Cosmic event horizons and the light-speed limit for relative radial
  motion
  Cosmic event horizons separate spacetime into disjoint regions: those regions
whose light signals can reach us, and more distant regions we cannot, even in
principle, observe. For one type of cosmic horizon, associated with universes
that keep expanding forever, there is a simple intuitive picture of where the
cosmic horizon is located, in terms of the relative speed between a distant
galaxy and our own approaching the speed of light: Where the light-speed limit
is reached, light signals from that distant galaxy will not be able to catch up
with our own galaxy; that galaxy and more distant galaxies are behind the
cosmic horizon. Applied to the usual recession speeds of galaxies, that simple
picture turns out to be wrong. But there is another relevant concept of speed,
derived from the relativistic relative velocity of galaxies, which in turn is
defined via the parallel transport of four-velocities. This article shows how,
using this concept of relative velocity, key elements of the intuitive picture
are valid, and can be used in a simplified explanation for cosmic horizons.
When phrased as a statement about relative radial velocities, the notion of
light not being able to catch up with our own galaxy is consistent with the
calculations, and the cosmic event horizon is indeed the limit where the
relative radial velocity of distant Hubble-flow galaxies reaches the speed of
light. While the derivation itself requires advanced concepts of general
relativity, we also discuss consequences of this result for teaching about
cosmic event horizons in less advanced settings, where students are not
introduced to the full formalism of general relativity.
