Analytical time-like geodesics

Abstract

Time-like orbits in Schwarzschild space-time are presented and classified in a very transparent and straightforward way into four types. The analytical solutions to orbit, time, and proper time equations are given for all orbit types in the form r=r(\lambda), t=t(\chi), and \tau=\tau(\chi), where \lambda\ is the true anomaly and \chi\ is a parameter along the orbit. A very simple relation between \lambda\ and \chi\ is also shown. These solutions are very useful for modeling temporal evolution of transient phenomena near black holes since they are expressed with Jacobi elliptic functions and elliptic integrals, which can be calculated very efficiently and accurately.