Objective of this work is to study whether some of the known non-gravitational phenomena can be explained as motion on a straight line as gravity is treated within General Relativity. To do that, we explore a metric field on a complexified manifold with holomorphic coordinates. Specifically we look into the behaviour of geodesics on such a smooth complex manifold and the path traced out by its real component. This yields a family of equations of motions in real coordinates which is shown to have deviations from usual geodesic equation and in that way expands the geodesic to capture contributions from additional fields and interactions beyond the mere gravitational ones as a function of the metric field.