Remarks on 't Hooft's brick wall model

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. The brick wall model proposed by 't Hooft has been used for the purpose of deriving the entropy of black hole and other quantities [1] [2], and has been extensively studied (an incomplete list, see Refs.[3] [4]). In the model, the thickness of the brick wall near the horizon of Schwarzschild black hole was set to bewhere r H is the radius at which the horizon is located, l p = √ G (in this letterh = c = 1) the Planck length and N ′ the number of the multiplet of the quantum field in the model. Eq. (1) is a prior hypothesis of the brick wall model. Actually, only under this hypothesis, the thermodynamic properties of a black hole can be reproduced correctly. Namely, this model can lead to the correct Bekenstein-Hawking entropy formulawhere S BH is Bekenstein-Hawking entropy [5] [6] and A is the horizon area . In this letter, we try to derive the brick wall thickness by a semi-classical argument and to reveal some underlying physics related to this hypothesis. For the sake of definiteness, we study the 3 + 1 dimensional Schwarzschild black hole. In this case, ∂ t is the time Killing vector. Its global energy (or the mass of the black hole) is E BH = M = r H /2G. Rather than thinking black hole as a classical object, we treat it as a quantum state with high degeneracy and its degrees of freedom are located near by the horizon. Treating the energy E BH and its conjugate time coordinate as operators, quantum mechanics tells that these two quantities can not be measured simultaneously. In other words, E BH and t satisfy Heisenberg

The quantum horizon of the Garfinkle—Horowitz—Strominger dilatonic blackhole

“…How to solve this dilemma? Recently Yans' works [5][6][7] solved the dilemma using the semi-classical effect at the horizon, which naturally led to the space-time noncommutative at the horizon, called as quantum horizon. In this treatment the classical horizon spreads, so the trouble due to the singularity on the horizon disappears.…”

“…(29), we can see that the integration of Eq. (24) is rigorous, so that the considerations to quantum event horizon presented in Yans' paper [5,6,7] have indeed solved the puzzle caused by the singularity at the classical horizon in Hawking radiation calculations naturally. At the new horizon r H + (ω) the field has an infinite red-shift, an observer can't detect the at r r H + (ω), so we can calculate the entropy of the black hole in this quantum horizon model using the BWM method proposed by 't Hooft.…”

BLACK HOLE EVAPORATION BASED UPON A q-DEFORMATION DESCRIPTION

“…However, the investigation for testing those noncommutative effects near black hole is extremely difficult due to the fact that a consistent theory of quantum gravity is not available at present, let alone a theory of noncommutative quantum gravity. Still, some efforts have been made on this subject [7][8][9]. It should be mentioned that by considering a semi-classical theory of noncommutative fields, the authors of Ref.…”

“…It should be mentioned that by considering a semi-classical theory of noncommutative fields, the authors of Ref. [8] give some remarks on 't Hooft's brick wall model [10].…”