Numerical techniques for differential geometry : the development of GRworkbench for investigation of manifolds of arbitrary spacetimes

Abstract

A numerical tool for differential geometry is implemented as a package for an existing computer algebra system, and demonstrated by its application to the investigation of the Curzon space-time and the maximally-extended Schwarzschild solution. Improvements to this tool, named GRworkbench, have included modifying the design to more closely emulate the mathematical formalism, for example in the internal representation of vectors, tensors and coordinate fields. Impediments to the loading of new charts have been removed by the change in software architecture. Limitations on the order of obtainable dervitatives have been avoided by the use of functional and symbolic computation. Mathematica has been employed to replace a less practical previous scripting-interface, and to add further capabilities. Greater reliance on the capabilities of existing software has significantly reduced the maintenance burden, and improved portability. GRworkbench has been used to project simulations onto Penrose diagrams and other visualisations. By automatically ray-tracing null geodesics across chart boundaries, images of the sky have been produced from the perspective of an observer in the interior region of the maximally extended Schwarzschild model of a black-hole. In the Curzon space-time, accurate visualisations have been produced for diagrams which previous authors only sketched. This has revealed features of the Scott-Szekeres coordinates which had not previously been observed. Show more