[Paper Review] Thermal Fluctuations Of Stable Quantum ADS Kerr-Newman Black Hole

Aloke Kumar Sinha|arXiv (Cornell University)|Aug 30, 2016

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper calculates thermal fluctuations of horizon area, charge, and angular momentum in stable Anti-de Sitter (AdS) Kerr-Newman black holes using loop quantum gravity and grand canonical ensemble statistical mechanics. In the large horizon area limit, it finds that leading-order fluctuations of charge and angular momentum are independent of their equilibrium values, a result tied to quantum spacetime corrections beyond the Bekenstein-Hawking area law.

ABSTRACT

We have already derived the Criteria for thermal stability of charged rotating black holes in any dimension , for horizon areas that are large relative to the Planck area (in these dimensions). The derivation is done by using results of loop quantum gravity and equilibrium statistical mechanics of the Grand Canonical ensemble. It is also shown there [1] that in four dimensional spacetime, ADS Kerr-Newman Black hole is thermally stable. In this paper, the expectation values of fluctuations of horizon area,charge and angular momentum of stable ADS black hole are calculated. Interestingly, it is found that leading order fluctuations of charge and angular momentum , in large horizon area limit , are independent of the values of charge and angular momentum at equilibrium.

Motivation & Objective

• To investigate thermal fluctuations of horizon area, charge, and angular momentum in stable quantum AdS Kerr-Newman black holes.
• To determine whether quantum corrections from loop quantum gravity affect the stability and fluctuation behavior of charged, rotating black holes.
• To analyze how fluctuations depend on equilibrium values of charge and angular momentum in the large horizon area regime.
• To derive explicit expressions for the variance of area, charge, and angular momentum fluctuations using statistical mechanics and quantum gravity inputs.
• To explore the implications of charge and angular momentum fluctuations being independent of their equilibrium values in the large area limit.

Proposed method

• Utilizes the grand canonical partition function derived from loop quantum gravity and equilibrium statistical mechanics.
• Applies saddle point approximation to the partition function to compute fluctuations around equilibrium values of area, charge, and angular momentum.
• Employs the Hessian matrix of the effective action to compute variances of fluctuations using second derivatives of the mass function.
• Uses the generalized Smarr formula for AdS Kerr-Newman black holes to express mass as a function of area, charge, and angular momentum.
• Performs asymptotic expansion in the large horizon area limit (A ≫ l², A² ≫ 4J² + Q⁴) to extract leading-order fluctuation terms.
• Derives fluctuation variances via functional derivatives of the logarithm of the partition function with respect to curvature terms in the action.

Experimental results

Research questions

• RQ1 How do thermal fluctuations of charge and angular momentum behave in stable AdS Kerr-Newman black holes in the large horizon area limit?
• RQ2 Are the leading-order fluctuations of charge and angular momentum dependent on their equilibrium values?
• RQ3 What is the role of quantum spacetime corrections (beyond Bekenstein-Hawking) in determining the magnitude of thermal fluctuations?
• RQ4 How do the variances of area, charge, and angular momentum fluctuations scale with horizon area in the large area regime?
• RQ5 What does the independence of charge and angular momentum fluctuations from their equilibrium values imply for quantum gravity and black hole thermodynamics?

Key findings

• In the large horizon area limit, the leading-order fluctuation of charge, $(\Delta Q)^2$, is approximately $\frac{3A_p A}{16\pi^2 l^2}$, independent of the equilibrium charge $Q$.
• The leading-order fluctuation of angular momentum, $(\Delta J)^2$, is approximately $\frac{3A_p A^2}{128\pi^3 l^2}$, independent of the equilibrium angular momentum $J$.
• The area fluctuation, $(\Delta A)^2$, scales as $8A_p A$, consistent with the area law and quantum corrections from loop quantum gravity.
• The independence of $\Delta Q^2$ and $\Delta J^2$ from $Q$ and $J$ implies that even black holes with negligible charge or angular momentum exhibit finite quantum fluctuations.
• These results emerge from the interplay of the generalized Smarr formula and quantum corrections to entropy, particularly the $-3/2$ logarithmic correction from loop quantum gravity.
• The findings suggest a deep quantum nature of black hole thermodynamics, where fluctuations are governed by spacetime quantum geometry rather than classical parameters.