Relative velocity in pseudo-Riemannian spacetime

Abstract

We give a coordinate independent definition of relative velocity of test particle in pseudo-Riemannian spacetime as measured along the observer’s line-of-sight in general and several instructive cases. In doing this, the test particle is considered as a luminous object, otherwise, if it is not, we assume that a light source is attached to it, which has neither mass nor volume. Then we utilize the general solution of independent definition of relative velocity of a luminous source in generic pseudo-Riemannian spacetime. As a corollary, we discuss the implications for the Minkowski metric, the test particle and observer at rest in an arbitrary stationary metric, the uniform gravitational field, the rotating reference frame, the Schwarzschild metric, the Kerr-type metrics, and the spatially homogeneous and isotropic Robertson-Walker (RW) spacetime of standard cosmological model. In the last case, it leads to 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 measure of carrying away of a galaxy at redshift z by the expansion of space, which proves, in particular, that cosmological expansion of a flat 3D–space is fundamentally different from a kinematics of galaxies moving in a non-expanding flat 3D-space. So, it is impossible to mimic the true cosmological redshift by a Doppler effect caused by motion of galaxies in a non-expanding 3D-space, flat or curved. We also give a reappraisal of the `standard ́ kinematic interpretation of redshifts in RW spacetime as accumulated Doppler-shifts.