bibkey: regge1961coordinates
authors: Tullio Regge
year: 1961
title: General relativity without coordinates
doi: 10.1007/BF02733251
url: https://doi.org/10.1007/BF02733251
claim: Regge describes approximating curved spaces by higher-dimensional polyhedra; flat local pieces require gluing and transport data to describe global geometry.
strata_touched: []
license: citation-only
triage: anchor

Flat pieces, gluing and transport

Source boundary

The publisher record identifies T. Regge, Il Nuovo Cimento 19, 558–571 (1961). Its accessible summary describes a coordinate-free approach that approximates curved spaces by higher-dimensional polyhedra and uses simplicial decompositions. The article body was not verified here; no precise action, field equation or theorem number from it is imported. The two-dimensional illustration below is elementary exposition, not a quotation or a new result attributed to a particular passage.

Elementary angle deficit

Take Euclidean triangles glued isometrically along matching edges, forming an oriented triangulated surface near an interior vertex. Suppose its incident triangle angles at that vertex are ${\theta }_1,\dots ,{\theta }_k$. The total intrinsic angle is ${\sum }_j{\theta }_j$ and the deficit is

$$\delta =2\pi -\sum _{j=1}^k{\theta }_j.$$

To see the transport interpretation, cut along one radial edge and unfold the incident triangles into a Euclidean sector. Parallel transport is constant in that unfolded plane, but identifying the two bounding rays closes a sector of angle ${\sum }_j{\theta }_j$ into the surface. The resulting rotation is the missing angle $\delta$ modulo $2\pi$, with sign reversed if the orientation of traversal is reversed. Flat interiors therefore coexist with nontrivial transport around a vertex. Boundary vertices and singular nonmanifold gluings need different local contracts.

The local shapes alone do not determine the glued surface: edge identifications, lengths and orientations are additional data. Similarly, the static circuit contraction identity retains port order and complex matrix entries rather than only local probabilities. This is a correspondence between descriptions of composition, not a claim that the geometries or their resource costs coincide.

For AURIC, the occupancy pyramid is a three-dimensional convex set of local means. A four-dimensional simplex is a different object. Relating a network of AURIC ports to a spacetime triangulation would require a specified map preserving the relevant incidence, metric, causal and transport data. Neither the common use of local building blocks nor the angle-deficit illustration supplies that map or establishes a physical ontology.

This note contains original citation-only exposition and no copied article text or figures.