Remarks on 't Hooft's Brick Wall Model

TL;DR

This paper explores how non-commutative geometry near a black hole horizon can explain the brick wall boundary condition, derive the black hole entropy proportional to horizon area, and connect to Hawking temperature, offering insights beyond the traditional model.

Contribution

It introduces a noncommutative scalar field model near the horizon to interpret the brick wall hypothesis and derive black hole entropy and temperature from this framework.

Findings

Noncommutativity near the horizon leads to a new effective metric with an additional horizon.

The boundary condition for the scalar field is explained by the noncommutative space range.

Black hole entropy is proportional to the horizon area, consistent with previous results.

Abstract

A semi-classical reasoning leads to the non-commutativity of the space and time coordinates near the horizon of Schwarzschild black hole. This non-commutativity in turn provides a mechanism to interpret the brick wall thickness hypothesis in 't Hooft's brick wall model as well as the boundary condition imposed for the field considered. For concreteness, we consider a noncommutative scalar field model near the horizon and derive the effective metric via the equation of motion of noncommutative scalar field. This metric displays a new horizon in addition to the original one associated with the Schwarzschild black hole. The infinite red-shifting of the scalar field on the new horizon determines the range of the noncommutativ space and explains the relevant boundary condition for the field. This range enables us to calculate the entropy of black hole as proportional to the area of its original horizon along the same line as in 't Hooft's model, and the thickness of the brick wall is found to be proportional to the thermal average of the noncommutative space-time range. The Hawking temperature has been derived in this formalism. The study here represents an attempt to reveal some physics beyond the brick wall model.