Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1061Fig. 6.9 Positive (a) and negative (b) space–like deficit angles δ (adapted from [Loll(2001); Loll (1998)]).6.4.3 Regge CalculusThe use of simplicial methods in general relativity goes back to the pioneeringwork of Regge [Regge (1961)]. In classical applications one tries toapproximate a classical space–time geometry by a triangulation, that is, apiecewise linear space get by gluing together flat simplicial building blocks,which in dimension d are dD generalizations of triangles. By ‘flat’ we meanthat they are isometric to a subspace of dD Euclidean or Minkowski space.We will only be interested in gluings leading to genuine manifolds, whichtherefore look locally like an R d . A nice feature of such simplicial manifoldsis that their geometric properties are completely described by the discreteset {li 2 } of the squared lengths of their edges. Note that this amounts toa description of geometry without the use of coordinates. There is nothingto prevent us from re–introducing coordinate patches covering the piecewiselinear manifold, for example, on each individual simplex, with suitabletransition functions between patches. In such a coordinate system the metrictensor will then assume a definite form. However, for the purposes offormulating the path integral we will not be interested in doing this, butrather work with the edge lengths, which constitute a direct, regularizedparametrization of the space Geom(M) of geometries.How precisely is the intrinsic geometry of a simplicial space, most importantly,its curvature, encoded in its edge lengths? A useful example tokeep in mind is the case of dimension two, which can easily be visualized.A 2D piecewise linear space is a triangulation, and its scalar curvatureR(x) coincides with the Gaussian curvature (see section 3.10.1.3 above).One way of measuring this curvature is by parallel–transporting a vectoraround closed curves in the manifold. In our piecewise–flat manifold such avector will always return to its original orientation unless it has surroundedlattice vertices v at which the surrounding angles did not add up to 2π, but