  In this article we will discuss a Lorentzian sector calculation of the
entropy 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 and
the presence of the ergosphere. Nevertheless, 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
nonsuperradiant modes is found to be proportional to the area of the horizon
and is logarithmically divergent. Thus, to the leading order, the entropy of a
three-dimensional system in the near-horizon region is proportional to the
boundary surface. This aspect is also valid in the Schwarzschild black hole.
The corresponding internal energy remains finite if the entropy is chosen to be
of the order of the black hole entropy itself. For a fixed value of the brick
wall cut-off, the leading order entropy in the Kerr black hole is found to be
half of the corresponding term in the Schwarzschild black hole. This is due to
rotation and is consistent with the preferential emission of particles in the
Kerr black hole with azimuthal angular momentum in the same direction as that
of the black hole itself. However, we can obtain the Schwarzschild case
expression by including a subleading term and taking the appropriate slow
rotation limit. In this version, we will discuss a sub-leading term for the
scalar field entropy in Schwarzschild black hole. This term reduces to an
expression similar to the corresponding flat spacetime entropy in the limit
when radius of event horizon is vanishing.
