Supposed i wanted to do a relativistic simulation of charged point particles moving at different velocities and interacting with each other. My simulation would give me the x,y,z coordinates of each particle seen from an arbitrary observer's point of view, at a given t. The t given however, is the simultaneity axis. Basically a 3D slice of 4D spacetime. The problem i would be facing, as it seems to me, is that when i compute the next slice, i would use charged particles located on that simultaneity axis. But would that be the right way to go about it? Creating a simulation for the twin paradox, it appeared to me, that the simultaneity axis, while mathematically and physically well defined, does not really represent the "present" or what we call as "now" but only serves as a point(axis) of reference. Events on the simultaneity axis do happen, but only the even at x=0 is an even which happens in the "now". When computing the next 3D slice of my simulation however, i would have to use events that are in the "present" or wouldn't I? Maybe my approach is corrupted altogether, so maybe someone can shine some light on this. How would a physicists go about creating the most accurate relativistic simulation of charged particles in empty space?