[Paper Review] Quantum correction to the entropy of noncommutative BTZ black hole

M. A. Anacleto, F. A. Brito+3 more|arXiv (Cornell University)|Oct 28, 2015

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper applies the Hamilton-Jacobi tunneling method with generalized uncertainty principle (GUP) corrections to compute quantum-corrected Hawking temperature, entropy, and specific heat for noncommutative BTZ black holes. It derives logarithmic corrections to the area law and shows the black hole becomes stable and forms a remnant at minimum mass, suggesting a resolution to the information paradox via noncommutative and quantum gravity effects.

ABSTRACT

In this paper we consider the generalized uncertainty principle (GUP) in the tunneling formalism via Hamilton-Jacobi method to determine the quantum-corrected Hawking temperature and entropy for noncommutative BTZ black hole. In our results we obtain several types of corrections including the expected logarithmic correction to the area entropy associated with the noncommutative BTZ black holes. We also show that the area entropy product of the noncommutative BTZ black holes is dependent on mass and by analyzing the nature of the specific heat capacity we have observed that the noncommutative BTZ black hole is stable at some range of parameters.

Motivation & Objective

• To investigate quantum-corrected thermodynamics of noncommutative BTZ black holes using the generalized uncertainty principle (GUP).
• To determine how noncommutativity and GUP modify Hawking temperature, entropy, and specific heat capacity.
• To analyze the stability of noncommutative BTZ black holes via specific heat behavior and identify conditions for remnant formation.
• To explore the statistical origin of entropy corrections, particularly logarithmic terms, in a 2+1 dimensional gravity framework.

Proposed method

• Employ the Hamilton-Jacobi method to compute the imaginary part of the action for particle tunneling across the horizon.
• Incorporate the generalized uncertainty principle (GUP) via a modified momentum-space structure to account for quantum gravity effects near the horizon.
• Use the GUP-corrected energy relation to derive quantum-corrected expressions for Hawking temperature and entropy.
• Apply the tunneling formalism to compute the specific heat capacity and analyze its sign and behavior across mass ranges.
• Introduce a scaling ansatz Δx = αlpf(M) to derive an analytical expression for the specific heat capacity in terms of mass-dependent functions.
• Use the Stefan-Boltzmann law in 2+1 dimensions to relate mass loss rate to the GUP-corrected temperature.

Experimental results

Research questions

• RQ1 How do noncommutative and GUP corrections modify the Hawking temperature of a BTZ black hole?
• RQ2 What is the form of the quantum-corrected entropy, and does it include logarithmic corrections to the area law?
• RQ3 How does the specific heat capacity of the noncommutative BTZ black hole behave, and what does it imply for thermodynamic stability?
• RQ4 Does the GUP-corrected model predict the formation of black hole remnants as a final state?
• RQ5 How does the entropy product depend on the black hole mass and noncommutative parameter θ?

Key findings

• The quantum-corrected entropy includes a logarithmic correction to the area law, consistent with universal quantum gravity corrections.
• The entropy product of the noncommutative BTZ black hole depends explicitly on the mass parameter M, becoming mass-independent only when θ = 0.
• The specific heat capacity C_GUP becomes zero at a critical radius when Δx = αlp, indicating the black hole ceases to evaporate and forms a stable remnant.
• For θB = 0.1 and α = 0.5, the specific heat remains positive, indicating thermodynamic stability in that parameter range.
• For θB = 0.2 and α = 0.5, two points of vanishing specific heat appear, with an unphysical region in between, suggesting a phase transition or remnant formation.
• The minimum mass of the black hole is M_min = (α²lp² / 2l²)(1 - 2θB / α²lp²), achieved when Δx = αlp, marking the end of evaporation.