Geometric Analysis, Wave Optics, and Geometric Gravity Models

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AbstractAward: DMS-0707003Principal Investigator: Arlie O. PettersThe goal of this research project is to develop a mathematicaltheory of the shadow patterns cast in the cosmos by the action ofgravity on light. The project aims to characterize the pointwisetotal magnification and magnification cross section functions ofthese shadow patterns, both in the geometric and wave opticslimits for certain key gravity models. This study requires ageometric analysis of caustic singularities, diffractionintegrals, and asymptotics in a curved spacetime setting. Theproposed research would lay a platform for generalization a studyof shadow patterns to arbitrary Lorentzian manifolds, which inturn would lead to a new exploration in Lorentzian geometry. Thefindings will also yield a detailed gravitational lensingcomparison of current key geometric gravity models like generalrelativity, PPN models, and braneworld gravity, allowing forpredictions that may test these gravity models.Shadow patterns are all around us. We drive through them on theway to work and swim among them in pools. Similar patterns arealso cast throughout the universe by the gravitational fields ofstars, galaxies, black holes, dark matter, and dark energy.Perhaps even more mysterious is that these shadow patterns areconnected with one of the profound questions about the nature ofour physical world -- namely, whether physical space has morethan the familiar three spatial dimensions of length, width, andheight. This research will employ gravity's cosmic shadowpatterns to explore the nature of black holes and dark matter,and the dimensionality of physical space. The work will alsoformulate predictions that test Einstein's general theory ofrelativity and other gravity models. The project isinterdisciplinary and creates a synergistic interaction betweenmathematics, physics, and astronomy.