Black hole solutions surrounded by an anisotropic fluid in a Kalb--Ramond two--form background

Abstract: We investigate static, spherically symmetric black hole spacetimes induced by the spontaneous Lorentz symmetry breaking of a Kalb--Ramond (KR) two-form field non--minimally coupled to gravity, coexisting with an anisotropic fluid. By adopting a general equation of state where the radial pressure relates to the energy density via $w_1 = -1$ and the tangential pressure via an arbitrary parameter $w_2$, we derive exact analytical solutions representing black holes surrounded by diverse matter fields, including dust ($w_2=0$), radiation ($w_2=1/3$), and dark energy-like distributions ($w_2=-1/2$). A rigorous analysis of curvature invariants confirms a genuine core singularity, while the global geometry and adherence to standard energy conditions are shown to be highly sensitive to the interplay between the KR coupling ($\ell$), the fluid density parameter ($K$), and $w_2$. Furthermore, we analyse null geodesics in detail to determine the photon sphere and shadow radii. Using the Gibbons--Werner geometrical approach and the Gauss-Bonnet theorem applied to the optical metric, we compute the weak deflection angle of light, demonstrating that both the KR field and the anisotropic fluid significantly enhance light bending, particularly in dark-energy-like backgrounds. Finally, we evaluate strong deflection limit (SDL) observables for the supermassive black holes Sgr A$^*$ and M87$^*$, revealing quantifiable deviations from standard Schwarzschild geometries. These results offer novel astrophysical signatures for constraining string-inspired KR gravity and anisotropic dark matter halos using current and future observations.