The Program CRACK for Solving PDEs in General Relativity

Summary

In the introduction an approach to solving differential equations is motivated in which nonlinear DEs are not attacked directly but properties like infinitesimal symmetries or the existence of an equivalent variational principle are investigated. In the course of such investigations over-determined PDE systems are generated, which are to be solved (where the term ‘over-determined’ just stands for ‘more conditions than free functions’). In Sect. 2 algorithms for simplifying and solving over-determined PDE systems are given together with examples. References for more details of the corresponding program CRACK, written by A. Brand and myself, are given.

In Sects.3-5 applications of the program CRACK are discussed. The first appli-cation is the investigation of symmetries of spacetime metrics by solving Killing equations for Killing vectors and Killing tensors and their integrability conditions. A program CLASSYM that formulates these equations, written by G. Grebot, is briefly described.

In Sect. 4 an example of the original application of CRACK is discussed, which is the determination of symmetries of a PDE system. The problem is to find the symmetries of an unusual unified field theory of gravitational and hadronic inter-actions.

The application of symmetries with a program APPLYSYM is the content of Sect. 5, where an ODE, resulting from an attempt to generalize Weyl’s class of solutions of Einsteins field equations, is solved.

The final section is devoted to future work on, first, making a general PDE solver more flexible and effective, and second, on applying it to more advanced applications. This section contains as yet unpublished work. An example requir-ing the extension of CRACK to deal with nonpolynomial nonlinearities results from an investigation of interior solutions of Einstein’s field equations for a spherically symmetric perfect fluid in shear-free motion by H. Stephani. A possible future ap-plication of CRACK is the determination of Killing tensors of higher rank. In the last subsection an algorithm for formulating corresponding integrability conditions has been sketched. The maximal number of Killing tensors of rank r in a n-dimensional Riemannian space has been found to be ${1 \over {r + 1}}\left( {\matrix{ {n + r -1} \cr r \cr } } \right)\left( {\matrix{ {n + r} \cr r \cr } } \right)$.