Geometric Analysis Applied to General Relativity

几何分析应用于广义相对论

• 批准号: 0706794
• 负责人: Hubert Bray
• 金额: $ 20.3万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706794&HistoricalAwards=false
• 关键词: Geometric; Analysis; Applied; General; Relativity

Professor Bray studies geometric analysis problems that relate to scalar curvature, many of which are motivated by fundamental questions in General Relativity. Recently, Marcus Khuri and the PI have made important progress on the Penrose Conjecture for asymptotically-flat space-like slices of spacetimes by reducing the conjecture to interesting existence questions for certain naturally motivated systems of p.d.e.'s. One of these existence questions is similar to the one solved by Huisken-Ilmanen to prove the existence of the inverse mean curvature flow with jumps, but for a system of two equations instead of one equation. The physical interpretation of the Penrose Conjecture is the natural idea that the total mass of a spacetime with nonnegative energy density should be at least the mass contributed by the black holes in the spacetime. In 1973, Roger Penrose was able to turn the above statement into a precise geometric conjecture about the Cauchy data of an asymptotically flat space-like slice (itself a Riemannian 3-manifold) of a spacetime. The time-symmetric case, known as the Riemannian Penrose Conjecture, was proved by the PI in 1999, and by Huisken-Ilmanen in 1997 for a single black hole. In this case, the energy density of the spacetime equals the scalar curvature of the slice, the total mass is a parameter describing the rate at which the Riemannian manifold is becoming flat at infinity, and apparent horizons of black holes are area-outerminimizing minimal surfaces. The Riemannian Penrose Inequality is the statement that the total mass is greater than or equal to the square root of the surface area of the apparent horizons of the black holes divided by 16 pi. When we drop the assumption that the Riemannian manifold is time-symmetric in the spacetime and allow the second fundamental form of the slice to be anything, a generalization of this statement is known simply as the Penrose Conjecture. The PI also studies negative point mass singularities in General Relativity and questions relating to quasi-local mass.As acclaimed a theory as General Relativity is, fundamental aspects of the theory are still not understood. For example, given the state of the universe at one instant of (coordinate) time, it is not currently known if the relevant equations, including the Einstein equation, can be solved forward in time other than for a very short period. Yet the universe, as we observe daily, exists without interruption. Hence, understanding the existence theory of the Cauchy Problem in General Relativity, as this problem is called, is a major question. If General Relativity does not has a physical existence theory, this will be a major hint as to how the theory needs to be modified. If General Relativity does have an existence theory corresponding to the physical universe, then this will be one more piece of evidence supporting the theory. One thing that is clear is that black holes and singularities play an important role in this question. In any case, understanding fundamental theoretical questions about General Relativity will not only deepen our understanding of the theory, but also help current and future researchers develop the next generation of physical theories.

布雷教授研究与标量曲率有关的几何分析问题，其中许多问题是由广义相对论中的基本问题引起的。最近，Marcus Khuri和PI通过将彭罗斯猜想简化为某些自然驱动的p.d.e系统的有趣的存在性问题，在关于时空渐近平坦类空片的彭罗斯猜想上取得了重要进展。其中一个存在性问题类似于Huisken-Ilmanen为证明带跳跃的逆平均曲率流的存在性而解决的问题，但这是一个由两个方程而不是一个方程组成的系统。彭罗斯猜想的物理解释是一个自然的想法，即具有非负能量密度的时空的总质量至少应该是时空中黑洞所贡献的质量。1973年，罗杰·彭罗斯（Roger Penrose）能够将上述陈述转化为关于时空的渐近平坦类空片（本身是黎曼3流形）的柯西数据的精确几何猜想。时间对称的情况，被称为黎曼彭罗斯猜想，在1999年由PI证明，在1997年由Huisken-Ilmanen证明了一个黑洞。在这种情况下，时空的能量密度等于薄片的标量曲率，总质量是描述黎曼流形在无穷远处变平的速率的参数，黑洞的视视界是面积极限的最小表面。黎曼彭罗斯不等式是关于总质量大于等于黑洞视界表面积的平方根除以16的表述。当我们放弃黎曼流形在时空中是时间对称的假设，并允许片的第二种基本形式是任何东西时，这一陈述的推广被简单地称为彭罗斯猜想。PI还研究广义相对论中的负质量点奇点以及与准局部质量有关的问题。广义相对论作为一种广受赞誉的理论，其基本方面仍未被理解。例如，给定宇宙在某一时刻（坐标）的状态，目前还不知道相关方程，包括爱因斯坦方程，是否可以在时间上向前求解，而不是在很短的时间内。然而，正如我们每天所观察到的那样，宇宙是不间断地存在的。因此，理解广义相对论中柯西问题的存在论是一个重大问题。如果广义相对论没有一个物理存在理论，这将是一个重要的提示，说明该理论需要如何修改。如果广义相对论确实有一个与物理宇宙相对应的存在理论，那么这将是支持该理论的又一个证据。有一件事是明确的，黑洞和奇点在这个问题中扮演着重要的角色。无论如何，理解广义相对论的基本理论问题不仅会加深我们对这一理论的理解，还会帮助当前和未来的研究人员发展下一代物理理论。