Inverse problems for Einstein equations and related topics of Lorentzian geometry

点击翻译按钮获取中文摘要

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.