Can the “brick wall” model present the same results in different coordinate representations?

Abstract

By using the 't Hooft's "brick wall" model and the Pauli-Villars regularization scheme we calculate the statistical-mechanical entropies arising from the quantum scalar field in different coordinate settings, such as the Painlevé and Lemaitre coordinates. At first glance, it seems that the entropies would be different from that in the standard Schwarzschild coordinate since the metrics in both the Painlevé and Lemaitre coordinates do not possess the singularity at the event horizon as that in the Schwarzschild-like coordinate. However, after an exact calculation we find that, up to the subleading correction, the statistical-mechanical entropies in these coordinates are equivalent to that in the Schwarzschild-like coordinate. The result is not only valid for black holes and de Sitter spaces, but also for the case that the quantum field exerts back reaction on the gravitational field provided that the back reaction does not alter the symmetry of the spacetime.

Detect black holes using photons for coupling model of electromagnetic and gravitational fields

“…The corresponding free energy, F , can be defined in term of the one-particle spectrum. One of the ways to calculate F is "brick wall" model (BWM) proposed by 't Hooft [34] which has been used extensively [35,36,37,38,40]. In this model, in order to eliminate divergence which appears due to the infinite growth of the density of states close to the horizon, 't Hooft introduces a "brick wall" cutoff.…”

Entropy of the Schwarzschild-de Sitter black hole due to arbitrary spin fields in different coordinates

“…The time coordinate transformation from the Schwarzschild-like coordinate (1.1) to the Painlevé coordinate [7] is…”

“…It is therefore interesting to study the following question: can we obtain the same statistical mechanical entropies of the black hole in the Painlevé and the Lemaitre coordinates due to the fact that in this question arises naturally after Shankaranarayanan et al [19,20] studied the Hawking temperature of the Schwarzschild black hole in the Painlevé and the Lemaitre coordinates by using the method of complex paths and they showed that the results are equal to those in the Schwarzschild-like coordinate. For the massless scalar field in the general static black hole, Jing [7] find that the entropies in the Painlevé and Lemaitre coordinates are exactly equivalent to that in the Schwarzschild-like coordinate. However, whether the entropies of the Schwarzschild-de Sitter black hole due to arbitrary spin fields are the same in the Painlevé, the Lemaitre and Schwarzschild-like coordinates is still an open question.…”