0 On the use of algebraic programming in the general relativity 1

2 Using Maple in the study of the canonical formalism of general relativity 17 2. 1 2 CONTENTS Preview This is a review devoted to some results of Algebraic Programming (Computer Algebra-CA) used in treating several problems of general relativity, based mainly on already published articles (see the references). The first chapter presents procedures in REDUCE language using EXCALC package for algebraic programming of the Hamiltonian formulation of general relativity (ADM formalism). The procedures calculate the dynamic and the constraint equations and, in addition, we have extended the obtained procedures in order to perform a complete ADM reductional procedure. Several versions of the procedures were realized for the canonical treatment of pure gravity, gravity in interaction with material fields, inflationary models (based on a scalar field non-minimally coupled with gravity) and theories with higher order lagrangians. The second chapter is devoted to the same problem as the first one, but now we use Maple V platform. In this purpose we present Maple procedures using the GrTensorII package adapted for the study of the canonical version of the general relativity based on the ADM formalism. Last chapter presents CA procedures and routines applied to the Dirac field on curved spacetimes. The main part of the procedures is devoted to the construction of Pauli and Dirac matrices algebra on an anholonomic orthonormal reference frame. Then we shall present how these algebraic programming sequences are used in the study of the Dirac equation on curved spacetimes and noninertial reference frames. A comparative review of such procedures obtained for the two mentioned CA platforms (REDUCE + EXCALC and MAPLE + GRTen-sorII) is presented. Applications for the calculus of Dirac equation 3 4 CONTENTS