  In this article, we will discuss a Lorentzian sector calculation of the
entropy of a minimally coupled scalar field in the Schwarzschild black hole
background using the brick wall model of 't Hooft. In the original article, the
WKB approximation was used for the modes that are globally stationary. In a
previous article, we found that the WKB quantization rule together with a
proper counting of the states, leads to a new expression of the scalar field
entropy which is not proportional to the area of the horizon. The expression of
the entropy is logarithmically divergent in the brick wall cut-off parameter in
contrast to an inverse power divergence obtained earlier. In this article, we
will consider the entropy for a thin shell of matter field of a given thickness
surrounding the black hole horizon. The thickness is chosen to be large
compared with the Planck length and is of the order of the atomic scale. When
expressed in terms of a covariant cut-off parameter, the entropy of a thin
shell of matter field of a given thickness and surrounding the horizon in the
Schwarzschild black hole background is given by an expression proportional to
the area of the black hole horizon. This leading order divergent term in the
cut-off parameter remains to be logarithmically divergent. The logarithmic
divergence is expected from the nature of the solution in the near-horizon
region. We will find that these discussions are significant in the context of
the continuation to the Euclidean sector and the corresponding regularization
schemes used to evaluate the thermodynamical properties of matter fields in
curved spaces. These are related with the geometric aspects of curved spaces.
The above discussions are also important in presence of cosmological event
horizon.
