Analysis of Inverse Problems in General Relativity

In spite of the rapid developments into inverse problems (IP) for the hyperbolic systems, existing results stay well short of dealing with either equations with time-dependent coefficients or non-linear hyperbolic equations which describe multiple phenomena in our world. In particular, there are no approaches to study IP for Einstein's equations of general relativity (GR) which are one of the most fundamental and mathematically difficult models in modern physics. These equations connect the geometric properties of the Universe, expressed by the Einstein tensor, with its material properties, expressed by the stress-energy tensor.
In this project we will mathematically rigorously study IP of GR. Having started this research a couple of years ago, in collaboration with M. Lassas (Finland) and G. Uhlmann (USA) (supported by a small EPSRC grant), we showed that, having a large number of passive,
kinematic-type observations, which correspond to the light observations from emerging stars, e.g. quasars, supernovas, etc, it is possible to get significant information about the geometry of the reachable part of the Universe. However, since these stars are not dense, to get more accurate information about the Universe, we suggest to supplement passive measurements by active ones, where we produce sources to probe the Universe. But what would this IP tell us about the world around?
1. One of the most challenging problems in modern physics is that of the dark matter. If successful, our research will provide a tool to identify the existence of dark matter in a particular part of the Universe. Indeed, our eventual goal is the geometry of Universe which defines Einstein's tensor and, by the equations of GR, the stress-energy. As, in turn, this stress-energy tensor depends on the dark matter this gives information about the latter.
2. Another fundamental question in physics related to our research is finding the topology of the Universe. Is it solid like an apple or has holes like a donut? The answer to this question have serious repercussions for physical models and our research would answer this question for the reachable part of the Universe.
With our study of IP with passive sources being successful, in this project we'd concentrate on IP with active sources. This means that, in our part of the Universe, i.e. the one where we live, we would model and analyse some special, "primary sources". We would use them to probe the Universe beyond our part. In doing so, we would make an extensive use of the non-linearity of Einstein's equations. This non-linearity would allow us to use the "primary sources" to generate the "secondary sources" which lie outside our part of the Universe and have properties mimicking those studied in the case of passive observations and looking like tiny stars.
Clearly, we would never achieve the infinite precision and never be able to have infinitely many measurements. These make it necessary to analyse the stability of our IP, i.e. its robustness with respect to the finiteness and error-proneness of the data. This is also our goal in this project.
In addition, having developed a method for IP of GR, we would look at IP for some other quasi-linear, time-dependent hyperbolic systems, in particular, IP for elastography which is an emerging important modality in medical imaging.
Although the study of IP for GR requires new ideas and methods, we have already developed some important ones to start attacking this problem. This is, firstly, our analysis of the case of passive observations where we use a strong parallelism between Riemannian geodesics and light-like geodesics on Lorentzian manifolds. Secondly, this is the analysis of clean intersections of Lagrangians and propagation of conormal singularities in our work on IP in optical tomography. Thirdly, this is our preliminary study of the Minkowski case dealing with the linearized conservation laws intrinsic for Einstein's equations.

The goal of the project is to develop a new rigorous method to solving non-linear hyperbolic inverse problems. This method would be superior to the existing ones with respect to its ability to deal with more general, non-linear equations with time-dependent coefficients, including those of general relativity as well as non-linear elastography, which are untreatable by the existing methods. It will also address the issues of stability in inverse problem of general relativity.

Inverse problems pertains a wide range of practical applications, from medical imaging to process monitoring, to oil and other mineral resources exploration, to meterological observations, to non-destructive testing. In many 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, the emerging potentially very promising modality of elastography imaging, seismological observations, oceanological research, including monitoring of fish stocks, weather monitoring, etc. Developing new mathematical methods which are robust with respect to the variation of the parameters of the medium with time and non-linear interaction inside the medium, will open new horizons in practical applications of inverse problems. For example, there is much activity now towards linear elastography. However, by its nature, this phenomenon is non-linear and, therefore, solving the related inverse problem in the non-linear context, which is one of the objectives of this research, will be important to getting better imaging. Similarly, if we consider meteorology/weather predictions, the processes in the atmosphere are rapidly changing with a high-level of interaction and, therefore, require models involving the non-linearity and time-dependence of parameters. As for the equations of general relativity, which are the primary goal of this project, they are intrinsically non-linear and, in the large scale, time-dependent. Also, the coordinate-invariant nature of the Einstein equations, describing the space-time as a 4-manifold, requires from a mathematician the use of invariant methods while analysing stability of the inverse problem of general relativity-necessary to make physical/cosmological conclusions compatible with measurements' errors -- requires stability analysis on manifolds.

The goal of this project, which is the area of the mathematical analysis, is not to create algorithms workable for a wide range of possible applications although we would undertake a numerical testing of the reconstruction algorithm developed for elastography. Rather, this project would eventually benefit practitioners in a variety of areas dealing with non-linear, time-dependent hyperbolic models (see above) by providing them a mathematical techniques to rigorously address inverse problems occurring in their areas of applications. When turning to general relativity, the project would provide a possibility, within the chosen mathematical model of the physical reality, to describe the topology and geometry of the reachable part of the universe and get information about the structure of the stress-energy tensor (which is related, e.g. with dark matter).