Unique definition of relative speed along the line of sight of a luminous object in a Riemannian space-time

Abstract

Using a way of separating the spectral shifts into infinitesimally displaced `relative´ spectral bins and sum over them, we overcome the ambiguity of the parallel transport of four-velocity, in order to give an unique definition of the so-called kinetic relative velocity of luminous source as measured along the observer’s line-of-sight in a generic pseudo-Riemannian space-time. The ubiquitous relationship between the spectral shift and the kinetic relative velocity is utterly distinct from a familiar global Doppler shift rule (Synge, 1960). Such a performance of having found a kinetic relative velocity of luminous source, without subjecting it to a parallel transport, manifests its virtue in particular case when adjacent observers are being in free fall and populated along the null geodesic, so that it is reduced to a global Doppler velocity as studied by Synge. We discuss the implications for the instructive case of spatially homogeneous and isotropic Robertson-Walker space-time, which leads to cosmological consequences that the resulting kinetic recession velocity of a galaxy is always subluminal even for large redshifts of order one or more, and thus, it does not violate the fundamental physical principle of causality.