Inverse problems for Einstein equations and related topics of Lorentzian geometry

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.

The ultimate goal of research started by this project is to develop a new rigorous method to solve hyperbolic IP which would be able to deal with more general, non-linear equations with time-dependent coefficients, including those of general relativity, which are untreatable by the existing methods. In this respect, the new method, if successful, would be superior to the existing inversion techniques.
In this respect, there are four facets of the impact of the proposed research.
A. An Immediate Impact on a Wider Research Community. Our preliminary considerations show that to succeed in building up the new method to tackle IP for non-linear equations with time-dependent coefficients, significant further advancements are needed not only in IP but also in such areas of mathematics as Lorentzian geometry, microlocal analysis and quasi-linear PDE-theory. When focusing the method to general relativity, it would require construction of new solutions to Einstein's equation with a prescribed behaviour of singularities. Therefore, the results and methods developed during the project would be beneficial for the theory of IP, microlocal analysis, hyperbolic non-linear PDE's and general relativity in its both mathematical and physical aspects.
B. Longer-Term Impact on Practical Applications of IP. IP pertain a wide range of practical applications, from medical imaging to process monitoring, to oil and other mineral resources exploration, to metereological observations, to non-destructive testing. In many of those applications we deal with dynamical, time-dependent processes which are described mathematically by hyperbolic equations. This is the case of ultrasound and optical imaging, seismological observations, oceanological research, including monitoring of fish stocks, weather monitoring, etc. Moreover, in many cases, non-linear interactions in the media play crucial role so that the inverse method should accommodate for such non-linearity. Therefore, developing a new mathematical method to solve IP which is robust with respect to the variation of the parameters of the medium with time and non-linear interactions inside the medium, will open new horizons in practical applications of IP, increasing accuracy and reliability of inversion methods. This is clearly of critical necessity for such applications as medical imaging, atmospheric and, in general, metereological research, process monitoring in aggressive media, etc. Thus, in the longer run, the results of the project and its follow-ups would provide new methods to solve IP occuring in practical applications.
C. Impact on General Relativity. Achieving the ultimate goal of the research started in this project, ie providing a solution to IP for Einstein's equation would much impact our understanding of general relativity and cosmology, eg providing a tool to analyse the remote parts of the universe.
D. Training of Young Researchers. Aiming at a new powerful method to solve IP, we intend to make it a research tool for the next generation of researchers into IP. To this end, we will prepare, although outside the 12-months duration of the project, a monograph devoted to the method developed. In this monograph we would not only describe the new method itself and other results obtained but also provide the necessary prerequisites, i.e. topics from Lorentzian geometry, analysis of singularities and hyperbolic quasi-linear PDE's. This would make the monograph accessible for a wide range of researchers and postgraduate students in IP and applications.