Relative velocities for radial motion in expanding Robertson-Walker spacetimes

TL;DR

This paper compares different notions of relative velocity for radially moving test particles in expanding Robertson-Walker spacetimes, revealing conditions for superluminal speeds and their geometric implications.

Contribution

It provides analytical and numerical comparisons of four types of relative velocities in various cosmological models, clarifying their properties and limitations.

Findings

Astrometric velocity cannot be superluminal in expanding RW spacetimes.

Conditions for superluminal Fermi speeds are established.

Different velocity notions reveal geometric properties of the universe.

Abstract

The expansion of space, and other geometric properties of cosmological models, can be studied using geometrically defined notions of relative velocity. In this paper, we consider test particles undergoing radial motion relative to comoving (geodesic) observers in Robertson-Walker cosmologies, whose scale factors are increasing functions of cosmological time. Analytical and numerical comparisons of the Fermi, kinematic, astrometric, and the spectroscopic relative velocities of test particles are given under general circumstances. Examples include recessional comoving test particles in the de Sitter universe, the radiation-dominated universe, and the matter-dominated universe. Three distinct coordinate charts, each with different notions of simultaneity, are employed in the calculations. It is shown that the astrometric relative velocity of a radially receding test particle cannot be superluminal in any expanding Robertson-Walker spacetime. However, necessary and sufficient conditions are given for the existence of superluminal Fermi speeds, and it is shown how the four concepts of relative velocity determine geometric properties of the spacetime.