Editing Regge calculus (section)

==Overview==
The starting point for Regge's work is the fact that every four dimensional time orientable [[Lorentzian manifold]] admits a [[Triangulation (geometry)|triangulation]] into [[simplices]].  Furthermore, the [[spacetime]] [[curvature]] can be expressed in terms of [[Defect (geometry)|deficit angles]] associated with ''2-faces'' where arrangements of ''4-simplices'' meet.  These 2-faces play the same role as the [[vertex (geometry)|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 ''positive'' [[Gaussian 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 (geometry)|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 (mathematics)|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]].