Relativity and gravitational theory Definition

Regge calculus

Builds on Simplicial complex Dihedral angle Einstein–Hilbert action

Definition

Let $T$ be a simplicial complex that triangulates an $n$ -manifold, $n \ge 2$ . Assign a positive length $l_e$ to each edge $e$ . Assume every simplex of $T$ can be realized as a simplex in Euclidean space $\R^n$ (or Minkowski space) with these edge lengths. This makes $T$ a piecewise flat manifold. For each $(n-2)$ -simplex $h$ of $T$ (a hinge), the deficit angle is $\varepsilon_h = 2\pi - \sum_{\sigma \supset h} \theta_{\sigma, h}.$ The sum runs over the $n$ -simplices $\sigma$ containing $h$ , and $\theta_{\sigma, h}$ is the dihedral angle of $\sigma$ at $h$ . Regge calculus is the theory of the edge lengths with the Regge action $S_R = \frac{1}{8 \pi G} \sum_h \operatorname{vol}_{n-2}(h) \, \varepsilon_h,$ whose field equations are $\partial S_R / \partial l_e = 0$ for every edge $e$ .

Remarks

The action approximates the Einstein–Hilbert action $\frac{1}{16\pi G} \int R \sqrt{g}$ , because the curvature of a piecewise flat manifold is a sum of delta functions on the hinges with $\int R \sqrt{g} = 2 \sum_h \operatorname{vol}(h) \varepsilon_h$ . By the Schläfli identity $\sum_h \operatorname{vol}(h) \, d\varepsilon_h = 0$ , the field equations reduce to $\sum_h \varepsilon_h \, \partial \operatorname{vol}(h) / \partial l_e = 0$ . Example ( $n = 2$ ): the surface of a regular tetrahedron has hinges at its four vertices. At each vertex three equilateral triangles meet with angles $\pi/3$ , so $\varepsilon = 2\pi - \pi = \pi$ . The total $4\pi$ equals $2\pi \chi(S^2)$ , as the Gauss–Bonnet theorem requires. Non-example: lengths $1, 1, 3$ on a triangle are not allowed, since they violate the triangle inequality and no flat triangle has these sides.

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Tag	`VBAD0NBY0M`
URI	`m2://VBAD0NBY0M`
Kind	definition
Writing	W0 stub
MSC	83C27, 52B70
Defines	Regge calculus
Uses	Simplicial complex, Dihedral angle, Einstein–Hilbert action
Changed	3 November 2026

Regge calculus

Definition

Remarks

Reading path

Further reading

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