Geometric Analysis, Wave Optics, and Geometric Gravity Models

摘要：dms -0707003项目负责人：Arlie O. petters本研究项目的目标是建立引力对光作用在宇宙中投射的阴影模式的数学理论。该项目旨在描述这些阴影模式的点扭曲、总放大倍率和放大倍率截面函数，包括几何和波光学限制的某些关键重力模型。这项研究需要在弯曲时空中对焦性奇点、衍射积分和渐近性进行几何分析。本文的研究为将阴影模式的研究推广到任意洛伦兹流形奠定了基础，从而对洛伦兹几何进行了新的探索。这些发现还将对当前关键的几何引力模型（如广义相对论、PPN模型和膜世界引力）进行详细的引力透镜比较，从而允许对这些引力模型进行预测。阴影图案在我们周围随处可见。我们在上班的路上开车经过他们，在他们中间的游泳池里游泳。恒星、星系、黑洞、暗物质和暗能量的引力场也在整个宇宙中投射出类似的图案。也许更神秘的是，这些阴影图案与我们物理世界本质的一个深刻问题有关——即，物理空间是否不止我们所熟悉的长、宽、高三个空间维度。这项研究将利用引力的宇宙阴影模式来探索黑洞和暗物质的本质，以及物理空间的维度。这项工作还将制定预测，以检验爱因斯坦的广义相对论和其他引力模型。该项目是跨学科的，在数学、物理和天文学之间创造了协同互动。

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.