New approaches to the study of gravitation in relativistic astrophysics

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The applicant, along with several students and over a period of many years, developed the widely used computer algebra system GRTensor, an essential tool in the area of applied differential geometry. My proposed research program makes use of this expertise in new ways. Here I will concentrate on just one aspect of this program, gradient fields of curvature. Our work involves the novel use of both computer algebra and numerical routines with the objective being the development of a fresh view of "curvature" - a view that reveals detailed structure and insights previously unavailable. Given any invariant, one can always construct the associated gradient field and study the resultant flow. Essentially, we have turned the abstract idea of curvature into a hypothetical fluid the properties of which can be easily visualized. Although this research is in its early stages, I can already cite the following: (i) The flow distinguishes different types of singularities. For example, whereas the flow seeks out the singular ring in the equatorial plane of the Kerr metric (the real singularity), the flow completely ignores "shell focusing singularities", substantiating my claim, made many years ago, that such singularities are not "gravitational". (ii) In the area of inhomogeneous cosmology, it has been incorrectly claimed that the divergence of the "d'Alembertian of the Ricci scalar" signifies a "weak singularity". Such a divergence is actually a caustic in the gradient flow - and artificial singularity of no consequence. We expect that this fresh approach to the visualization of curvature will evolve into a standard approach much to the benefit to all researchers in the field. Whereas much work remains to be done, one way to summarize this approach is to note that when Killing vectors (directions of symmetry) are available, it is wise to make use of them. Gradient flows are, by construction, orthogonal to Killing flows and are always available - algorithmically. It should be noted that the use of gradient flows is not restricted to Einstein's theory nor to four dimensions. This aspect of my overall research program offers exceptional opportunities for the training of HQP and involves two graduate students and a summer research assistant.