Non-commutative geometry and thermodynamics of the Schwarzschild-AdS black hole

TL;DR

The paper analyzes Schwarzschild–AdS black holes in noncommutative geometry by introducing a NC parameter $\Theta$ via the Bopp shift and Moyal product, deriving first-order corrections to the mass $M$, temperature $\hat{T}$, and entropy $\hat{S}$, and verifying the first law $dM = \hat{T} d\hat{S} + V dP$ with $P = -\Lambda/(8\pi)$. The NC equation of state exhibits a van der Waals–like phase structure with a critical point given by $\frac{1}{T_c} = 4\pi \Theta L$, $v_c = \frac{8}{3}L\Theta$, $\frac{1}{P_c} = \frac{64}{3}\pi \Theta^2 L^2$, and a modified compact form $(P + \frac{1}{2\pi v^2})(v - \frac{\Theta L}{3}) = \hat{T} - \frac{2 \Theta L}{3 v^2}$. Notably, noncommutativity regularizes the Hawking temperature, introduces a finite minimal mass $M_0 = \Theta/4$, and the parameter $\Theta$ acts as a Planck-scale thermodynamic variable that influences stability and phase transitions between small and large black holes. Overall, the work demonstrates how quantum gravitational corrections from NC geometry imprint discernible thermodynamic effects on AdS black holes.

Abstract

We investigate the thermodynamic properties of a Schwarzschild-AdS black hole within the framework of noncommutative geometry. We derive and analyze the black hole's thermodynamic functions, showing that they depend critically on the noncommutativity parameter denoted as Θ, while still satisfying the first law of thermodynamics. Stability analysis reveals that the noncommutative Schwarzschild-AdS black hole undergoes a phase transition at a critical point. Moreover, the thermodynamic behavior closely resembles that of a van der Waals fluid, with the noncommutativity introducing a correction term to the black hole's surface temperature. Our results indicate that the noncommutativity parameter Θ is of the order of the Planck scale and functions as a novel thermodynamic variable within the system.

Figures (7)

• Figure 1: The black hole mass $M(r_{h})$ as a function of the horizon radius for different values of the noncommutative parameter $\Theta$. We set $\Lambda = -7$ and $L = 1$.
• Figure 2: The Hawking temperature $\hat{T}(r_{h})$ of a Schwarzschild–AdS black hole in noncommutative geometry. Parameters are set to $\Lambda = -7$ and $\Theta = 0, 0.1, 0.02, 0.03$.
• Figure 3: Entropy $\hat{S}(r_{h})$ of a Schwarzschild–AdS black hole in noncommutative geometry, plotted for $\Lambda = -7$, $L = 1$, and $\Theta = 0, 0.1, 0.02, 0.03$.
• Figure 4: Pressure $\hat{P}(v)$ of a Schwarzschild–AdS black hole in noncommutative geometry, where $v$ denotes the specific volume. Each curve corresponds to a different Hawking temperature, illustrating the variation of thermodynamic behavior with temperature.
• Figure 5: The heat capacity $\hat{C}(r_{h})$ of a Schwarzschild-AdS black hole in noncommutative geometry for $\Lambda = -7$ and $\Theta = 0,\, 0.01,\, 0.02$.