CAREER: The Wave Equation on Black Hole Backgrounds

职业：黑洞背景上的波动方程

• 批准号: 1054289
• 负责人: Jason Metcalfe
• 金额: $ 41.09万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1054289&HistoricalAwards=false
• 关键词: CAREER; Wave; Equation; Black; Hole

The primary goal of the proposed research is to improve the understanding of the decay for solutions to the wave equation on black hole backgrounds. Kerr spacetimes and perturbations of such are of particular interest, due to aspirations to contribute to a proof of the stability of this family of solutions to Einstein's equations. Recent work has focused on proving localized energy estimates and Strichartz estimates, which are both measures of the dispersive nature of the wave equation which are known to be fairly robust. The former have played a key role in Tataru's recent proof of the long conjectured Price's law, which asserts a certain decay rate for solutions to the wave equation on the Schwarzschild and Kerr backgrounds. A key feature of these blackhole spacetimes which demands extra attention is the existence of trapped rays. In flat space, light travels on lines which escape to infinity. Thus packets of a solution which are initially overlapping but traveling in even slightly different directions quickly spread out, and this spreading promotes decay. On Schwarzschild, the paths on which light travels are dictated by the geometry, and in particular, there is a region, called the photon sphere, where photons can orbit the blackhole. Such trapping, where the rays remain in a compact set, is a known obstacle to certain dispersive estimates, such as localized energy estimates. Trapping also occurs on the Kerr family of spacetimes, though its geometry is more complicated. General relativity asserts that the universe is a (1+3) dimensional curved space and that gravity corresponds to the curvature of this space. A common description is to think of the universe as a trampoline. A mass, such as a bowling ball, which is placed on the trampoline causes it to curve in such a way as to attract other objects on the surface. Einstein's equations model the curvature and evolution of such curvature of universes. Though Einstein's equations are quite nonlinear, a few special solutions are known. These are typically found by imposing many symmetries to simplify the equations. Of particular relevance to this proposal are the Minkowski space time, the family of Schwarzschild space times, and the Kerr family of space times. These correspond to the flat solution, to spherically symmetric black holes, and to rotating black holes respectively. A natural question to ask is whether these solutions are stable. That is, if one starts close to, say, a member of the Kerr family of space times, will it necessarily remain close to a member of the Kerr family. The only rigorous proofs of (nonlinear) stability are for the Minkowski space time, which began with the seminal work of Christodoulou and Klainerman. The stability of the Kerr family of space times is a major open problem in mathematical relativity which has been garnering much interest recently. A thorough understanding of the decay properties of the wave equation on such backgrounds is considered to be prerequisite knowledge to any proof of such stability. The studies in this proposal are expected to directly contribute to this. The teaching components of this proposal consist primarily of the development of a course on general relativity as well as a first year seminar course. The research and teaching components are integrated through reading seminars and directed research projects.

拟议研究的主要目标是提高对黑洞背景下波动方程解的衰减的理解。 克尔时空和扰动等是特别感兴趣的，由于愿望有助于证明这一家庭的解决方案，爱因斯坦方程的稳定性。 最近的工作集中在证明本地化的能量估计和Eschenhartz估计，这两个措施的色散性质的波动方程是已知的是相当强大的。 前者在Tataru最近证明的长期约束的普莱斯定律中发挥了关键作用，该定律断言史瓦西和克尔背景下波动方程的解具有一定的衰减率。 这些黑洞时空的一个需要特别注意的关键特征是被捕获的射线的存在。 在平坦的空间中，光沿着逃逸到无限远的直线传播。 因此，最初重叠但方向略有不同的溶液包迅速扩散，这种扩散促进了衰变。 在史瓦西上，光的传播路径是由几何决定的，特别是有一个称为光子球的区域，光子可以围绕黑洞旋转。 这样的捕获，其中射线保持在一个紧凑的集合，是一个已知的障碍，某些色散估计，如局部能量估计。 陷获也发生在克尔族时空中，尽管它的几何形状更为复杂。 广义相对论认为宇宙是一个（1+3）维的弯曲空间，引力对应于这个空间的曲率。 一个常见的描述是把宇宙想象成一个蹦床。 一个质量，如保龄球，这是放在蹦床上，使它弯曲的方式，以吸引其他物体的表面。 爱因斯坦方程模拟了宇宙的曲率和这种曲率的演化。 虽然爱因斯坦方程是相当非线性的，但有几个特殊的解是已知的。 这些通常是通过施加许多对称性来简化方程来发现的。 与这个提议特别相关的是闵可夫斯基时空、史瓦西时空家族和克尔时空家族。 它们分别对应于平坦解、球对称黑洞和旋转黑洞。 一个自然的问题是这些解是否稳定。 也就是说，如果一个开始接近，比如说，克尔族时空的一个成员，它一定会保持接近克尔族的一个成员。 唯一严格证明（非线性）稳定性的是闵可夫斯基时空，它始于Christodoulou和Klainerman的开创性工作。 Kerr时空族的稳定性是数学相对论中一个重要的开放问题，近年来引起了人们的广泛兴趣。 深入了解这种背景下波动方程的衰减性质被认为是证明这种稳定性的先决条件。 预计本提案中的研究将直接为此作出贡献。 该建议的教学内容主要包括广义相对论课程的开发以及第一年的研讨课程。 研究和教学部分通过阅读研讨会和定向研究项目相结合。