Regge Calculus

Regge calculus is a formulation of general relativity in which spacetime is approximated by a piecewise flat manifold built from simplices — triangles in two dimensions, tetrahedra in three, and their higher-dimensional analogues. Developed by Tullio Regge in 1961, it provides a discretization of the Einstein field equations that is exact in the limit of infinitely fine triangulation. The curvature of spacetime is concentrated at the hinges — the lower-dimensional faces where simplices meet — rather than being distributed smoothly as in the continuum theory.

Regge calculus is the mathematical ancestor of causal dynamical triangulation. Where CDT uses Regge's simplicial discretization as its starting point and adds a global causal structure, Regge calculus itself makes no commitment to causality and can be applied to both Euclidean and Lorentzian geometries. The Einstein-Hilbert action in Regge calculus becomes a sum over the deficits angles at hinges, weighted by their volumes, and the field equations are replaced by variational conditions on the edge lengths.

The framework has found applications in numerical relativity, quantum gravity, and computational geometry. In numerical relativity, it provides a coordinate-independent way to evolve initial data. In quantum gravity, it underlies the simplicial approaches — CDT, dynamical triangulations, and spin foam models — that attempt to define the gravitational path integral on a discrete structure. The central question in all these applications is whether the continuum limit exists and whether it reproduces classical general relativity.

Regge calculus is the proof that general relativity does not need the continuum. The smooth manifold of Einstein's theory is a convenience, not a necessity. Curvature can live on hinges, gravity can propagate through simplices, and the geometry of the universe can be built from flat pieces like a geodesic dome. The question is not whether this works — it does — but whether the universe itself is built this way, or whether the simplices are merely a computational scaffold we erect because we cannot solve the continuum equations.