Hubble's law and faster than light expansion speeds

TL;DR

Hubble's law is only meaningful for nearby objects with non-relativistic speeds, as the concept of relative velocity at cosmological distances is geometrically ill-defined in curved spacetime.

Contribution

The paper clarifies the geometrical interpretation of Hubble's receding speed using relativistic geometry and shows its limitations at large cosmological distances.

Findings

Hubble's law is valid only for non-relativistic receding speeds.

Relative velocity in cosmology cannot be straightforwardly defined at large distances.

Hubble's speed corresponds to rapidity in Lorentz transformations when properly interpreted.

Abstract

Naively applying Hubble's law to a sufficiently distant object gives a receding velocity larger than the speed of light. By discussing a very similar situation in special relativity, we argue that Hubble's law is meaningful only for nearby objects with non-relativistic receding speeds. To support this claim, we note that in a curved spacetime manifold it is not possible to directly compare tangent vectors at different points, and thus there is no natural definition of relative velocity between two spatially separated objects in cosmology. We clarify the geometrical meaning of the Hubble's receding speed v by showing that in a Friedmann-Robertson-Walker spacetime if the four-velocity vector of a comoving object is parallel-transported along the straight line in flat comoving coordinates to the position of a second comoving object, then v/c actually becomes the rapidity of the local Lorentz transformation, which maps the fixed four-velocity vector to the transported one.