CAREER: Geometric-Analytic Investigations of Spacetimes and their Nonlinear Phenomena

职业：时空及其非线性现象的几何分析研究

• 批准号: 1253149
• 负责人: Lydia Bieri
• 金额: $ 41.05万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-05-15 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1253149&HistoricalAwards=false
• 关键词: CAREER; Geometric; Analytic; Investigations; Spacetimes

This project concerns the investigation of nonlinear partial differential equations (PDEs), namely the Einstein equations in general relativity, the geometry of the solution spacetimes and their implications in physics and astrophysics. One of the main aims of the proposed research is to derive a complete theory for the nonlinear memory effect of gravitational waves displacing test masses permanently. This will include answering the questions whether there exists such an effect for neutrino radiation as present in binary neutron star mergers and if there is a cosmological memory effect. The goal is to give a detailed geometric-analytic description in the various scenarios and to apply the mathematics to experiments. The PI has already made contributions in this direction in the recent past. The emerging results of the project are expected to answer fundamental physical questions about gravitational radiation, the source objects in astrophysics and the very early period of our universe. Since the pioneering article by D. Christodoulou, where he derived the nonlinear memory effect (also called the Christodoulou effect), it had been an outstanding problem if electromagnetic fields coupled to the Einstein equations affect this nonlinear phenomenon. The PI with collaborators P. Chen and S.-T. Yau solved the problem, proving that the electromagnetic field in the Einstein-Maxwell (EM) equations enlarges the Christodoulou effect. Working with Christodoulou's geometric-analytic approach for the Einstein vacuum equations and N. Zipser's analysis of the EM equations, the PI, Chen and Yau established novel results and methods in the geometric analysis of the EM equations. The geometric-analytic investigations by Christodoulou as well as by the PI with Chen and Yau allowed the authors to deduce exact solutions describing the related effects precisely and for all data. In particular, the results hold for large data such as binary black hole or binary neutron star mergers. Geometric analysis has since proven to be the most powerful method to tackle these problems. In the study of the nonlinear memory effect of gravitational waves, null hypersurfaces in Lorentzian geometry play a crucial role. This is due to the fact that gravitational waves travel along such null hypersurfaces and experiments are performed at null infinity. Therefore, the asymptotic behavior of the latter has to be understood. D. Christodoulou and S. Klainerman in their work "The global nonlinear stability of the Minkowski space" developed new techniques to investigate those. The PI generalized their work and established the borderline case from the point of view of decay of the data. In a research monograph of 300 pages the PI obtained numerous new results that can be applied to other problems in geometry and analysis. The main method and its developments have recently led to the solution of some of the most challenging problems in mathematical physics. The new project will use new ideas from the PI's recent work and it is expected to yield further new results in general relativity (GR), geometry and PDE analysis. The educational component of the proposal focuses on disseminating knowledge of mathematics, physics and astrophysics as well as historic background to the public. Aiming at the creation of an exhibit at the Museum of Natural History at the PI's home University combining geometry and analysis with astrophysics, the educational project will include student work at undergraduate and graduate levels. Together with the PI and the museum staff, they will be involved in teaching at high schools working on the project. At public events organized by the museum the PI will collaborate with the museum staff to present activities to enhance learning. Research results will be presented in a simplified way and adapted to the audience. The project is designed as to reach out to a very broad public including minority groups, families with small kids, schools at all levels and will have an appealing component for everyone. Some of the long-lasting impacts will be new activities for high school classes to teach mathematics by introducing concepts of physics combined with history and art. The geometric nature of general relativity singles out geometric analysis to be the perfect research field to answer the many physical questions. The laws of GR are the Einstein equations, linking the curvature of spacetime to its matter content. Exploring these equations will lead to a better understanding of the universe as a whole and of isolated systems such as galaxies, binary black holes or binary neutron stars. To unravel the complex interplay between geometry, analysis and physics is one of the main goals in GR. This project aims to investigate this beautiful interaction. Along the way, the mathematical tools developed promise to bear fruit in the analysis of the many structurally similar nonlinear PDEs. Through the educational component, the PI's research will also have direct impact in a broader sense via the afore-mentioned activities. The PI will complement the outreach by attending conferences to communicate her results. She will also make her results available via internet and publications.

