Evaluation of plasma on deflection angle and shadow by a black hole solution immersed in perfect fluid in Rastall theory

The present study examines the gravitational deflection of particles in curved space-times immersed in perfect fluid in the context of Rastall theory, applying the Gibbons-Werner useful technique. In its application to integral areas inside a four-dimensional space-time, the Gauss-Bonnet theorem studies the computation expression of the deflection angle. The Gibbons-Werner technique has two limitations in Rastall theory for space-times immersed in perfect fluid: the integral region is generally infinite and the integral complicates calculation, particularly for complex space-times and extremely accurate solutions. An infinite region approach to Gibbons-Werner is proposed to avoid singularity. For demonstrating the Gibbons-Werner method, we use a complete Rastall theory framework. It studies black hole solutions like the dust field, radiation field, quintessence field, cosmological constant field and phantom field using the Gibbons-Werner approach. Additionally, we check the deflection angle from these space-times under the influence of plasma. Furthermore, we compute analytically the influence of a plasma on a black hole shadow by using a ray-tracing approach. In the Hamiltonian equation, our model describes the plasma, so the light ray motion equations are independent of the plasma's velocity. It is assumed that plasma is a dispersive medium, pressure-less and non-magnetized. We investigate the perfect fluid in Rastall theory in further depth, where the plasma particle density corresponds to particle accumulation. The supermassive black hole shadows are explored in the case when plasma falls radially from infinity onto a black hole.