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On the Behavior of Solutions of Einstein's Equations and Solutions of Geometric Heat Flow Systems

爱因斯坦方程组解和几何热流系统解的行为

基本信息

批准号：
1707427
负责人：
James Isenberg
金额：
$ 12万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-15 至 2023-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1707427&HistoricalAwards=false
关键词：
Behavior Solutions Einstein Equations Geometric

项目摘要

Einstein's gravitational field theory provides a beautiful and remarkably accurate means for modeling gravitational physics on both the astrophysical and cosmological scales. It can be used to predict the observational consequences (both in terms of electromagnetic and gravitational radiation) of black holes and neutron stars and ordinary stars colliding, and it can also be used to discern from observations what the Big Bang was like. One particularly useful way to work with Einstein's theory is by formulating it as an initial value problem: According to this formulation, to construct spacetimes of use in modeling gravitational physics, one first chooses an initial state for the spacetime (at some arbitrary time of interest), and one then constructs the spacetime by evolving both into the past and the future of this initial state. Einstein's equations (at the heart of Einstein's theory) both control possible choices of the initial state, and determine how the evolution proceeds. The research supported in this proposal involves how to make choices of the initial state which satisfy the Einstein constraint equations; it also involves determining the generic behavior of solutions as one approaches "singular regions" of the spacetime (near the Big Bang, for example). Besides studies of the behavior of solutions of Einstein's equations, this grant also supports studies of solutions of geometric heat flow equations such as the Ricci flow and mean curvature flow. Here, the interest is in the mathematical relationship between topological spaces and the types of curvature that they can support. Remarkably, some of the techniques used in the study of geometric heat flow solutions are also useful in studying solutions of Einstein's equations.Among the specific projects supported by this grant are the following:1) Solutions of the Einstein constraint equations: Two approaches have been developed for constructing and studying solutions of the constraints: The first of these, the conformal method, works beautifully for constant mean curvature ("CMC") and near-CMC solutions of the vacuum or electrovac Einstein constraints (with nonpositive cosmological constant), but appears to have major problems otherwise. This grant supports work which studies these problems---non-existence and non-uniqueness of solutions---in a number of cases, including asymptotically Euclidean ("AE") and asymptotically hyperbolic ("AH") solutions, as well as solutions on closed manifolds. The second approach, gluing, allows known solutions of the constraints to be joined to produce new ones--e.g., N-body initial data sets. AH initial data must be "shear-free" if it is to be used to produce asymptotically flat spacetimes; hence this grant supports work to develop gluing techniques which allow the joining at infinity of a pair of shear-free AH solutions, thereby producing a new shear-free AH solution (with a single asymptotic region).2) Strong Cosmic Censorship: For almost 50 years, one of the major questions in mathematical relativity has been if the ubiquitous geodesic incompleteness in maximal spacetime developments predicted by the Hawking-Penrose "singularity theorems" is generically accompanied by spacetime curvature blowup. Geodesically incomplete solutions of Einstein's equations with bounded curvature (allowing extensions across a Cauchy horizon) are known; but the "Strong Cosmic Censorship ("SCC") conjecture suggests that this does not happen generically. Model versions of SCC have been proven for families of solutions, such as the Gowdy spacetimes. In these proofs, verifying "AVTD" behavior (dominance of time derivatives over space derivatives near the singular region) has been a crucial tool. The PI and collaborators has developed the singular initial value problem as a way of identifying AVTD behavior, and proposes to use it to find non-analytic AVTD solutions among vacuum solutions with one Killing field, and among Einstein-scalar solutions with no Killing fields. Other supported works seeks to show that the AVTD behavior of Kasner solutions is stable among solutions with two Killing fields. 3) Expanding Cosmologies: The PI and his collaborators propose to use a combination of numerical and analytical studies to explore the expanding direction of model cosmological spacetimes. There is good evidence for strongly attracting "entropic" behavior. This grant supports work to verify and explore this behavior. 4) Ricci Flow Near Kahler Geometries: For certain even-dimensional manifolds M, the set of Kahler geometries on M forms a subspace of the space of all Riemannian geometries on M. A Ricci flow solution which begins at a Kahler geometry remains Kahler. Are there Ricci flow solutions which begin outside the set of Kahler geometries but asymptotically approach it? The PI and his collaborators are working to show that this is the case, for a certain class of geometries.5) Stability of Neckpinch Behavior in Geometric Heat Flows: Neckpinch behavior in Ricci flow and mean curvature flow is well-understood for rotationally symmetric geometries and embeddings. There is evidence, both numerical and analytical, that such behavior is stable. The PI and his collaborators propose further work to verify this stability.

