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Inverse problems for Einstein equations and related topics of Lorentzian geometry

爱因斯坦方程的反问题和洛伦兹几何的相关主题

基本信息

批准号：
EP/J006564/1
负责人：
Yaroslav Kurylev
金额：
$ 4.58万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ006564%2F1
关键词：
Inverse problems Einstein equations related

项目摘要

In spite of the rapid developments into the theory of inverse problems (IP) for the hyperbolic systems, existing results stay well short of dealing with either equations with time-dependent coefficients (except for the case of analytic dependence), or non-linear equations. In particular, there exist no approaches to study IP for the fundamental equations of general relativity. In this project we intend to start addressing these deficiencies in the theory of IP. Although, clearly, the study of IP for the non-linear hyperbolic equation with time-dependent coefficients would require some principally new ideas and methods, we believe we have some important ideas to start tackling these problems. They involve analysis of rigidity of the broken light-like geodesics on Lorentzian manifold. They also involve the use of non-linearity to generate secondary waves with desired conormal singularities and microlocal analysis of the secondary waves which are generated through the interaction of the incoming singular waves. We aim later to combine these two ideas in order to extract information about the behaviour of the broken light-like geodesics from observations of inverse data . Therefore, research into this project would consist of two principal parts; i) Study into rigidity of the broken light-like geodesics flow on Lorentzian manifolds. Here we believe that first publishable results will appear by the end of the 12-months duration of the project. ii) Analysis of the interaction of singular incoming waves and study of the propagation of conormal singularities generated through this interaction. Here we expect that the first preliminary results, eg in the preprint format, would appear by the end of the project.

尽管双曲型方程组的反问题（IP）理论发展迅速，但现有的结果对于系数随时间变化的方程（解析依赖的情况除外）或非线性方程的反问题都没有得到很好的解决。特别是，目前还没有研究广义相对论基本方程的IP的方法。在这个项目中，我们打算开始解决知识产权理论中的这些缺陷。虽然，很明显，研究IP的非线性双曲型方程与时间相关的系数将需要一些主要的新的想法和方法，我们相信我们有一些重要的想法，开始解决这些问题。其中包括洛伦兹流形上破缺类光测地线的刚性分析。它们还涉及使用非线性生成具有所需的余法线奇异性的二次波，并对通过入射奇异波的相互作用生成的二次波进行微局部分析。我们的目标是以后联合收割机这两个想法，以提取有关的行为的信息，打破光样测地线从观测的逆数据。因此，本计画的研究将包含两个主要部分：i）洛伦兹流形上破缺类光测地线流的刚性研究。在此，我们相信，第一批可验证的结果将在12个月的项目期结束时出现。ii）分析奇异入射波的相互作用，并研究通过这种相互作用产生的余法向奇点的传播。在这里，我们希望第一个初步的结果，例如在预印格式，将出现在项目结束。