[Paper Review] Thermodynamics of noncommutative BTZ black hole

Rashida Jahangir, K. Saifullah|arXiv (Cornell University)|Jan 31, 2011

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper investigates the thermodynamics of the (2+1)-dimensional BTZ black hole in noncommutative spacetime using a noncommutative deformation of the Chern-Simons gauge theory. It derives the Hawking temperature and entropy, showing they reduce to the commutative BTZ results in the limit θ→0. The key finding is that the Hawking area law fails for small outer horizons, with noncommutativity introducing a finite, non-zero entropy at the origin and modifying the entropy-area relation via a θ-dependent correction term.

ABSTRACT

Thermodynamics of the BTZ black hole in noncommutative geometry is studied. We work out the Hawking temperature and entropy which reduce to their commutative limits when the noncommutativity parameter tends to zero. We also discuss the range of validity of the Hawking area law in the noncommutative case and provide graphical analysis. We see that the law is not valid unless the outer horizon is very large.

Motivation & Objective

• To study the thermodynamic properties of the BTZ black hole in a noncommutative spacetime framework.
• To examine the validity of the Hawking area law in the noncommutative regime.
• To derive corrections to the Hawking temperature and entropy due to spacetime noncommutativity.
• To analyze the behavior of mass, temperature, and entropy as functions of the noncommutativity parameter βθ.

Proposed method

• Noncommutativity is introduced via the Moyal star product, replacing pointwise multiplication in the action and field equations.
• The noncommutative BTZ metric is derived from a deformed Chern-Simons gauge theory with U(1,1)×U(1,1) gauge fields and a noncommutative parameter θ.
• The Seiberg-Witten map is applied to relate noncommutative gauge fields to their commutative counterparts, enabling perturbative expansion in θ.
• The Hawking temperature is computed from the surface gravity of the noncommutative black hole metric, valid to O(θ).
• Entropy is derived by integrating the inverse temperature over the horizon, leading to a modified expression involving the outer horizon radius and θ.
• Graphical analysis is performed for mass, temperature, and entropy as functions of the event horizon radius for varying βθ.

Experimental results

Research questions

• RQ1 How does noncommutativity affect the Hawking temperature of the BTZ black hole?
• RQ2 What is the form of the entropy for the noncommutative BTZ black hole, and how does it differ from the commutative case?
• RQ3 Under what conditions does the Hawking area law remain valid in the noncommutative BTZ black hole?
• RQ4 How does the noncommutativity parameter βθ influence the mass and thermodynamic properties of the black hole?

Key findings

• The Hawking temperature of the noncommutative BTZ black hole is derived as T = (1/2π) × [1 - (βθJ²l²)/(r²(4r⁴ - J²l²))]^(1/2) × [1 - βθ(1/2 + J²l²/(8r⁴)) × (4r²/(4r⁴ - J²l²))]^(-1/2), valid to O(θ).
• The entropy of the noncommutative BTZ black hole is S = 4π(r₊ + βθ/4 - βθ/(r₊ + βθ/4)), which reduces to the standard 4πr₊ in the limit θ → 0.
• The noncommutative entropy correction term 4π(βθ/4 - βθ/(r₊ + βθ/4)) is finite and non-zero even at r₊ = 0, indicating a non-vanishing entropy at the origin.
• The Hawking area law is violated for small outer horizons; it only holds when r₊ is very large, as confirmed by graphical analysis of entropy vs. horizon radius.
• The mass of the noncommutative BTZ black hole remains finite at r → 0, unlike the commutative case where it diverges, due to the Gaussian-like mass distribution induced by noncommutativity.
• Graphs of mass, temperature, and entropy show monotonic increases with βθ for fixed J, l, and Q, with all quantities approaching their commutative limits as βθ → 0.