The starting point for Regge's work is the fact that every four dimensional time orientable Lorentzian manifold admits a triangulation into simplices. Furthermore, the spacetimecurvature can be expressed in terms of deficit angles associated with 2-faces where arrangements of 4-simplices meet. These 2-faces play the same role as the vertices where arrangements of triangles meet in a triangulation of a 2-manifold, which is easier to visualize. Here a vertex with a positive angular deficit represents a concentration of positiveGaussian curvature, whereas a vertex with a negative angular deficit represents a concentration of negative Gaussian curvature.
The deficit angles can be computed directly from the various edge lengths in the triangulation, which is equivalent to saying that the Riemann curvature tensor can be computed from the metric tensor of a Lorentzian manifold. Regge showed that the vacuum field equations can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initial spacelike hyperslice according to the vacuum field equation.
The result is that, starting with a triangulation of some spacelike hyperslice (which must itself satisfy a certain constraint equation), one can eventually obtain a simplicial approximation to a vacuum solution. This can be applied to difficult problems in numerical relativity such as simulating the collision of two black holes.
The elegant idea behind Regge calculus has motivated the construction of further generalizations of this idea. In particular, Regge calculus has been adapted to study quantum gravity.
John Archibald Wheeler (1965). "Geometrodynamics and the Issue of the Final State, in "Relativity Groups and Topology"". Les Houches Lecture Notes 1963, Gordon and Breach. {{cite journal}}: Cite journal requires |journal= (help)
Misner, Charles W. Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN978-0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link) See chapter 42.
Herbert W. Hamber (1984). "Simplicial Quantum Gravity, in the Les Houches Summer School on Critical Phenomena, Random Systems and Gauge Theories, Session XLIII". North Holland Elsevier: 375–439. {{cite journal}}: Cite journal requires |journal= (help)[4]
