of the Hawking radiation and the black hole is taken to be zero. We will next consider the entropy for a thin shell of matter field surrounding the black hole horizon. When expressed in terms of a covariant cut-off parameter, the entropy of a thin shell of matter field surrounding the horizon in the non-extreme Reissner-Nordstrom black hole background is given by an expression proportional to the area of the black hole horizon. We will briefly explain the significance of this result.