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.

爱因斯坦的重力场理论为在天体物理和宇宙学量表上建模引力物理学提供了一种美丽而准确的手段。它可用于预测黑洞和中子星和普通恒星碰撞的观察结果（在电磁和重力辐射方面），也可以用来辨别观察结果。与爱因斯坦理论合作的一种特别有用的方法是将其作为一个初始价值问题：根据这种表述，构建在建模引力物理学中的使用时代，首先选择了时空的初始状态（在某种任意的兴趣时期），然后通过将时间构建到该初始状态的过去和未来。爱因斯坦的方程式（爱因斯坦理论的核心）都控制着初始状态的可能选择，并确定进化如何进行。该提案中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；它还涉及确定解决方案的通用行为，因为一个人接近时空的“奇异区域”（例如，在大爆炸附近）。除了对爱因斯坦方程解决方案溶液的行为的研究外，该赠款还支持对几何热流动方程（如RICCI流量和平均曲率流量）的解决方案的研究。在这里，兴趣在于拓扑空间与它们可以支持的曲率类型之间的数学关系。 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真空或电动爱因斯坦约束的接近CMC解决方案（具有非阳性宇宙常数），但否则似乎存在重大问题。这项赠款支持研究这些问题的工作 - 在许多情况下，包括渐近欧几里得（“ AE”）和渐近双曲线（“ AH”）解决方案以及封闭形式的解决方案，在许多情况下，解决方案的不存在和非唯一性。第二种方法，即粘合，可以将已知解决方案连接起来，以生成新的解决方案 - 例如n体初始数据集。 AH初始数据必须“无剪切”，如果要用于产生渐近平坦的空间；因此，这项赠款支持工作以开发胶合技术，从而使一对无剪切AH解决方案的无穷大，从而产生一种新的无剪切AH解决方案（具有一个单一的渐近区域）。霍金 - 柔性“奇异定理”通常伴随着时空曲率爆炸。已知的爱因斯坦方程的地质不完整解（允许在凯奇（Cauchy）地平线上延伸）。但是，“强大的宇宙审查制度（“ SCC”）的猜想表明，这并不是一般发生的。SCC模型版本已被证明是针对解决方案的家族（例如Gowdy spaceTime。在这些证明中，验证“ AVTD”行为（avtd“行为）（占用时间衍生品的占主导地位）在附近的空间衍生物中的占主导地位是一个cripart的工具。识别AVTD的行为，并建议使用一个杀人领域的真空解决方案中找到非分析的AVTD解决方案，而Einstein-Scalar解决方案中没有杀戮领域的其他支持的工作，以表明Kasner解决方案的AVTD行为稳定，可以在两种杀戮领域中稳定。分析研究以探索模型宇宙学空间的扩展方向。该赠款支持验证和探索这种行为的工作。 4）RICCI在Kahler几何形状附近流动：对于某些均匀的歧管M，M上的Kahler几何形状构成了M.所有Riemannian几何形状的空间的子空间。M。Ricci流动溶液上的Ricci流动溶液，该溶液始于Kahler几何形状，在Kahler的几何形状上仍然是Kahler。是否有RICCI流动解决方案在Kahler几何形状的集合之外开始，但渐近地接近它？ PI和他的合作者正在努力表明，对于某些几何形状，几何形状的稳定性在几何热流中的稳定性：RICCI流动和平均曲率流中的颈钉行为是旋转对称的对称的地质和嵌入的。有证据表明，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。