On a Bipolar Model of Hyperbolic Geometry and its Relation to Hyperbolic
  Robertson-Walker Space
  Negatively curved, or hyperbolic, regions of space in an FRW universe are a
realistic possibility. These regions might occur in voids where there is no
dark matter with only dark energy present. Hyperbolic space is strange and
various "models" of hyperbolic space have been introduced, each offering some
enlightened view. In the present work we develop a new bipolar model of
hyperbolic geometry, closely related to an existing model - the band model -
and show that it provides new insights toward an understanding of hyperbolic as
well as elliptic Robertson-Walker space and the meaning of its isometries. In
particular, we show that the circular geodesics of a hyperbolic
Robertson-Walker space can be referenced to two real centers - a Euclidean
center and an offset hyperbolic center. These are not the Euclidean center or
poles of the bipolar coordinate system but rather refer to two distinct centers
for circular orbits of particles in such systems. Considering the physics of
elliptic RW space is so well confirmed in the Lambda-CDM model with respect to
Euclidean coordinates from a Euclidean center, it is likely that the hyperbolic
center plays a physical role in regions of hyperbolic space.
