Near-Horizon Geometry and the Entropy of a Minimally Coupled Scalar Field in the Asymptotically Flat Black Holes

Abstract

In this article we will discuss a Lorentzian sector calculation of the entropy of a minimally coupled scalar field in the Kerr family of black hole backgrounds. We will use the brick wall model of ’t Hooft. In a Kerr black hole, it is possible to calculate the entropy of a thin shell of matter field in the near-horizon region using the brick wall model. The corresponding leading-order entropy of the scalar field is found to be proportional to the area of the horizon and is logarithmically divergent. Thus, the entropy of a three-dimensional system in the near-horizon region is proportional to the boundary surface. The entropy is found to be a decreasing function of the thickness of the thin shell. The above aspects are also valid in a Schwarzschild black hole and is similar to that of the black hole entropy itself. We will discuss the significance of the results with reference to the holography principle and the entropy bound.