Relative velocities, geometry, and expansion of space
  What does it mean to say that space expands? One approach to this question is
the study of relative velocities. In this context, a non local test particle is
"superluminal" if its relative velocity exceeds the local speed of light of the
observer. The existence of superluminal relative velocities of receding test
particles, in a particular cosmological model, suggests itself as a possible
criterion for expansion of space in that model. In this point of view,
superluminal velocities of distant receding galaxy clusters result from the
expansion of space between the observer and the clusters. However, there is a
fundamental ambiguity that must be resolved before this approach can be
meaningful. The notion of relative velocity of a nonlocal object depends on the
choice of coordinates, and this ambiguity suggests the need for coordinate
independent definitions. In this work, we review four (inequivalent)
geometrically defined and universal notions of relative velocity: Fermi,
kinematic, astrometric, and spectroscopic relative velocities. We apply this
formalism to test particles undergoing radial motion relative to comoving
observers in expanding Robertson-Walker cosmologies, and include previously
unpublished results on Fermi coordinates for a class of inflationary
cosmologies. We compare relative velocities to each other, and show how pairs
of them determine geometric properties of the spacetime, including the scale
factor with sufficient data. Necessary and sufficient conditions are given for
the existence of superluminal recessional Fermi speeds in general
Robertson-Walker cosmologies. We conclude with a discussion of expansion of
space.
