Gravitational Lensing Geometry and Optics

摘要 提案：DMS-9734586主要研究者：Arlie Petters 这个项目的中心是两个数学程序，处理引力场产生的光学焦散。 其中一个程序研究了一般多平面弱场引力透镜系统的焦散曲线网络的放大截面。 另一个程序研究时空中光学焦散面的全局几何之间的联系（例如，沿着我们过去的光锥或关于旋转的黑洞）和相关时空度规的奇点。 这些问题指出了一个新的和有前途的方向，在光学焦散线的数学理论。 此外，他们的严格治疗也可能导致更深入的了解引力透镜类星体的通量增强和它们周围的前景星系的过密度，以及活动星系核中的快速X射线变化。 同样重要的是，这些研究可能揭示理解和组织引力透镜基本物理概念的新的和强大的方法，并可以为数值模拟的结果提供理论检验。 引力透镜是来自远距离光源的光的偏转（例如，类星体，扩展星系）通过介入物质分布（例如，星系，星系团）。 当前景星系放大背景类星体时，它会在某些类星体样本中引起“放大偏差”。 这导致了引力透镜中一个数学上困难而物理上重要的问题：确定光源被放大超过某个特定量的概率。 对这个问题的严格研究直接关系到“放大偏倚”问题。另一个重要的问题是通过动态透镜来研究引力透镜（例如， 旋转黑洞）。 这是物理相关的，因为大多数星系似乎在它们的中心有黑洞。 此外，旋转黑洞的引力透镜效应可以解释活动星系核中X射线的快速变化。 上述问题将引导我与天体物理学家和天文学家进行物理讨论和最新观测，计算机科学家在高性能计算机上运行复杂的数值代码，数学家在奇点理论和微分几何问题上进行互动。 此外，这个总的目标隐含着分离和研究引力透镜所调用的基本数学结构。 一旦确定，这些结构就可以作为其他物理系统（可能是流体中的激波形成）的潜在数学工具。

Abstract Proposal: DMS-9734586 Principal Investigator: Arlie Petters This project centers on two mathematical programs dealing with the optical caustics created by gravitational fields. One program investigates the magnification cross sections of caustic-curve networks due to generic multiplane weak-field gravitational lens systems. The other program studies the links between the global geometry of the optical caustic surfaces in a spacetime (e.g., along our past light cone or about rotating black holes) and the singularities of the associated spacetime metric. These issues point to a new and promising direction in the mathematical theory of optical caustics. Furthermore, their rigorous treatment may also lead to a deeper understanding of the flux enhancement of gravitationally lensed quasars and their overdensity around foreground galaxies, and rapid X-ray variability in active galactic nuclei. Equally important, such investigations may reveal new and powerful ways of understanding and organizing fundamental physical concepts in gravitational lensing, and can provide a theoretical check on results from numerical simulations. Gravitational lensing is the deflection of light from a distant source (e.g., quasar, extended galaxy) by an intervening matter distribution (e.g., galaxy, cluster of galaxies). When foreground galaxies magnify background quasars, it causes an ``amplification bias'' in certain quasar samples. This has led to a mathematically difficult and physically important problem in gravitational lensing: Determine the probability that a light source is magnified greater than some specified amount. A rigorous study of this problem directly bears on the the issue of ``magnification bias.'' Another important problem is to investigate gravitational lensing by dynamical lenses (e.g., rotating black holes). This is physically relevant since most galaxies seem to have black holes at their centers. Moreover, gravitational lensing by a rotatin g black hole may account for rapid X-ray variability in active galactic nuclei. The aforementioned problems will lead me to interact with astrophysicists and astronomers for physics discussions and latest observations, computer scientists for sophisticated numerical codes running on high-performance computers, and mathematicians for issues in singularity theory and differential geometry. Also, implicit with this general aim is isolating and studying the underlying mathematical structures invoked by gravitational lensing. Once determined, these structures can then be made available as potential mathematical tools for other physical systems (possibly shock formation in fluids) that may call upon them.