Photon orbits and phase transition for gravitational decoupled Kerr anti-de Sitter black holes

Abstract: Interpreting the cosmological constant as the energy of the vacuum and using a gravitational decoupling approach leads to a new Kerr--anti-de Sitter (AdS) black hole. The metric of the new Kerr--AdS is simpler than the standard Kerr--AdS and exhibits richer geometry, where the effects of rotation appear as warped curvature. We investigate the relationship between unstable photon orbits and thermodynamic phase transitions in this new Kerr--AdS black hole background. We derive an exact expression for various thermodynamic properties, including mass ($M$), Hawking temperature ($T$), entropy ($S$), heat capacity ($C$), and free energy ($G$), by relating the negative cosmological constant to positive pressure through the equation $P = -\Lambda/(8 \pi) = 3/(8 \pi l^2)$, where $l$ represents the horizon radius, and by introducing its conjugate variable as the thermodynamic volume $V$. When $P < P_c$, black holes with $C_P > 0$ are thermodynamically stable, while those with $C_P \leq 0$ are unstable. Our analysis of the Gibbs free energy reveals a phase transition from small, globally unstable black holes to large, globally stable ones. Additionally, investigating the system's $P$-$V$ criticality and determining the critical exponents shows that our system shares similarities with a Van der Waals (vdW) fluid. In the reduced parameter space, we observe non-monotonic behavior of the photon sphere radius and the critical impact parameter when the pressure is below its critical value. Furthermore, we present the distribution of critical points in parameter space and derive a fitting formula for the coexistence curve.