Coordinate-Independent Definition of Relative Velocity in Pseudo-Riemannian Space-Time: Implications for Special Cases

Using the general solution that we recently obtained for the coordinate-independent definition of a relative velocity of a luminous source as measured along the observer’s line of sight in generic pseudo-Riemannian space-time, in the present article we invoke important implications for test particles and observers in several instructive cases. We consider a test particle as a luminous object, otherwise, if it is not, we assume that a luminous source is attached to it, which has neither mass nor volume. We calculate the relative velocities in special metrics: the Minkowski metric, the test particle and observer at rest in an arbitrary stationary metric, a uniform gravitational field, a rotating reference frame, the Schwarzschild metric, a Kerr-type metrics, and the spatially homogeneous and isotropic Robertson–Walker space-time of the standard cosmological model. In the last case, it leads to a remarkable cosmological consequence that the resulting, so-called, kinetic recession velocity of an astronomical object is always subluminal even for large redshifts of order one or more, so that it does not violate the fundamental physical principle of causality. We also calculate the carrying-away measure of a galaxy at redshift \(z\) by the expansion of space, which proves, in particular, that the cosmological expansion of a flat 3D space is fundamentally different from the kinematics of galaxies moving in a nonexpanding flat 3D space.