New geometrical perspectives in general relativity

广义相对论中的新几何观点

• 批准号: RGPIN-2022-03440
• 负责人: Woolgar, Eric
• 金额: $ 2.04万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758369
• 关键词: New; geometrical; perspectives; general; relativity

Two driving forces have fueled general relativity research in the past few decades. One is the onslaught of data from ``big science'', such as the cosmic microwave background (CMB) observations and LIGO gravitational wave observations. The other is the explosive growth of geometric analysis, one of the most successful fields in mathematics in the last half-century. This proposal aims to leverage geometric analytical tools and to apply them to problems in general relativity. One such problem is that of the possible topology of black hole horizons. This in turn is part of the black hole uniqueness problem. While these problems are mostly well understood in 4 spacetime dimensions, the theory will not be complete until we understand them in all dimensions and for nonzero cosmological constant. Two tools to exploit here are the equations of static vacuum Einstein metrics and the near horizon geometry equations. These are both types of Bakry-Émery Ricci "quasi-Einstein" equations, as are the Ricci soliton equations which are important in the theory of Ricci flow. I propose to study these equations from a unified viewpoint, which will enable me to borrow techniques from one kind of Bakry-Émery Ricci equation and exploit it to study another. This strategy has already yielded early results, including a surprising application to interpretation of CMB data: Galloway, Khuri, and I have been able to show that the allowed topologies for the Universe can be constrained whenever the Universe is closed, even if the Universe does not achieve closure density (which is a sufficient but not necessary condition for a closed Universe). A second, related part of the proposal is the study of asymptotically hyperbolic spacetimes, especially static vacuum ones. A vexing problem is an old conjecture of Horowitz and Myers that there exists a kind of positive mass theorem (allowing some negative mass, but with lower bound) for isolated systems whose conformal infinity is that of a torus rather than a sphere, if the cosmological constant is negative. Recent progress has shown that the conjecture holds in some quite special circumstances, and I propose to extend those results. But the main test of the conjecture will come when special symmetry assumptions are removed. In particular, can the conjecture hold when conformal infinity is a torus but the spacetime contains an array of topologically spherical black holes whose interaction potential energy may contribute negative mass-energy? Finally, I propose to study soliton solutions of the curve shortening flow (CSF) and related extrinsic geometric flows. A curve evolves by curve shortening if it deforms in the normal direction with speed given by its curvature. Solitons are curves that evolve self-similarly under the flow. But little is known of CSF solitons in curved manifolds. This will be a rich source of research projects for undergraduates. I expect to apply the results to cosmic string evolution in cosmology.

过去几十年来，有两种驱动力推动了广义相对论的研究。一是来自“大科学”的数据冲击，例如宇宙微波背景（CMB）观测和LIGO引力波观测。另一个是几何分析的爆炸性增长，这是过去半个世纪数学中最成功的领域之一。该提案旨在利用几何分析工具并将其应用于广义相对论中的问题。其中一个问题是黑洞视界的可能拓扑问题。这又是黑洞唯一性问题的一部分。虽然这些问题大多在 4 个时空维度上得到了很好的理解，但只有我们在所有维度和非零宇宙学常数上理解它们，这个理论才会完整。这里要利用的两个工具是静态真空爱因斯坦度量方程和近地平线几何方程。这些都是 Bakry-Émery Ricci“准爱因斯坦”方程的类型，在 Ricci 流理论中很重要的 Ricci 孤子方程也是如此。我建议从统一的角度研究这些方程，这将使我能够借用一种 Bakry-Émery Ricci 方程的技术，并利用它来研究另一种方程。这一策略已经取得了早期成果，包括对 CMB 数据解释的令人惊讶的应用：加洛韦、库里和我已经能够证明，只要宇宙是封闭的，宇宙允许的拓扑就可以受到限制，即使宇宙没有达到封闭密度（这是封闭宇宙的充分但不是必要条件）。该提案的第二个相关部分是渐近双曲时空的研究，特别是静态真空时空。一个令人烦恼的问题是霍洛维茨和迈尔斯的一个古老猜想，即如果宇宙常数为负，则对于共形无穷大是环面而不是球体的孤立系统，存在一种正质量定理（允许一些负质量，但有下限）。最近的进展表明，这个猜想在一些非常特殊的情况下是成立的，我建议扩展这些结果。但当特殊的对称性假设被消除时，该猜想的主要检验就会到来。特别是，当共形无穷大是一个环面，但时空包含一系列拓扑球形黑洞，其相互作用势能可能贡献负质能时，该猜想是否成立？最后，我建议研究曲线缩短流（CSF）和相关的外在几何流的孤子解。如果曲线以曲率给定的速度沿法线方向变形，则曲线会因曲线缩短而演化。孤子是在流动下自相似演化的曲线。但人们对弯曲流形中的 CSF 孤子知之甚少。这将成为本科生研究项目的丰富来源。我希望将这些结果应用于宇宙学中的宇宙弦演化。