35,405 research outputs found
Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations
Causal Dynamical Triangulations is a non-perturbative quantum gravity model,
defined with a lattice cut-off. The model can be viewed as defined with a
proper time but with no reference to any three-dimensional spatial background
geometry. It has four phases, depending on the parameters (the coupling
constants) of the model. The particularly interesting behavior is observed in
the so-called de Sitter phase, where the spatial three-volume distribution as a
function of proper time has a semi-classical behavior which can be obtained
from an effective mini-superspace action. In the case of the three-sphere
spatial topology, it has been difficult to extend the effective semi-classical
description in terms of proper time and spatial three-volume to include genuine
spatial coordinates, partially because of the background independence inherent
in the model. However, if the spatial topology is that of a three-torus, it is
possible to define a number of new observables that might serve as spatial
coordinates as well as new observables related to the winding numbers of the
three-dimensional torus. The present paper outlines how to define the
observables, and how they can be used in numerical simulations of the model.Comment: 26 pages, 15 figure
Fundamental Domains in Lorentzian Geometry
We consider discrete subgroups Gamma of the simply connected Lie group
SU~(1,1), the universal cover of SU(1,1), of finite level, i.e. the subgroup
intersects the centre of SU~(1,1) in a subgroup of finite index, this index is
called the level of the group. The Killing form induces a Lorentzian metric of
constant curvature on the Lie group SU~(1,1). The discrete subgroup Gamma acts
on SU~(1,1) by left translations. We describe the Lorentz space form
SU~(1,1)/Gamma by constructing a fundamental domain F for Gamma. We want F to
be a polyhedron with totally geodesic faces. We construct such F for all Gamma
satisfying the following condition: The image of Gamma in PSU(1,1) has a fixed
point u in the unit disk of order larger than the index of Gamma. The
construction depends on the group Gamma and on the orbit Gamma(u) of the fixed
point u.Comment: 16 pages with 5 figures; typos corrected; introduction complete
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