Tullio Regge proposed that an n dimensional manifold could be decomposed into simplexes in order to perform geometrical and physical calculations where the manifold’s curvature was concentrated in bones consisting of the set of n–2 dimensional simplexes distributed thru out the manifold. Regge observed that the lengths of the shared edges of the simplexes determine the shape of each simplex and thus the metric properties of the entire manifold. We concentrate on Riemannian (metric) manifolds here. A central problem is to relate coordinate systems across the boundaries between simplexes. Here is some math to do that.

I call one of the n-simplexes into which the manifold has been partitioned, a zone, for we must speak of simplexes of lesser dimension as well. It is natural to use barycentric coordinates (bcc) in a simplex. A problem arises at the interface between two zones, called a facet here. A facet is an n–1 dimensional simplex.

To move a vector or tensor across a facet to a neighboring zone we conceptually adopt a new intermediate facet coordinate system whose bases are n–1 edge vectors in the facet eminating from some vertex of the facet, and a unit vector normal to the facet. This representation is neutral to the two zones, except for the normal’s orientation. To move now to a corresponding facet coordinate system in the neighboring zone we must change the sign of the normal component, and permute the order of the other components to match the vertex numbering plan of the neighbor. After this we transform to the new zone’s bc coordinates.

Pythag: All of these computations may be carried out if we can only calculate the inner product between vectors. This is supported directly if we have the metric tensor in whatever coordinate system we have expressed our vectors. Fortunately it is trivial to compute the covariant metric tensor in bcc given the squared edge lengths of the zone. The dot product of sides a and b of a triangle is (a2 + b2 – c2)/2. Note that I use the terms “length”, “vector”, “normal”, and especially “dot product” (as in “a • b”) in a Euclidean geometry sense in distinction to any algebraic sense. For each zone, choose some vertex of the zone as the origin and take the n edges emanating from the origin as bases for the barycentric coördinate system.

gij = edgei • edgej
x • y = gijxiyj
Thus the i and j in equations above each take on n values. Compute the n(n+1)/2 elements of the covariant metric tensor gij directly. To express v as the sum of a vector in the facet and a multiple of the unit normal, N, we write v = (v–vN) + vN, where vN = (v • N)N. The vector (v–vn) is in the facet. This requires expressing N in bcc.

The rows of the contravariant metric tensor are vectors expressed in bcc, normal to the facets of the zone. Those normals need only to be divided by their length to become the unit normals. The contravariant metric tensor is computed by inverting the covariant metric tensor.

The general curvature tensor may now be computed centered at the (n–2)-simplexes by carrying a frame around each (n–2)-simplex. This passes thru the set of (n–1)-simplexes which are incident on the (n–2)-simplex.

Here is an another method of doing this calculation which I currently think is inferior.

Here are fragments of ideas of applying this to General Relativity.

If we assemble simplexes of some fixed dimension into some complex we must match edge lengths; or more properly, perhaps, squared edge lengths. When the length2 of a shared edge is computed in the metric of each of the zones sharing that edge, the calculations must agree in order to say we have assembled a complex from simplexes. If our geometry is locally Minkowskian squared edge lengths may be positive, zero or negative. Matching squared edge lengths thus leads to inserting causality. See remark beginning “Simply put:” here. From my perspective it hard to see how simplexes could be sensibly united to complexes without such a rule. I had not noticed this before reading some CDT literature. It makes their causality rule look mathematical rather than a physical axiom.

Another note is that any set of n*(n–1) edge length squares from ℝn*(n–1) defines a pseudo Riemannian metric and conversely. Indeed the n*(n–1) edge length squares are linear in the n*(n–1) independent elements of the symmetric covariant metric tensor. Beware there is nothing left of the triangle inequality in Minkowskian space. Picking zones whose metric has signature (+ - - -) is merely to apply a few homogenious inequalities to the edge length squares.

The subject of Causal dynamic triangulation proposes to extend ideas such as these to a theory of everything. Nexus

Here is a zoo of simplexes to avoid.