Coordinates in General Relativity: Orbit, Velocity, and Time form Perihelion to Aphelion

In general relativity, coordinate transformations are often made to simplify calculations, and theoretical predictions are calculated in some specific coordinates. We take the test particle’s motion in Schwarzschild space-time as an example, to illustrate that the solutions for orbit and velocity as well as the time from perihelion to aphelion depend on the coordinates employed for the calculations, even if they are formulated in terms of orbital energy and angular momentum. The aim of this work is to demonstrate that coordinate transformations may change the solutions, and solutions achieved in specific coordinates may not be the final answer and should be mapped into the observer’s reference frame for being compared with observations.