该项目涉及非线性偏微分方程（PDE）的研究，即广义相对论中的爱因斯坦方程，解时空的几何及其在物理学和天体物理学中的含义。提出的研究的主要目的之一是推导出一个完整的理论的非线性记忆效应的引力波位移测试质量永久。这将包括回答这样的问题是否存在这样的影响中微子辐射目前在双中子星星合并，如果有一个宇宙记忆效应。我们的目标是在各种情况下给出详细的几何分析描述，并将数学应用于实验。不久前，PI已经在这方面做出了贡献。该项目的新成果预计将回答有关引力辐射、天体物理学中的源物体和我们宇宙的非常早期的基本物理问题。自从D. Christodoulou，在那里他推导出非线性记忆效应（也称为Christodoulou效应），它一直是一个突出的问题，如果电磁场耦合到爱因斯坦方程影响这种非线性现象。PI与合作者P. Chen和S.- T.丘解决了这个问题，证明了爱因斯坦-麦克斯韦（EM）方程中的电磁场放大了克里斯托杜卢效应。利用Christodoulou的几何分析方法求解爱因斯坦真空方程和N。Zipser的EM方程分析，PI，Chen和Yau在EM方程的几何分析中建立了新的结果和方法。Christodoulou以及PI与Chen和Yau的几何分析调查使作者能够推导出精确的解，精确地描述了所有数据的相关效应。特别是，结果适用于大数据，如双黑洞或双中子星星合并。几何分析已被证明是解决这些问题的最强大的方法。在引力波非线性记忆效应的研究中，洛伦兹几何中的零超曲面起着至关重要的作用。 这是因为引力波沿着这样的零超曲面传播，并且实验是在零无穷远处进行的。 因此，必须理解后者的渐近行为。D. Christodoulou和S. Klainerman在他们的工作“闵可夫斯基空间的全局非线性稳定性”中开发了新的技术来研究这些问题。PI从数据衰减的角度概括了他们的工作并建立了边界情况。在一个研究专着300页的PI获得了许多新的成果，可以适用于其他问题的几何和分析。主要方法及其发展最近导致了一些最具挑战性的问题的解决方案在数学物理。新项目将使用PI最近工作的新想法，预计将在广义相对论（GR），几何和PDE分析方面产生进一步的新成果。 该提案的教育部分侧重于向公众传播数学、物理学和天体物理学知识以及历史背景。该教育项目的目标是在PI所在大学的自然历史博物馆举办一个展览，将几何和分析与天体物理学相结合，该项目将包括本科生和研究生的学生作品。他们将与PI和博物馆工作人员一起参与该项目的高中教学。在博物馆举办的公共活动中，PI将与博物馆工作人员合作，介绍活动，以促进学习。研究结果将以简化的方式呈现，并根据受众进行调整。该项目旨在接触到非常广泛的公众，包括少数群体、有小孩的家庭、各级学校，并将对每个人都有吸引力。一些长期的影响将是高中班级通过引入物理概念结合历史和艺术来教授数学的新活动。 广义相对论的几何性质使几何分析成为回答许多物理问题的完美研究领域。 GR定律是爱因斯坦方程，将时空的曲率与其物质含量联系起来。探索这些方程将导致更好地了解整个宇宙和孤立的系统，如星系，双黑洞或双中子星。为了解开几何，分析和物理之间的复杂的相互作用是在GR的主要目标之一。这个项目旨在研究这种美丽的相互作用。沿着这条路，数学工具的发展承诺在许多结构相似的非线性偏微分方程的分析中结出硕果。通过教育部分，PI的研究也将通过上述活动产生更广泛的直接影响。PI将通过参加会议来传达其结果，以补充外联活动。她还将通过互联网和出版物提供她的结果。