Scalar Curvature, the Penrose Conjecture, and the Axioms of General Relativity

标量曲率、彭罗斯猜想和广义相对论公理

• 批准号: 1007063
• 负责人: Hubert Bray
• 金额: $ 32.9万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007063&HistoricalAwards=false
• 关键词: Scalar; Curvature; Penrose; Conjecture; Axioms; Scalar; Curvature; Penrose; Conjecture; Axioms

First, the PI will continue his research on scalar curvature, especially on 3 manifolds. Prior results by the PI in this area include a joint work with Andre Neves in 2002 that classifies prime 3-manifolds with Yamabe invariant greater than RP^3 and a 2008 paper with Pengzi Miao that gives an upper bound on the capacity of surfaces in 3-manifolds with nonnegative scalar curvature. In 2009, the PI's joint paper with Simon Brendle, Michael Eichmair, and AndreNeves proves that A_{min}R_{min} \le 12\pi on compact 3-manifolds which contain embedded incompressible RP^2, where A_{min} is the area of the minimal RP^2 and R_{min} is the minimum value of the scalar curvature. Using Ricci flow, they show that the 3-manifold is a spherical space form in the case of equality. Second, the PI will continue to work toward a proof of the full Penrose conjecture. The PI's 2001 paper proved the Riemannian Penrose conjecture in dimension 3, improving the case of one black hole proved by Huisken and Ilmanen to any number of black holes using a different technique. Since then, the PI proved a similar type of inequality for zero area singularities in 2005 (with some additional hypotheses), the Riemannian Penrose conjecture in dimensions less than 8 in a joint work with Dan Lee in 2007, and showed that the full Penrose conjecture on Cauchy data (M^3,g,k) reduces to the Riemannian case whenever certain systems of p.d.e.s can be solved in a joint work with Marcus Khuri in 2009. These systems of p.d.e.s rely on a new identity that they proved called the Generalized Schoen-Yau identity, which they believe will be a very useful identity for a broad range of problems in mathematical relativity. Third, the PI is opening up a new research direction for himself as he examines the axioms of general relativity to see how they may be modified as little as possible to account for the widely accepted existence of dark matter.Einstein's theory of general relativity was made possible by Gauss and Riemann, both mathematicians, who developed the field of mathematics called differential geometry decades before. Since then, advances in differential geometry have played a crucial role in understanding the implications of Einstein's theory. Einstein used differential geometry to make the qualitative statement ``matter curves spacetime'' precise, thereby showing that gravity results as a consequence of this fundamental idea. By contrast, Newton's inverse square law for gravity has been shown to be false by measuring the precession of the orbit of Mercury. Hence, understanding gravity correctly would appear to require understanding the properties of curvature, currently pursued most directly by mathematicians studying geometric analysis. Black holes, predicted by general relativity and now known to exist, are fundamentally geometric objects, and have been the focus of much of the PI's efforts, resulting in theorems which yield a deeper physical insight into these fascinating phenomena. In light of this rich history of geometric analysis playing a crucial role in understanding the large scale structure of the universe, the PI is now looking to geometric motivations to try to understand the nature of dark matter. While dark matter is known to make up 23% of the mass of the universe and hence has very important gravitational effects, it is otherwise invisible. A geometric idea observed by the PI, as well as other motivations, leads to considering a real-valued scalar field as a model for dark matter, described by the Einstein Klein-Gordon equations. Astrophysicists have already observed that this model for dark matter is consistent with the flat rotation curves of galaxies. The PI is studying the idea that density waves in this scalar field dark matter produce density waves in regular matter, resulting in star formation and both bars and spiral patterns in some galaxies, an exciting possibility supported by preliminary simulations. If correct, this would suggest that while dark matter itself is invisible, its gravitational effects may be quite dramatic.

首先，PI将继续他对标态曲率的研究，尤其是在3个歧管上。 PI在该领域的先前结果包括2002年与安德烈·尼维斯（Andre Neves）的联合作品，该合作将yamabe不变性的素数大于RP^3和2008年的Pengzi Miao纸分类，该纸张具有pengzi miao的上限，该纸在3个manifolds中表面的能力具有上限，该表面具有非负量表曲线的3个manifolds。 In 2009, the PI's joint paper with Simon Brendle, Michael Eichmair, and AndreNeves proves that A_{min}R_{min} \le 12\pi on compact 3-manifolds which contain embedded incompressible RP^2, where A_{min} is the area of​​ the minimal RP^2 and R_{min} is the minimum value of the scalar curvature.使用RICCI流，他们表明3个manifold是平等情况下的球形空间形式。其次，PI将继续朝着完整的Penrose猜想的证明努力。 PI的2001年论文证明了Riemannian Penrose在维度3中的猜想，改善了Huisken和Ilmanen证明了一个黑洞的情况，并使用其他技术来改善任何数量的黑洞。 Since then, the PI proved a similar type of inequality for zero area singularities in 2005 (with some additional hypotheses), the Riemannian Penrose conjecture in dimensions less than 8 in a joint work with Dan Lee in 2007, and showed that the full Penrose conjecture on Cauchy data (M^3,g,k) reduces to the Riemannian case whenever certain systems of p.d.e.s can be solved in 2009年与马库斯·库里（Marcus Khuri）的联合合作。这些P.D.E.S的系统依赖于他们被证明称为广义的Schoen-Yau身份的新身份，他们认为这对于数学相对性的广泛问题来说将是非常有用的身份。第三，PI正在研究一般相对论的公理，以了解如何将它们的修改尽可能少地修改，以说明深色物质的广泛接受理论。从那时起，差异几何形状的进步在理解爱因斯坦理论的含义中起着至关重要的作用。爱因斯坦使用差异几何形状来确切地陈述``物质曲线''的精确陈述，从而表明由于这种基本思想而导致重力结果。相比之下，牛顿的重力反向定律已通过测量汞轨道的进攻证明是错误的。因此，正确理解重力似乎需要了解曲率的特性，目前，数学家研究几何分析最直接地追求。一般相对论预测的黑洞现在已经存在，从根本上是几何对象，并且已成为PI大部分努力的重点，从而导致定理对这些迷人现象产生更深入的物理见解。鉴于这种丰富的几何分析历史在理解宇宙的大规模结构中起着至关重要的作用，PI现在正在寻求几何动机来尝试理解暗物质的本质。虽然众所周知，暗物质占宇宙质量的23％，因此具有非常重要的引力效应，但否则是看不见的。 PI观察到的几何思想以及其他动机，导致将真实价值的标量场视为暗物质的模型，由爱因斯坦·克莱恩·戈登方程描述。天体物理学家已经观察到，暗物质模型与星系的平坦旋转曲线一致。 PI正在研究以下想法：在这种标量场中，密度波在常规物质中产生密度波，导致恒星形成以及某些星系中的条形和螺旋模式，这是由初步模拟支持的令人兴奋的可能性。如果正确的话，这表明虽然暗物质本身是看不见的，但其引力效应可能会非常引人注目。