Friedmann cosmologies via the Regge calculus

The detailed construction of six Regge spacetimes, each being an approximation to a time symmetric Friedmann dust-filled Universe, are presented. These spacetimes are a generalisation of those originally constructed by Collins and Williams (1973). This paper presents new methods for the subdivision of each Cauchy surface into a set of tetrahedra, for the construction of the general four-dimensional block and for the implementation of the constraints of homogeneity and isotropy. A new action sum for pure dust in a Regge spacetime is also presented. The evolution of the Regge spaces is seen to terminate prior to the full collapse of the Universe. This is shown to occur when the particle horizon for an observer at the centre of one tetrahedron has contracted so as to just touch the vertices of that tetrahedron. It is argued that this is a generic feature and will occur in any Regge spacetime whenever the local curvature becomes too large.