Noncommutative black holes in extended anti-de Sitter phase space

Abstract

We study thermodynamic aspects of ordinary and lower dimensional noncommutative black holes within an extended anti-de Sitter phase space by treating the negative cosmological constant and the minimal cut-off length as thermodynamic variables representing the pressure and tension of the system, respectively. In four-dimensional spacetime, the regular black hole exhibits a small/large black hole phase transition analogous to the liquid/gas transition of a Van der Waals gas. The three-dimensional case demonstrates global and local thermodynamic stability, while the two-dimensional case reveals a novel type of transition referred to as the anti-Hawking-Page transition.

Figures (21)

• Figure 1: We set $l=20\sqrt{\theta}$ and we plot the metric potential \ref{['4dmetric']} for $M/l=0.075$ (blue solid curve), for $M_0/l\approx 0.097$ (black dashed curve) and for $M/l=0.125$ (red solid curve).
• Figure 2: The temperature of a ($3+1$)-dimensional noncommutative black hole is displayed for $l=18 \sqrt{\theta}\,$ (red solid curve), for $l = l_{\mathrm{c}} \approx 11.15 \sqrt{\theta}\,$ (black dashed curve) and for $l=8 \sqrt{\theta}\,$ (blue solid curve). The dotted curves represent the conventional Hawking-Page temperatures HaP83.
• Figure 3: The entropy is plotted for $\theta=1\,$. The red dots represent the noncommutative black hole entropy, while the blue dots represent the usual area law.
• Figure 4: The black hole volume is displayed for $\theta=1\,.$ The red solid curve represents the volume of the noncommutative black hole, while the blue dashed curve represents the conventional volume $V_{\mathrm{AdS}}$ of a singular black hole. The black dot spots the volume and the radius of the extremal black hole.
• Figure 5: The function $\frac{2}{\sqrt{\pi}}\Gamma(r_+)\,$vs$r_+^2/4\theta\,$.