Properties of dynamical fractal geometries in the model of causal dynamical triangulations

We investigate the geometry of a quantum universe with the topology of the four-torus. The study of noncontractible geodesic loops reveals that a typical quantum geometry consists of a small semiclassical toroidal bulk part, dressed with many outgrowths, which contain most of the four-volume and which have almost spherical topologies, but nevertheless are quite fractal.

Figure 1

Up: Illustration of a torus with outgrowths. The blue and red lines represent two nonequivalent and noncontractible loops. The green loop is the shortest loop passing through the green point in the same direction as the blue line. Down: Embedding of a triangulation of the two-torus consisting of 150000 triangles into the Euclidean plane (picture from [21]). Shown in red is the shortest noncontractible loop.

Figure 2

Distributions of a sum of distances from simplices to the two opposite boundaries in the $x$ direction (red), the $y$ direction (green) and $z$ direction (blue) for systems with $N^{\{4,1\}}=80\mathrm{k}$ (up) and $N^{\{4,1\}}=160\mathrm{k}$ (down). The distributions scale consistently with the Hausdorff dimension $d_H=4$. The dotted lines refer to simplices adjacent to the boundary, $x=1$ or $x^{\prime }=1$, respectively, for the two sides of the boundary.

Figure 3

A schematic view of a single time slice in the initial configuration. Visible are the $4^3$ hypercubes (each of which is divided into 16 simplices) and the starting position of the boundaries.

Figure 4

Distributions of distances from simplices to their copies in neighboring elementary cells (heights) for systems with $N^{\{4,1\}}=80\mathrm{k}$ (up) and $N^{\{4,1\}}=160\mathrm{k}$ (down). These are lengths of minimal loops with winding numbers $\{1,0,0,0\}$ (red), $\{0,1,0,0\}$ (green) and $\{0,0,1,0\}$ (blue) shifted in $r$ by a shift of order one. The dotted lines refer to simplices adjacent to the boundary.

Figure 5

The dual-lattice graphs of all the connections between simplices of lowest x-heights: 18 (purple), 19 (red), 20 (yellow) and 21 (blue). In general, loops longer than the minimal length almost always contain fragments belonging to shorter loops, but for small heights there exist also a small number of loops built only of simplices with equal x-height.

Figure 6

Distance from a starting simplex to its copy in cell $\{n,0,0,0\}$ minus distance from the same simplex to its copy in cell $\{n-1,0,0,0\}$, averaged over all the simplices of the configuration. Already for $n=8$ the minimal value of 18 is reached.

Figure 7

Heights in the 4 basic directions of consecutive simplices along loops of winding numbers $\{1,0,0,0\}$ (blue), $\{2,0,0,0\}$ (orange), $\{3,0,0,0\}$ (green), $\{6,0,0,0\}$ (red) starting from a simplex in an outgrowth. The dashed lines indicate the minimal and maximal heights in the configuration, and the dotted line indicates the height of the initial simplex.

Figure 8

The four basic heights of simplices belonging to one of the loops from Table 1: a loop of length 27 with winding numbers $\{2,-1,1,0\}$.

Figure 9

The four basic heights of simplices belonging to one of the loops from Table 1: a loop of length 43 with winding numbers $\{-2,3,-1,1\}$.

Figure 10

The four basic heights of simplices belonging to one of the loops from Table 1: a loop of length 47 with winding numbers $\{2,-1,3,0\}$.

Figure 11

A log-log scale histogram of the number of simplices crossed by a given number of shortest loops.

Figure 12

The x-, y-, and z-heights of simplices. The simplices were sorted in the order of descending number of loops passing through them, and then the heights were averaged over blocks containing 1000 simplices each. The fitted functions are $h_x=21.98k^{0.0732}$, $h_y=21.66k^{0.0723}$ and $h_z=22.74k^{0.0706}$.

Figure 13

Six concentric shells around a simplex of height equal to 16. The colors and the numbers indicate the height of a simplex, and the size of a vertex its distance from the starting simplex. The central simplex lies in the bulk region.

Figure 14

Six concentric shells around a simplex of height equal to 40. The colors and the numbers indicate the height of a simplex, and the size of a vertex its distance from the starting simplex. The central simplex lies near the deep end of an outgrowth.

Figure 15

Six concentric shells around a simplex of height equal to 30. The colors and the numbers indicate the height of a simplex, and the size of a vertex its distance from the starting simplex. The central simplex lies in the middle of an outgrowth.

Figure 16

The shortest $\{1,-1,1,0\}$ loop together with its neighborhood. The colors and the numbers indicate the height of a simplex, and the size of a vertex its distance from the loop.