CAREER: The Wave Equation on Black Hole Backgrounds

提出的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。克尔时空和这样的扰动是特别有趣的，因为渴望有助于证明爱因斯坦方程的这个解族的稳定性。最近的工作集中在证明局域能量估计和Strichartz估计上，它们都是波动方程色散性质的测量，已知波动方程相当稳健。前者在Tataru最近对长期猜测的Price定律的证明中发挥了关键作用，Price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。这些黑洞时空需要特别注意的一个关键特征是被捕获的射线的存在。在平坦空间中，光沿着逃逸到无限远的线传播。因此，最初重叠但方向稍有不同的溶液包会迅速扩散，而这种扩散会促进衰变。在史瓦西，光的传播路径是由几何形状决定的，特别是有一个区域，叫做光子球，光子可以在那里绕黑洞运行。这种捕获，即射线保持在一个紧凑的集合中，是对某些色散估计的已知障碍，例如局部能量估计。捕获也发生在克尔时空族上，尽管它的几何结构更为复杂。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。一个常见的描述是把宇宙想象成一个蹦床。放置在蹦床上的物体，如保龄球，使蹦床以某种方式弯曲，从而吸引表面上的其他物体。爱因斯坦的方程模拟了宇宙的曲率和这种曲率的演化。虽然爱因斯坦的方程是非常非线性的，但有一些特殊的解是已知的。这些通常是通过施加许多对称性来简化方程而得到的。与这一建议特别相关的是闵可夫斯基时空、史瓦西时空族和克尔时空族。它们分别对应于平解、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，时空克尔族的一个成员，它一定会一直接近克尔族的一个成员吗。（非线性）稳定性的唯一严格证明是闵可夫斯基时空，它始于Christodoulou和Klainerman的开创性工作。克尔时空族的稳定性是数学相对论中一个重要的开放性问题，近年来引起了人们的极大兴趣。彻底了解波动方程在这种背景下的衰减特性，被认为是证明这种稳定性的先决条件。本提案中的研究预计将对此作出直接贡献。本建议的教学内容主要包括开设一门广义相对论课程以及一年级的研讨会课程。研究和教学部分通过阅读研讨会和指导研究项目相结合。

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.