Generalized Brick Wall Method for Stationary Axisymmetric Spacetimes

TL;DR

This paper introduces a generalized brick wall method to compute the statistical entropy of scalar fields in stationary axisymmetric spacetimes, unifying the approach for various black hole solutions and confirming the area law in complex backgrounds.

Contribution

It develops a unified formalism for calculating black hole entropy using the brick wall method across diverse geometries without re-deriving wave equations.

Findings

Recovered Bekenstein-Hawking area law for Kerr black hole.

Extended analysis to Kerr-Newman-AdS and black holes with matter fields.

Confirmed the area law holds with proper cutoff renormalization in complex backgrounds.

Abstract

The microscopic origin of black hole entropy remains one of the central puzzles in quantum gravity. In this work, we investigate the statistical entropy of scalar fields propagating in stationary axisymmetric spacetimes using the thin-film modification of the 't Hooft brick wall method. We derive a generalized expression for the free energy of both superradiant and non-superradiant modes, expressed explicitly in terms of generic metric components. This unified formalism allows for a systematic evaluation of entropy across a diverse class of black holes without re-deriving the wave equation for each specific case. We validate our approach by recovering the Bekenstein-Hawking area law for the standard Kerr black hole. Subsequently, we extend the analysis to the Kerr-Newman-AdS geometry and, finally, to the novel case of a Kerr-Newman-AdS black hole surrounded by quintessence and a cloud of strings. Our results confirm that the area law holds even in the presence of these complex background matter fields, provided the cutoff parameter is appropriately renormalized.