Regge calculus

Regge calculus
Name	Regge calculus
Field	General relativity, Differential geometry
Introduced	1961
Founder	Tullio Regge
Notable applications	Numerical relativity, Quantum gravity, Loop quantum gravity

Regge calculus is a discrete formulation of the General theory of relativity introduced to approximate continuous spacetime by a piecewise-linear assembly of simplices. It replaces the smooth metric of Einstein field equations with edge lengths on a simplicial complex so that curvature is concentrated on lower-dimensional subsimplices, enabling combinatorial and numerical treatments of black hole dynamics, cosmology, and attempts at quantizing gravity. The approach has influenced work in Canonical quantization, Path integral formulation, and discrete models connected to Loop quantum gravity and Causal dynamical triangulations.

Introduction

Regge calculus was proposed by Tullio Regge in 1961 to provide a coordinate-free discretization of the Einstein field equations suitable for approximation and quantization. The method models a four-dimensional manifold by gluing together flat simplices (4-simplices) so that curvature resides on triangular faces (2-simplices), allowing use of algebraic and combinatorial techniques related to Simplicial homology, Pachner moves, and PL topology. Its formulation connects to the variational principles of Hilbert action and has spurred numerical applications in Numerical relativity, comparisons with ADM formalism, and interactions with discrete approaches such as Spin foam models and Causal sets.

Mathematical formulation

The mathematical core assigns squared edge lengths as fundamental variables on a simplicial complex built from vertices, edges, triangles, tetrahedra, and 4-simplices; this mirrors coordinate-independent constructions in Riemannian geometry and Lorentzian geometry. The metric in each simplex is flat and determined algebraically by edge lengths via Cayley–Menger determinants, echoing matrix constructions from Gram determinant and techniques used in Euclidean geometry and Minkowski space. Gluing conditions impose matching of induced metrics on shared faces, analogous to boundary conditions in the Hamiltonian formulation of general relativity and constraints seen in Dirac quantization.

Discretization of spacetime and simplicial complexes

Spacetime is approximated by a finite or infinite simplicial complex assembled from 4-simplices whose combinatorics relate to triangulations studied in Algebraic topology and Piecewise-linear topology. Choices of triangulation draw on algorithms and theorems from Delaunay triangulation, Voronoi diagram, and bistellar flips associated with Pachner moves. In practical computations one employs triangulations influenced by techniques developed in Finite element method and mesh generation used in Computational geometry and Computer-aided design, while respecting causal structure when constructing Lorentzian complexes akin to those in Causal dynamical triangulations.

Curvature and deficit angles

Curvature is concentrated on codimension-2 simplices (triangles in 4D) and quantified by deficit angles computed from the dihedral angles of adjacent 4-simplices, paralleling angle deficit concepts from Euclidean geometry and earlier work on angular defects in Crystallography. The deficit angle at a hinge equals 2π minus the sum of dihedral angles, an idea related to holonomy studied in Gauge theory and Fiber bundles. This discrete curvature replaces the continuous Riemann curvature tensor and connects to scalar curvature when summing contributions, reflecting principles from Gauss–Bonnet theorem and integral geometry used in Geometric analysis.

Action principle and equations of motion

The Regge action is a discretized analogue of the Hilbert action, formed by summing the product of hinge areas and corresponding deficit angles, much as the continuum action integrates scalar curvature against the volume element. Variation of edge lengths yields discrete equations of motion that parallel the Einstein field equations and correspond to stationarity conditions similar to those in the Principle of least action. Boundary terms and coupling to matter can be incorporated by adapting techniques from Gibbons–Hawking–York boundary term and discrete matter models inspired by Lattice gauge theory and Ising model constructions.

Applications and numerical implementations

Regge calculus has been applied in Numerical relativity for evolving discretized spacetimes including model problems for Gravitational collapse, Cosmological models such as lattice cosmology, and discretized studies of Black hole thermodynamics. Numerical implementations use tools from Finite element method, adaptive mesh refinement ideas from Computational fluid dynamics, and algorithms developed in High-performance computing and Parallel computing. Comparisons with continuum codes employing the ADM formalism or BSSN formalism provide cross-validation, while Regge-based Monte Carlo approaches inform investigations in Euclidean quantum gravity and path integral approximations akin to those in Lattice quantum chromodynamics.

Extensions and relation to other approaches

Extensions include spin foam and group field theory formulations that relate Regge edge-length data to representation-theoretic variables in Loop quantum gravity and Spin networks, and Lorentzian adaptations that interface with Causal dynamical triangulations and Causal set theory. Connections to discrete differential geometry, Discrete exterior calculus, and statistical models derive from work in Mathematical physics and Combinatorial topology. The framework also informs semiclassical analyses linking to Asymptotic safety and proposals in Discrete quantum gravity while interacting with canonical approaches such as Dirac constraint quantization and path-integral techniques pioneered by Richard Feynman and developed further in studies by John Wheeler and Bryce DeWitt.

Category:General relativity Category:Quantum gravity Category:Tullio Regge