Abstract
In this paper we propose to use Lie sphere geometry as a new tool to systematically construct time-symmetric initial data for a wide variety of generalised black-hole configurations in lattice cosmology. These configurations are iteratively constructed analytically and may have any degree of geometric irregularity. We show that for negligible amounts of dust these solutions are similar to the swiss-cheese models at the moment of maximal expansion. As Lie sphere geometry has so far not received much attention in cosmology, we will devote a large part of this paper to explain its geometric background in a language familiar to general relativists.
Highlights
In their seminal paper [37] of 1957, Richard Lindquist and John Wheeler introduced the idea to approximate the global dynamics of homogeneous and isotropic cosmological models by latticelike configurations of vacuum Schwarzschild geometries
In this paper we propose to use Lie sphere geometry as a new tool to systematically construct time-symmetric initial data for a wide variety of generalised black-hole configurations in lattice cosmology
Our method presented in this paper can be seen as a significant generalisation of theirs, resting on a novel application of Lie sphere geometry, that so far does not seem to have enjoyed any application to cosmological model-building whatsoever
Summary
In their seminal paper [37] of 1957, Richard Lindquist and John Wheeler introduced the idea to approximate the global dynamics of homogeneous and isotropic cosmological models by latticelike configurations of vacuum Schwarzschild geometries. Inside the balls the geometry is strictly spherically symmetric, even though the distribution of blackholes around them on neighbouring vertices is only approximately so This is because the remaining dust just enforces this symmetry by construction. For regular lattices, it has been argued in [16] that the resulting local discrete rotation and reflection symmetries suffice to render the Einstein evolution equations ordinary (rather than partial) differential equations for points in the one-dimensional fixed-point set of these symmetries, effectively decoupling the evolution of the geometry at these points from that of their spatial neighbours If this were true, long term predictions for the geometry of these lower-dimensional structures could be made, as claimed in [16]. This paper is based in parts on [22]
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