**Analysis of Particle Trajectories Around Schwarzschild and Extreme Black Holes**

Einstein’s theory of general relativity is one of the most critically tested and verified modern theories of physical phenomena. General relativity has successfully described and quantified space-time surrounding massive objects, including black holes, which are among the most enigmatic features of the universe. For black holes with no angular momentum, the Schwarzschild metric can be used to describe the curved space-time surrounding the black hole. A Lagrangian can be constructed from the Schwarzschild metric to provide a system of first-order differential equations for describing the motion of a particle surrounding the black hole. Utilizing the computational accuracy of the Runge-Kutta method for differential equation iteration, trajectories were determined for a variety of initial radii and energies. From the Schwarzschild metric, other useful quantities including proper time and coordinate (wristwatch) time were calculated to provide clarification of gravitational time dilation in highly curved space-time regions.