CAREER: The Wave Equation on Black Hole Backgrounds

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变的理解。科尔时空和这种扰动特别令人感兴趣，因为他们希望能为证明爱因斯坦方程这一族解的稳定性做出贡献。最近的工作集中于证明局域能量估计和Strichartz估计，这两个估计都是对波动方程的色散性质的度量，众所周知，它们是相当稳健的。前者在Tataru最近对长期猜想的普赖斯定律的证明中发挥了关键作用，普赖斯定律断言在Schwarzschild和Kerr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存在被囚禁的射线。在平坦的空间中，光沿着逃逸到无限远的线传播。因此，最初重叠但方向甚至略有不同的溶液包迅速扩散，这种扩散促进了腐烂。在Schwarzschild上，光的传播路径是由几何形状决定的，特别是有一个区域，称为光子球，在那里光子可以绕黑洞运行。这样的陷阱，其中光线保持在一个紧凑的集合中，是某些色散估计的已知障碍，例如局部能量估计。陷阱也发生在科尔族的时空上，尽管它的几何结构更复杂。广义相对论断言，宇宙是一个(1+3)维的弯曲空间，重力对应于这个空间的曲率。一种常见的描述是把宇宙想象成一张蹦床。放置在蹦床上的物体，如保龄球，使其弯曲，以吸引表面上的其他物体。爱因斯坦的方程模拟了宇宙这种曲率的曲率和演化。虽然爱因斯坦的方程是非常非线性的，但也有一些特殊的解是已知的。这些通常是通过施加许多对称来简化方程来找到的。与这一提议特别相关的是Minkowski时空、Schwarzschild时空家族和Kerr时空家族。它们分别对应于平面解、球对称黑洞和旋转黑洞。一个自然的问题是，这些解决方案是否稳定。也就是说，如果一个人开始接近，比方说，空间时代科尔家族的成员，它必然会保持与科尔家族成员的密切关系。(非线性)稳定性的唯一严格证明是关于Minkowski时空的，该时空始于Christodoulou和Klaineman的开创性工作。科尔族时空的稳定性是数学相对论中的一个重大开放问题，近年来引起了人们的极大兴趣。对波动方程在这种背景下的衰减性质的透彻理解被认为是证明这种稳定性的先决条件。预计这项提案中的研究将直接促进这一点。这项建议的教学内容主要包括开发广义相对论课程和一年级研讨会课程。研究和教学部分通过阅读研讨会和指导性研究项目整合在一起。

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.