Application of Regge Calculus to the Schwarzschild and Reissner-Nordstro/m Geometries at the Moment of Time Symmetry

Abstract

Applied to 3-dimensional space, Regge calculus approximates a curved space by a collection of tetrahedrons or other simple solid blocks. Within each block the geometry is Euclidean. Curvature is idealized as concentrated at the edge common to two or more of these solids. We specialize to a static geometry endowed with spherical symmetry and to a radial electric field produced by the flux of electric lines of force trapped in a throat connecting two quasi-Euclidean regions of space. The one relevant Einstein field equation takes the form ∑ all edges which meet at a given vertex(length of edge of prism)(deficit between (1) sum of dihedral angles which meet at that edge and (2) normal value of 2π)=(factor proportional to square of electric field). In method (1) the space is decomposed into shells separated from one another by icosahedral surfaces, all having a common center. Method (2) is even simpler: Space is decomposed into successive spherical shells of area 4π ρi2 separated by a proper distance d. Regge calculus gives a recurrence relation relating the dimensions of the successive shells. The approximate geometries calculated by methods (1) and (2) are compared with the well-known exact Schwarzschild and Reissner-Nordstro/m geometries. Errors range from roughly 10% down to less than 1%, depending upon the method of analysis, the quantity under analysis, and the fineness of the subdivision.