Abstract
Abstract The discretized closed Friedmann–Lemaître–Robertson–Walker (FLRW) universe with positive cosmological constant is investigated by Regge calculus. According to the Collins–Williams formalism, a hyperspherical Cauchy surface is replaced with regular 4-polytopes. Numerical solutions to the Regge equations approximate well to the continuum solution during the era of small edge length. Unlike the expanding polyhedral universe in three dimensions, the 4-polytopal universes repeat expansions and contractions. To go beyond the approximation using regular 4-polytopes we introduce pseudo-regular 4-polytopes by averaging the dihedral angles of the tessellated regular 600-cell. The degree of precision of the tessellation is called the frequency. Regge equations for the pseudo-regular 4-polytope have simple and unique expressions for any frequency. In the infinite frequency limit, the pseudo-regular 4-polytope model approaches the continuum FLRW universe.
Highlights
Regge calculus was proposed in 1961 to formulate Einstein’s general relativity on piecewise linear manifolds [1, 2]
Regge calculus has been applied to the four-dimensional closed FLRW universe by Collins and Williams [4]
They considered regular polytopes (4-polytopes or polychora) 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
Summary
Regge calculus was proposed in 1961 to formulate Einstein’s general relativity on piecewise linear manifolds [1, 2]. Regge calculus has been applied to the four-dimensional closed FLRW universe by Collins and Williams [4] They considered regular polytopes (4-polytopes or polychora) 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. Liu and Williams have extensively studied the discrete FLRW universe [8,9,10] They found that a universe with regular 4-polytopes such as the Cauchy surfaces can reproduce the continuum FLRW universe to a certain degree of precision. Regge equations for the pseudo-regular polyhedron model turned out to approximate the corresponding geodesic dome universe well It is worth investigating whether a similar approach can be extended to higher dimensions. We consider all six types of regular 4-polytopes as the Cauchy surface in a unified way in terms of the Schlafli k=1
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