Recessional velocities and Hubble's law in Schwarzschild-de Sitter space
  We consider a spacetime with empty Schwarzschild-de Sitter exterior and
Schwarzschild-de Sitter interior metric for a spherical fluid with constant
density. The fluid interior may be taken to represent a galaxy supercluster,
for which the proper distance from the center of the supercluster to the
cosmological horizon has the same order of magnitude as the Hubble radius
derived from Friedmann-Robertson-Walker (FRW) cosmologies. The fluid interior
and surrounding vacuum may also be considered as a model of the Local Group of
galaxies in the far future. Particle motion is subject both to the attractive
gravity exerted by the fluid and the repelling cosmological constant. Using
global Fermi coordinates for the central observer within the fluid, the Fermi
velocity, the astrometric velocity, the kinematic velocity, and the
spectroscopic velocity, relative to the central (Fermi) observer, of a radially
receding test particle are calculated and compared. We find that the Fermi
relative velocity can exceed the speed of light in this model, but the presence
of a positive cosmological constant causes recessional speeds of distant high
energy particles to decrease rather than increase. We derive a version of
Hubble's law for this spacetime which might be applicable for the analysis of a
receding mass within a great void adjacent to a supercluster, relatively
isolated from gravitational sources other than the supercluster. We also
compare some of our results to related behavior in FRW cosmologies and consider
implications to arguments regarding the expansion of space.
