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.