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

TL;DR

This paper demonstrates that the statistical-mechanical entropy calculated via the 't Hooft's brick wall model remains consistent across different coordinate systems, such as Painlevé and Lemaitre, despite their differing metric singularities.

Contribution

It shows that the entropy results are invariant under coordinate transformations, extending the validity of the brick wall model beyond Schwarzschild coordinates.

Findings

Entropies in Painlevé and Lemaitre coordinates match Schwarzschild results after exact calculation.

The invariance holds for black holes, de Sitter spaces, and scenarios with quantum back reaction.

Subleading corrections do not affect the coordinate invariance of the entropy.

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 Painlev\'{e} 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 Painlev\'{e} 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.