First, find pairs of orthogonal geodesics.... then you can set up parallel transport....
Full Ricci tensor case, involving a tube traced out by balls:
[Finding an “average distance between balls” ; first, get balls with the same surface area ; then set up a bijection between the elements on the surfaces ; then find the bijection for which the average distance between corresponding points is minimized [or it could be something other than minimized] ] Construct the “tidal tensor” ... by comparing the average surface distance between balls with the distance between their centers. << Consider the bundle of geodesics between these balls ...
Riemann tensor : instead of transporting a whole ball, transport a Frenet frame like object
[ This definition is as independent of dimension as the definition of a geodesic ]
This is effectively a thick geodesic
Length scales: scale of the small balls scale of the distance between balls [which you swing around to get the Ricci scalar]
From the “frames” look at the bundles of geodesics; that gives Riemann tensor If you look at the shape of a cross-section of the bundle, gives you sectional curvature.
Homogeneous & Isotropic
E.g. a random sprinkling of h & i manifold will be h & i
[ n is the spatial topological dimension ]
Light cone structure is independent of time; determined by the structure of the characteristics for the solutions of these equations
The equations describe the time evolution of a spatial slice
In essence Hubble radius ~ a
Volume of universe: - number of nodes (n) - a^d [where a is the overall scale size of the universe] << - z [average graph distance between two points] >>
z ~ n^(1/d) [ n is total number of nodes ] But z ~ a a ~ n^(1/d) [ cf. Myhrheim-Meyer estimator ]
“It’s not a fractal” [ in the sense that its topological dimension is the same as its Hausdorff dimension ]
Rules exchange segments on neighboring geodesics
The dimension determines effectively the fanout of geodesics; “segment exchange” can be either the in fanlike direction, or in the cross-fiber direction
Cross-fiber is like moving material around in a given spherical shell; while fanlike is concentrating radially or not [or deforming the radius of shells]
Probability for a fanlike change is determined by effective topological dimension (probability of fanlike change is 1/(effective topological dimension) )