On a Bipolar Model of Hyperbolic Geometry and its Relation to Hyperbolic Friedmann-Robertson-Walker Space

Abstract: Negatively curved regions of space in a Friedmann-Robertson-Walker (FRW) universe are a realistic possibility. These regions would occur in voids in the large-scale structure of the universe where there is no dark matter with only dark energy present. Hyperbolic space is strange from a physical point of view and various models of hyperbolic space have been introduced, each offering a clarifying view. In the present work we develop a new bipolar model of hyperbolic geometry and show that it provides new insights toward an understanding of hyperbolic as well as elliptic FRW space. In particular, using the bipolar model, we show that the circular geodesics of an FRW space can be referenced to two real centers – a Euclidean center and a hyperbolic center. Considering the physics of elliptic FRW space is so well confirmed in the CDM model describing the expansion of the universe with respect to a Euclidean center, it is possible that the hyperbolic center also plays a physical role in regions of hyperbolic space.