Relativistic Signatures of Dark Matter Equations of State in Static Spherically Symmetric Spacetimes

We study how three physically motivated dark matter equations of state alter the spacetime geometry of static, spherically symmetric black holes in General Relativity. The models considered are anisotropic perfect fluid dark matter (PFDM), isotropic constant-$\omega$ dark matter, and Bose-Einstein condensate (BEC) dark matter with a polytropic equation of state (EoS). For each model, we derive the Einstein field equations and solve for the metric function analytically in the PFDM and constant-$\omega$ cases, and numerically via the Tolman-Oppenheimer-Volkoff equations for BEC dark matter. We then compute the key strong gravity observables: event horizon radius, photon sphere radius, black hole shadow radius, innermost stable circular orbit (ISCO), and circular orbital velocity profiles. The shadow radii obtained for each model are compared against the Event Horizon Telescope constraints to place bounds on the dark matter parameters. Physical viability is assessed through the null, weak, dominant, and strong energy conditions, along with causality requirements on the sound speed. Our results show that PFDM produces near Schwarzschild geometry with mild anisotropic corrections, the constant-$\omega$ model is strongly restricted by causality to $0 \leq \omega \leq 1$, and BEC dark matter generates smooth, bounded deviations governed by the bosonic self-interaction parameter K. Taken together, these findings show that strong gravity observables can serve as practical tools for distinguishing between competing dark matter models through their geometric imprints on black hole space times.