The Boundary Homotopy Retract on the Scalar Hairy Charged Black Hole Spacetime

In this paper, we investigate and define the topology of some astrophysical phenomena, like the hairy (scalarized) charged black hole spacetime, to improve our understanding of the kinematics and dynamics of their nature. We use the Lagrangian equation to find different types of geodesic equations. This can be done under some conditions for the variations of the Cosmological constant and Newton's constant. We show how to induce the two types (null and spacelike) of geodesics as boundary retractions, in order to obtain the boundary homotopy retract of the scalar charged black hole. These types are used the Lagrangian equation in a 4-D scalar charged black hole to explain the event horizon for this black hole.