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 timedependent coefficients or nonlinear 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 stressenergy 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,
kinematictype 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 stressenergy. As, in turn, this stressenergy 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 nonlinearity of Einstein's equations. This nonlinearity 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 errorproneness 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 quasilinear, timedependent 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 lightlike 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.
