Introducing quantum Ricci curvature

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian manifold by comparing the distance between pairs of spheres with that of their centers. The quantum Ricci curvature is designed for use on non-smooth and discrete metric spaces, and to satisfy the key criteria of scalability and computability. We test the prescription on a variety of regular and random piecewise flat spaces, mostly in two dimensions. This enables us to quantify its behavior for short lattices distances and compare its large-scale behavior with that of constantly curved model spaces. On the triangulated spaces considered, the quantum Ricci curvature has good averaging properties and reproduces classical characteristics on scales large compared to the discretization scale.

Figure 1

Two nearby spheres $S_p^{\epsilon }$ and $S_{p^{\prime }}^{\epsilon }$ of radius $\epsilon$ whose centers are a small distance $\delta$ along the unit vector $v$ apart. Parallel transport of a unit vector $w$ at $p$ along the geodesic of length $\delta$ connecting $p$ and $p^{\prime }$ yields another unit vector $w^{\prime }$. The distance between the points $q$ and $q^{\prime }$ in flat space is equal to $\delta$, while in the presence of curvature the lowest-order deviation from $\delta$ is given by formula (1).

Figure 2

Contour plots of the distance between two circles on a two-dimensional flat space, as function of the circle radius $\epsilon$ and the distance $\delta$ of their centers. Left: sphere distance (8). Right: average sphere distance (9).

Figure 3

Contour plots of the distance between two circles on a two-dimensional space of constant positive curvature, as function of the circle radius $\epsilon$ and the distance $\delta$ of their centers, both rescaled by the curvature radius $\rho$. Left: sphere distance (12). Right: average sphere distance (14).

Figure 4

Contour plots of the distance between two circles on a two-dimensional space of constant negative curvature, as function of the circle radius $\epsilon$ and the distance $\delta$ of their centers, both rescaled by the curvature radius $\rho$. Left: sphere distance (17). Right: average sphere distance (19).

Figure 5

Comparing sphere distance (left) and average sphere distance (right) for $\epsilon =\delta$, as function of $\delta \in [0,2\pi ]$, for the three constantly curved model spaces: hyperbolic (top), flat (middle) and spherical (bottom). The curvature radius $\rho$ has been set to 1.

Figure 6

Comparing the normalized versions of sphere distance (left) and average sphere distance (right) for $\epsilon =\delta$, as function of $\delta \in [0,2\pi ]$, for the three constantly curved model spaces: hyperbolic (top), flat (middle) and spherical (bottom). The curvature radius $\rho$ has been set to 1.

Figure 7

Two overlapping spheres $S_p^{\delta }$ and $S_{p^{\prime }}^{\delta }$ of radius $\delta =3$ on a square lattice, whose center vertices $p$ and $p^{\prime }$ are a link distance $\delta =3$ apart. Each sphere consists of 12 vertices. The diagonal edges between the sphere vertices are drawn for ease of visualization only and are not part of the spheres or the underlying lattice.

Figure 8

Normalized average sphere distance $\overline{d}(S_p^{\delta },S_{p^{\prime }}^{\delta })/\delta$ on the square, hexagonal and honeycomb lattices in two dimensions, marked by triangles, squares and crosses respectively, as function of $\delta$. The straight horizontal line is that of flat continuum space, and is included for comparison.

Figure 9

Normalized average sphere distance $\overline{d}(S_p^{\delta },S_{p^{\prime }}^{\delta })/\delta$ on two-dimensional flat lattices, averaged over lattice directions as described in the text. Triangles and squares mark the data points for the square and hexagonal lattices respectively. The straight horizontal line is that of flat continuum space.

Figure 10

A Delaunay triangulation in the plane, together with the circumcircles of its constituting triangles. By definition, no circumcircle contains any vertices of the triangulation in its interior.

Figure 11

Using the auxiliary square lattice: during the construction of the initial point set $P$, only the 20 cells shown surrounding the cell of a new candidate point $q$ need to be checked for the presence of other points within radius $d_{\min }$.

Figure 12

Probability distribution $p(\ell )$ of edge lengths $\ell$ in a Delaunay triangulation of flat space with 6.200 vertices, binned in intervals of 0.1 $d_{\min }$.

Figure 13

Distribution $p(n_v)$ of the vertex order $n_v$ of interior vertices of a Delaunay triangulation of a piece of flat space with 6.200 vertices.

Figure 14

The (averaged) size $\nu (\delta )$ of circles as a function of their radius $\delta$, on a geometry obtained by setting the edge lengths of a flat Delaunay triangulation to unity, including the best linear fit.

Figure 15

Normalized average sphere distance $\overline{d}/\delta$ as a function of the scale $\delta$, measured on random triangulations modeled on flat space (red data points with error bars). For comparison, we have included the corresponding data for the flat hexagonal lattice (blue dots) of Fig. 9 and the horizontal line marking the constant value of continuum flat space (grey).

Figure 16

Distribution $p(\mathrm{\Delta })$ of diameters $\mathrm{\Delta }$ of a sample spherical triangulation, generated using $d_{\min }=0.025$.

Figure 17

The (averaged) size $\nu (\delta )$ of circles as a function of their radius $\delta$ on spherical triangulations, for $d_{\min }=0.1$ (left) and $d_{\min }=0.05$ (right). We have included best fits to a function of the form $c{\rho }_{\mathrm{eff}}\sin (\frac{\delta }{{\rho }_{\mathrm{eff}}}+s)$ (grey curves), and to a linear function (blue curves).

Figure 18

Normalized average sphere distance $\overline{d}/\delta$ as a function of the scale $\delta$, measured on random triangulations modeled on continuum spheres of three different sizes. From top to bottom: “flat” random triangulation measured previously (for reference); large sphere ($d_{\min }=0.025$), medium-size sphere ($d_{\min }=0.05$), and small sphere ($d_{\min }=0.1$).

Figure 19

Measurements of $\overline{d}/\delta$ and best fit (using a multiplicative shift) to the corresponding data of a two-dimensional continuum sphere, for the combined and rescaled data of all three spheres. Error bars are smaller than dot sizes.

Figure 20

The (averaged) size $\nu (\delta )$ of circles as a function of their radius $\delta$ on geometries obtained by setting the edge lengths of Delaunay triangulations on a hyperboloid to unity, including best fits to a function of the form $\tilde{c}{\rho }_{\mathrm{eff}}\sinh (\frac{\delta }{{\rho }_{\mathrm{eff}}}+\tilde{s})$ (grey curve), and to a linear function (blue curve).

Figure 21

Normalized average sphere distance $\overline{d}/\delta$ as a function of the scale $\delta$, measured on random triangulations modeled on a continuum hyperboloid (blue), shown together with a best fit of the corresponding curve in the continuum (red).