爱因斯坦的引力场理论提供了一个美丽的和非常准确的手段，在天体物理和宇宙尺度上建模引力物理学。它可以用来预测黑洞、中子星和普通恒星碰撞的观测结果（包括电磁辐射和引力辐射），也可以用来从观测中辨别大爆炸是什么样的。一个特别有用的方法是将爱因斯坦的理论公式化为一个初始值问题：根据这个公式，要构造用于引力物理建模的时空，首先要为时空选择一个初始状态（在某个任意感兴趣的时间），然后通过演化到这个初始状态的过去和未来来构造时空。爱因斯坦的方程（爱因斯坦理论的核心）既控制了初始状态的可能选择，又决定了演化的进行方式。这个提议所支持的研究涉及如何选择满足爱因斯坦约束方程的初始状态;它还涉及当人们接近时空的“奇异区域”（例如大爆炸附近）时确定解的一般行为。除了研究爱因斯坦方程的解的行为，该补助金还支持几何热流方程的解的研究，如里奇流和平均曲率流。在这里，兴趣是在拓扑空间和它们可以支持的曲率类型之间的数学关系。值得注意的是，一些用于研究几何热流解的技术也可用于研究爱因斯坦方程的解。该基金资助的具体项目包括：1）爱因斯坦约束方程的解：开发了两种方法来构建和研究约束方程的解：其中第一个，共形方法，工程漂亮的恒定平均曲率（“CMC”）和近CMC的解决方案的真空或electrovac爱因斯坦约束（具有非正的宇宙学常数），但似乎有重大问题，否则。该补助金支持研究这些问题的工作-解的不存在性和非唯一性-在一些情况下，包括渐近欧几里德（“AE”）和渐近双曲（“AH”）的解决方案，以及封闭流形上的解决方案。第二种方法，胶合，允许已知的约束解决方案被连接以产生新的约束解决方案-例如，N体初始数据集。如果要用来产生渐近平坦的时空，那么所有的初始数据必须是“无剪切”的;因此，这项资助支持开发胶合技术的工作，这种技术允许在无穷远处连接一对无剪切的AH解，从而产生一个新的无剪切的AH解（具有单个渐近区域）。2）强宇宙审查：近50年来，数学相对论中的一个主要问题是，霍金-彭罗斯“奇点定理”所预言的最大时空发展中普遍存在的测地线不完备性，一般都伴随着时空曲率爆破。爱因斯坦方程的测地不完全解有界曲率（允许在柯西视界上扩展）是已知的;但“强宇宙审查（“SCC”）猜想表明，这并不普遍发生。SCC的模型版本已经被证明适用于解族，例如Gowdy时空。在这些证明中，验证“AVTD”行为（奇异区域附近的时间导数对空间导数的优势）一直是一个至关重要的工具。PI和合作者已经开发了奇异初值问题作为识别AVTD行为的一种方式，并建议使用它来寻找具有一个Killing场的真空解和没有Killing场的爱因斯坦标量解之间的非解析AVTD解。其他支持的作品试图表明，AVTD行为的Kasner解决方案是稳定的解决方案与两个Killing场。3)扩展宇宙学：PI和他的合作者建议使用数值和分析研究的结合来探索模型宇宙时空的扩展方向。有很好的证据，强烈吸引的“熵”的行为。这项拨款支持验证和探索这种行为的工作。4)Kahler几何附近的Ricci流：对于某些偶数维流形M，M上的Kahler几何集合形成M上所有黎曼几何空间的子空间。一个Ricci流动的解决方案，开始于一个Kahler几何仍然是Kahler。是否存在开始在Kahler几何集合之外但渐近地接近Kahler几何集合的Ricci流动解？ PI和他的合作者正在努力证明这是某种几何类型的情况。5）几何热流中颈缩行为的稳定性：Ricci流和平均曲率流中的颈缩行为对于旋转对称几何和嵌入是很好理解的。有证据表明，无论是数字和分析，这种行为是稳定的。PI和他的合作者提出了进一步的工作来验证这种稳定性。