Regge calculus is a finite element method utilized in numerical relativity in attempts of describing spacetimes with few or no symmetries by way of producing numerical solutions to the Einstein field equations (Khavari 2009). It was developed initially by Italian theoretical physicist Tullio Regge in the 1960s (Regge 1961).
Modern forays into Regge's method center on the triangulation of manifolds, particularly on the discrete approximation of 4-dimensional Riemannian manifolds and Lorentzian manifolds by way of simplicial complexes whose 4-dimensional triangular simplices share their boundary tetrahedra (i.e., 3-dimensional simplices) to enclose a flat piece of spacetime (Marinelli 2013). Worth noting is that Regge himself devised the framework in more generality, though noted that no such generality is lost by assuming a triangular approximation (Regge 1961).
The benefit of this technique is that the structures involved are rigid and hence are completely determined once their edge lengths have been specified (Khavari 2009). On the other hand, this method is inherently more difficult due to the fact that many of the fundamental differential-geometric properties of spacetime, such as its metric tensor and curvature, require a smooth structure. Its topology is part of the underlying topological manifold and is not determined by the smooth structure.
Intuitively, the gap between a crude spacetime approximation and the smoothness of the underlying manifold can be bridged by viewing the
manifold as the limit of a
sequence of these piecewise-linear approximations; this
is done by increasing the number of lower-dimensional simplices used to approximate and by passing to a limit.
With this in mind, the study of relativity by way of Regge calculus requires a discretization
of all spacetime-related structures (i.e., topology,
metric tensor, curvature,
etc.) to which some suitable limiting process can be applied in order to obtain the
smooth versions thereof. A key aspect is to notice that the curvature
of a smooth manifold is intuitively reflected
in the codimension-2 sub-simplices--called
hinges or bones within Regge's framework--of any simplicial approximation thereof.
Hinges in a 4-dimensional simplex 
are 2-dimensional simplices, e.g., triangles,
and the amount of curvature residing on a hinge 
is represented by its so-called deficit
angle (or deficiency) 
(Khavari 2009). A positive, respectively negative,
value of 
represents positive, respectively negative, manifold curvature. From here, Regge's theory develops by discretizing
the relativistic gravitational action, the vacuum Einstein
field equations, the Bianchi identities,
etc., and by adapting the discretization techniques for Riemannian
manifolds (as outlined above) to Lorentzian manifolds,
e.g., to Minkowski space.
Since Regge's original work, a number of improvements and extensions have been discovered, including explicit algorithms for computing dihedral angles and deficit angles for arbitrary discretized manifolds and generalizations intended to factor in more advanced physical phenomena.
Limiting behavior of the discretized theory has been studied by Brewin and Gentle (2001). Regge calculus has successfully reproduced known solutions of the Einstein field equations. Khavari (2009) distinguishes these successes from the comparatively unexplored evolution of arbitrary manifolds and their topology. Its role in classical and quantum gravity and in other mathematical applications is discussed by Williams (1991) and Marinelli (2013).
