Born–Infeld AdS black holes surrounded by perfect fluid dark matter

With an intent to study the combined effect of Born–Infeld field as a nonlinear electrodynamics (BI-NED) and perfect fluid dark matter (PFDM) to Einstein- Λ Λ theory of gravity, we obtain the exact charged AdS black hole solutions. Next, we study the effects of the parameters of PFDM and BI on the event horizon of these black holes. Then, we compute the conserved and thermodynamical quantities and evaluate that these thermodynamical quantities can satisfy the first law of thermodynamics. In addition, we investigate thermal stability conditions for these black hole solutions in context of canonical ensemble by using the heat capacity and the Helmholtz free energy. We study the effects of the parameters of PFDM and BI-NED on the local and global stability areas. Next, we extend our study to investigate thermodynamical properties of BI-NED AdS black holes surrounded by PFDM in extended phase space by considering the cosmological constant as a thermodynamics pressure. We, obtain the conserved and thermodynamic quantities which can satisfy the first law of thermodynamics in extended phase space. We then perform a rigorous analytical verification of the Ehrenfest equations to determine whether the critical behavior of the charged BI-NED AdS black holes surrounded by PFDM corresponds to a second-order phase transition. Our findings indicate that both Ehrenfest relations are satisfied, confirming that the black hole system undergoes a second-order phase transition at the critical point. we obtain heat engines corresponding to these black holes. The goal is to see how black holes’ parameters such as parameters of PFDM and BI-NED would affect efficiency of the heat engines. Finally, we study the geodesic structure of the spacetime by analyzing both timelike and null geodesics. The effective potential is examined to determine the conditions for stable and unstable circular orbits, as well as the behavior of the innermost stable circular orbit and the photon sphere. We show that the PFDM parameter significantly modifies the orbital structure and potential barrier, while the Born–Infeld parameter produces comparatively weaker corrections. The existence of bound, critical, and unstable trajectories highlights the dynamical impact of dark matter and nonlinear electrodynamics on particle and photon motion around the black hole.