Thermodynamics and Geometrical Optics of Reissner Nordstrom de Sitter Black Holes in Noncommutative Geometry

Abstract

We investigate the thermodynamic, optical, and dynamical properties of Reissner-Nordstrom-de Sitter black holes in a noncommutative spacetime with a minimal length scale Theta. Within a two-horizon framework, we formulate an effective first law of thermodynamics and introduce an entropy capturing correlations between the event and cosmological horizons. Imposing the lukewarm condition, where both horizons share a common temperature, uniquely determines the entropy correction and yields closed-form expressions for thermodynamic quantities. The analysis reveals a noncommutativity-induced second-order phase transition, emphasizing the role of short-distance structure. On the optical side, we study photon motion and weak gravitational lensing, showing that noncommutativity modifies the effective potential and critical impact parameter. Using the Gauss-Bonnet method, we derive the weak deflection angle and analyze the effects of charge and cosmological constant. We further connect geometry and dynamics through the Lyapunov exponent and quasinormal modes, showing systematic impacts on orbital instability and damping.

Figures (9)

• Figure 1: Physical region in the $(Q,\Theta)$ plane with $r_c=1$. The colour scale represents the numerical extremal horizon ratio $x_{\rm min}$, above which black hole horizon exists. The black and white curves represent the contour lines of constant $x_{\rm min}$.
• Figure 2: Behavior of the effective temperature $T_{\mathrm{eff}}(x)$ for the Schwarzschild--de Sitter black hole ($Q=0$) with $r_c=1$, shown for several values of the noncommutative parameter $\Theta$. For each curve, the marked point represents the minimum position $x_{\min}$.
• Figure 3: Behavior of the effective temperature $T_{\mathrm{eff}}$ as a function of $x$ for $Q = 0.1$, $r_c=1$, and different values of the noncommutative parameter $\Theta$. For each curve, the marked point represents the minimum position $x_{\min}$.
• Figure 4: Heat capacity $C_V$ as a function of $x$ and the effective temperature $T_{\mathrm{eff}}$ for fixed $Q=0.3$, $V=3$ with $\Theta = 0,\,0.005,\,0.010,\,0.015,$ and $0.020$.
• Figure 5: Entropy $S$ as a function of the effective temperature $T_{\mathrm{eff}}$ and the parameter $x$ for fixed charge $Q=0.3$, $V=3$ with $\Theta = 0,\,0.005,\,0.010,\,0.015,$ and $0.020$.