Abstract
The closed Friedmann--Lema\^itre--Robertson--Walker (FLRW) universe of Einstein gravity with positive cosmological constant in three dimensions is investigated by using the Collins--Williams formalism in Regge calculus. A spherical Cauchy surface is replaced with regular polyhedrons. The Regge equations are reduced to differential equations in the continuum time limit. Numerical solutions to the Regge equations approximate well the continuum FLRW universe during the era of small edge length. The deviation from the continuum solution becomes larger and larger with time. Unlike the continuum universe, the polyhedral universe expands to infinite within finite time. To remedy the shortcoming of the model universe we introduce geodesic domes and pseudo-regular polyhedrons. It is shown that the pseudo-regular polyhedron model can approximate well the results of the Regge calculus for the geodesic domes. The pseudo-regular polyhedron model approaches the continuum solution in the infinite frequency limit.
Highlights
Regge calculus was proposed to formulate Einstein’s general relativity on piecewise linear manifolds [1, 2]
In this note we investigate the Friedmann–Lemaıtre–Robertson–Walker (FLRW) universe of three-dimensional Einstein gravity with positive cosmological constant in Regge calculus by taking polyhedrons as the Cauchy surface
It is not difficult to carry out the Regge calculus for ν small, as we show in the appendix
Summary
Regge calculus was proposed to formulate Einstein’s general relativity on piecewise linear manifolds [1, 2]. The Regge action is written in coordinate-free form It is a highly complicated function of the edge length depending heavily on the triangulations of the space-time. This makes investigations of how the theory behaves with respect to refinement of the triangulation much more involved than the lattice gauge theory. Regge calculus has been applied to the four-dimensional closed FLRW universe by Collins and Williams [7] They considered regular polytopes as the Cauchy surfaces of the discrete FLRW universe and used, instead of simplices, truncated world-tubes evolving from one Cauchy surface to the as the building blocks of piecewise linear space-time. In Appendix A, the Regge calculus for the first two, simplest, geodesic domes is described
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