Analyzing the Influence of Geometrical Deformation on Photon Sphere and Shadow Radius: A New Analytical Approach -- Spherically Symmetric Spacetimes

In this paper, we introduce a new approach to study the behavior of the photon sphere and shadow radius. Our method uses extended gravitational decoupling and reveals two important analytic results. First, the additional matter field alters the photon sphere radius: it increases if $g'(r_{ph}^{(0)})>0$ and decreases if $g'(r_{ph}^{(0)})<0$ (where $g'$ represents the derivative of a specific metric function evaluated at the original photon sphere radius). Second, the presence of the matter field can modify the black hole shadow size. If $g\left(r_{ph}^{(0)}\right)>0$, the shadow shrinks, while it grows for $g\left(r_{ph}^{(0)}\right)<0$. These findings provide a deeper insight into how matter distribution influences the characteristics of black holes and their observable features. Through a systematic framework and various illustrative examples, our investigation not only clarifies these fundamental aspects but also significantly enhances the theoretical framework of black hole astrophysics.