# Near-Horizon Geometry and the Entropy of a Minimally Coupled Scalar Field in the Schwarzschild Black Hole - General Relativity and Quantum Cosmology

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Abstract: In this article, we will discuss a Lorentzian sector calculation of theentropy of a minimally coupled scalar field in the Schwarzschild black holebackground using the brick wall model of -t Hooft. In the original article, theWKB approximation was used for the modes that are globally stationary. In aprevious article, we found that the WKB quantization rule together with aproper counting of the states, leads to a new expression of the scalar fieldentropy which is not proportional to the area of the horizon. The expression ofthe entropy is logarithmically divergent in the brick wall cut-off parameter incontrast to an inverse power divergence obtained earlier. In this article, wewill consider the entropy for a thin shell of matter field of a given thicknesssurrounding the black hole horizon. The thickness is chosen to be largecompared with the Planck length and is of the order of the atomic scale. Whenexpressed in terms of a covariant cut-off parameter, the entropy of a thinshell of matter field of a given thickness and surrounding the horizon in theSchwarzschild black hole background is given by an expression proportional tothe area of the black hole horizon. This leading order divergent term in thecut-off parameter remains to be logarithmically divergent. The logarithmicdivergence is expected from the nature of the solution in the near-horizonregion. We will find that these discussions are significant in the context ofthe continuation to the Euclidean sector and the corresponding regularizationschemes used to evaluate the thermodynamical properties of matter fields incurved spaces. These are related with the geometric aspects of curved spaces.The above discussions are also important in presence of cosmological eventhorizon.

Author: ** Kaushik Ghosh**

Source: https://arxiv.org/