Stable Orbit of Massive Klein Gordon Field near Schwarzschild Black Hole and Its Quasi-Normal Modes

The findings of bound states of the massive Klein-Gordon field near the Schwarzschild black hole have provided certain conditions for a stable orbit. Unlike classical methods, I have obtained a specific relation between the mass of the scalar particle and the distance from the black hole. Furthermore, by applying a polynomial condition to the Heun function, more accurate predictions of quasi-normal modes have been made. Based on this, the time of a stable orbit and its relationship with mass, polynomial order, and distance from the Schwarzschild black hole have been calculated.