Analysis of Inverse Problems in General Relativity

尽管双曲型系统的逆问题得到了迅速的发展，但现有的结果仍然远远不能处理具有时间相关系数的方程或描述我们世界中多种现象的非线性双曲型方程。特别是，对于现代物理学中最基本、数学上最困难的模型之一爱因斯坦广义相对论方程（GR），没有研究IP的方法。这些方程将宇宙的几何特性（由爱因斯坦张量表示）与它的物质特性（由应力-能量张量表示）联系起来。在这个项目中，我们将在数学上严格地研究GR的IP。几年前开始这项研究，与M. Lassas（芬兰）和G. Uhlmann（美国）合作（由EPSRC小额资助），我们表明，有大量被动的，运动学型观测，对应于新兴恒星的光观测，例如类星体，超新星等，有可能获得关于宇宙可达部分的几何信息。然而，由于这些恒星的密度不高，为了获得关于宇宙的更准确的信息，我们建议用主动测量来补充被动测量，我们在主动测量中产生探测宇宙的源。但是这个IP能告诉我们关于周围世界的什么信息呢？现代物理学中最具挑战性的问题之一是暗物质。如果成功，我们的研究将提供一种工具来确定宇宙中某个特定部分暗物质的存在。事实上，我们的最终目标是宇宙的几何，它定义了爱因斯坦的张量，并通过GR方程，定义了应力-能量。反过来，这个应力-能量张量依赖于暗物质，这就提供了暗物质的信息。与我们的研究相关的另一个物理学基本问题是寻找宇宙的拓扑结构。它是像苹果一样实心的，还是像甜甜圈一样有洞？这个问题的答案对物理模型有严重的影响，我们的研究将回答宇宙中可到达部分的这个问题。随着我们对被动源IP的研究取得成功，在这个项目中，我们将专注于主动源IP。这意味着，在我们所处的宇宙中，也就是我们所处的宇宙中，我们将建模并分析一些特殊的“原始来源”。我们可以用它们来探测我们以外的宇宙。在这样做的过程中，我们将广泛地利用爱因斯坦方程的非线性。这种非线性将使我们能够使用“主要源”来产生“次要源”，这些“次要源”位于我们所处的宇宙之外，具有模仿被动观测中所研究的特性，看起来像微小的恒星。显然，我们不可能达到无限的精度，也不可能进行无限次的测量。这使得我们有必要分析IP的稳定性，即它相对于数据的有限性和错误性的稳健性。这也是我们这个项目的目标。此外，在开发了GR的IP方法后，我们将研究其他一些准线性，时间相关双曲系统的IP，特别是弹性成像的IP，这是医学成像中新兴的重要模式。虽然研究GR的IP需要新的思想和方法，但我们已经开发了一些重要的方法来开始解决这个问题。首先，这是我们对被动观测的分析，在被动观测中，我们使用了洛伦兹流形上黎曼测地线和类光测地线之间的强平行性。其次，本文分析了光学层析成像中拉格朗日的干净交叉点和法向奇点的传播。第三，这是我们对闵可夫斯基情况的初步研究，该情况涉及爱因斯坦方程固有的线性化守恒定律。

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.