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In this article we will discuss a Lorentzian sector calculation of theentropy of a minimally coupled scalar field in the Kerr black hole background.We will use the brick wall model of G.'t Hooft. In the Kerr black hole,complications arise due to the absence of a global timelike Killing field andthe presence of the ergosphere. Nevertheless, it is possible to calculate theentropy of a thin shell of matter field in the near-horizon region using thebrick wall model. The corresponding leading order entropy of thenonsuperradiant modes is found to be proportional to the area of the horizonand is logarithmically divergent. Thus, to the leading order, the entropy of athree-dimensional system in the near-horizon region is proportional to theboundary surface. This aspect is also valid in the Schwarzschild black hole.The corresponding internal energy remains finite if the entropy is chosen to beof the order of the black hole entropy itself. For a fixed value of the brickwall cut-off, the leading order entropy in the Kerr black hole is found to behalf of the corresponding term in the Schwarzschild black hole. This is due torotation and is consistent with the preferential emission of particles in theKerr black hole with azimuthal angular momentum in the same direction as thatof the black hole itself. However, we can obtain the Schwarzschild caseexpression by including a subleading term and taking the appropriate slowrotation limit. In this version, we will discuss a sub-leading term for thescalar field entropy in Schwarzschild black hole. This term reduces to anexpression similar to the corresponding flat spacetime entropy in the limitwhen radius of event horizon is vanishing.

Near-Horizon Geometry and the Entropy of a Minimally Coupled Scalar Field in the Kerr Black Hole