Mathematical relativity and asymptotically hyperbolic manifolds

数学相对论和渐近双曲流形

• 批准号: RGPIN-2017-04896
• 负责人: Woolgar, Eric
• 金额: $ 2.19万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668885
• 关键词: Mathematical; relativity; asymptotically; hyperbolic; manifolds

If there are two kinds of geometry generally familiar, they are the flat geometry of Euclid and the round geometry of spheres. A third kind, "saddle shaped'' hyperbolic geometry, is less familiar but was popularized by the work of the artist MC Escher. These three models represent the constant curvature geometries and exhibit maximal symmetry, looking the same in every direction and at every point. Asymptotically hyperbolic (AH) manifolds, and their close cousins the asymptotically anti-de Sitter (AAdS) spacetimes, as less symmetrical. They may be bumpy and irregular in the middle but resemble hyperbolic geometry more and more at large distances from this central region. They play an important role in the modern physics of the last 30 years. They are a natural arena for black hole thermodynamics, and appear in the AdS/CFT correspondence, which relates these geometries to quantum conformal field theory and manifests "holography'', the speculative notion that physics within a region is encoded by other physics on the boundary of that region.******The geometry of spacetime is governed by Einstein's general relativity. Einstein's equations admit a rich variety of AH and AAdS geometries. Some contain no matter and no black holes, yet have mass---indeed, negative mass!---and have unusual shape or topology. This research proposal seeks to study these fascinating geometries. Among other questions, it will ask when can the mass be negative, how negative can it be, and how is this related to the topology? It will also ask whether similar structure is seen in geometries governed by other geometric equations, namely fourth order partial differential equations.******Geometries can be deformed to be made smoother and more symmetrical. Such deformations are important tools for mathematicians and physicists. Part of this proposal concerns the study of AH geometries that are deformed by Ricci flow, a method used recently to prove the Poincaré conjecture. The proposal seeks to determine the detailed evolution of AH geometries deformed by Ricci flow.******Another part of this proposal involves a generalization of Einstein's equations in which the Ricci curvature tensor is replaced by a more general object, the Bakry-Émery-Ricci tensor. Here the question is whether the mathematical structure of Einstein's theory is really exclusive to the geometries that arise from that theory or is shared by other more general geometries as well.******In short, the proposal seeks answers to important questions in physics, both in the AdS/CFT correspondence and in general relativity, by leveraging recent mathematical advances in asymptotically hyperbolic manifolds, geometric flows, and manifolds-with-density with a Bakry-Émery-Ricci lower bound.

如果说有两种普遍熟悉的几何学，那就是欧几里得的平面几何学和球体的圆形几何学。第三种“鞍形”双曲几何不太为人所知，但因艺术家 MC Escher 的作品而普及。这三种模型代表恒定曲率几何并表现出最大对称性，在每个方向和每个点上看起来都相同。渐近双曲 (AH) 流形及其近亲渐近反德西特 (AAdS) 时空，如 对称的。它们的中间可能是凹凸不平且不规则的，但在距该中心区域较远的地方越来越类似于双曲几何。它们在过去 30 年的现代物理学中发挥着重要作用。它们是黑洞热力学的天然舞台，并出现在 AdS/CFT 对应关系中，该对应关系将这些几何形状与量子共形场论联系起来，并体现了“全息术”，这是一种推测性概念 一个区域内的物理现象是由该区域边界上的其他物理现象编码的。******时空的几何形状受爱因斯坦广义相对论的支配。爱因斯坦方程允许丰富多样的 AH 和 AAdS 几何形状。有些不包含任何物质，也不包含黑洞，但却具有质量——实际上是负质量！——并且具有不寻常的形状或拓扑。该研究计划旨在研究这些迷人的几何形状。除其他问题外，它还会问质量何时可以为负，负值可以有多大，以及这与拓扑有何关系？它还会询问在由其他几何方程（即四阶偏微分方程）控制的几何中是否可以看到类似的结构。******几何可以变形以变得更平滑和更对称。这种变形是数学家和物理学家的重要工具。该提案的一部分涉及研究由 Ricci 流变形的 AH 几何形状，这是最近用来证明庞加莱猜想的一种方法。该提案旨在确定由 Ricci 流变形的 AH 几何形状的详细演化。********该提案的另一部分涉及爱因斯坦方程的推广，其中 Ricci 曲率张量被更通用的对象 Bakry-Émery-Ricci 张量取代。这里的问题是，爱因斯坦理论的数学结构是否真的专属于该理论产生的几何结构，还是也被其他更一般的几何结构所共享。********简而言之，该提案通过利用渐近双曲流形、几何流和广义相对论中的最新数学进展，寻求物理学中重要问题的答案，无论是在 AdS/CFT 对应关系中还是在广义相对论中。 具有 Bakry-Émery-Ricci 下界的密度流形。