[Paper Review] Area Law in Noncommutative Schwarzschild Black Hole

Rabin Banerjee, Bibhas Ranjan Majhi+1 more|arXiv (Cornell University)|Feb 15, 2008

Black Holes and Theoretical Physics10 citations

TL;DR

This paper analytically demonstrates that the Bekenstein-Hawking area law for noncommutative Schwarzschild black holes remains valid as a standard noncommutative deformation of the classical law up to leading order in the noncommutative parameter θ, with exact validity for all orders when the horizon radius rh ≈ 4.8√θ. It further derives corrections to the area law for smaller horizon radii, revealing the full quantum-corrected entropy structure.

ABSTRACT

We show analytically that the semiclassical Bekenstein-Hawking area law for noncommutative Schwarzschild black hole in the regime r2 h 4θ>> 1 is a standard noncommutative deformation of the usual law, upto the leading order in the noncommutative parameter θ. A graphical analysis shows that for the horizon radius rh � 4.8 √ θ, this area law exactly holds for all orders of θ. Finally, we also give the corrections to the area law to get the exact nature of the Bekenstein-Hawking entropy when rh < 4.8 √ θ. Classical general relativity gives the concept of black hole from which nothing can escape. This picture was changed dramatically when Hawking [1, 2] incorporated the quantum nature into this classical problem. In fact he showed that black holes radiate a spectrum of particles which is quite analogous with a thermal black body radiation. Before this, Bekenstein [3, 4, 5, 6] proposed that a black hole has an entropy Sbh which is some finite multiple η of its area A. He was not able to determine the exact value of η, but gave heuristic arguments for conjecturing that it was ln2 8π. However, the first law of black hole mechanics imply that the black hole would have a temperature Th which is proportional to the surface gravity κ of the black hole. Therefore from Bekenstein’s argument and the first law of black hole mechanics one might say Th = ǫκ and Sbh = ηA with 8πηǫ = 1. Bekenstein proposed that η is finite and it is equal to ln2 1 8π. Then one would get ǫ = ln2 and so Th = κ ln2. Later on Hawking realised that Bekenstein’s idea was consistent. In fact, he found that the black hole temperature is Th = κ

Motivation & Objective

• To investigate whether the Bekenstein-Hawking area law holds in the context of noncommutative geometry for Schwarzschild black holes.
• To determine the nature of quantum corrections to the area law when the horizon radius is smaller than 4.8√θ.
• To examine the regime in which the noncommutative deformation of the area law becomes exact across all orders of θ.
• To derive the corrected form of the Bekenstein-Hawking entropy for noncommutative black holes beyond leading-order approximation.

Proposed method

• Analytical derivation of the area law in the semiclassical regime where r²h ≫ 4θ, treating θ as a small expansion parameter.
• Use of noncommutative general relativity to modify the Schwarzschild metric, introducing a deformation parameter θ.
• Application of the standard Bekenstein-Hawking entropy formula Sbh = ηA with η = ln2 / 8π, extended to noncommutative geometry.
• Graphical analysis of the area law behavior across different values of rh and θ to identify the critical threshold rh = 4.8√θ.
• Derivation of higher-order corrections to the area law for rh < 4.8√θ using perturbative methods in θ.
• Comparison of the noncommutative area law with the classical one to identify the nature of quantum corrections.

Experimental results

Research questions

• RQ1 Does the Bekenstein-Hawking area law persist in noncommutative Schwarzschild black holes, and if so, how is it deformed by the noncommutative parameter θ?
• RQ2 At what horizon radius does the noncommutative area law become exact for all orders of θ?
• RQ3 What are the explicit corrections to the area law when the horizon radius is smaller than 4.8√θ?
• RQ4 How does the noncommutative deformation affect the entropy of the black hole compared to the classical case?
• RQ5 Can the semiclassical limit of the noncommutative black hole reproduce the standard area law up to leading order in θ?

Key findings

• The area law for noncommutative Schwarzschild black holes is a standard noncommutative deformation of the classical law up to the leading order in θ.
• For horizon radii rh ≈ 4.8√θ, the area law holds exactly for all orders of the noncommutative parameter θ.
• When rh < 4.8√θ, the area law receives higher-order corrections that modify the Bekenstein-Hawking entropy formula.
• The graphical analysis confirms that the noncommutative area law is exact at the critical radius rh = 4.8√θ across all θ orders.
• The corrections to the area law are derived explicitly, revealing the full quantum-corrected entropy structure in the noncommutative regime.
• The semiclassical regime r²h ≫ 4θ supports the validity of the noncommutative deformation of the area law as a perturbative correction.