Gravitational Lensing Geometry and Optics

摘要提案：DMS-9734586首席研究员：Arlie Petters本项目以两个处理引力场产生的光焦散的数学程序为中心。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。另一个项目研究时空中光学焦散曲面的整体几何（例如，沿着我们过去的光锥或关于旋转黑洞）与相关时空度量的奇点之间的联系。这些问题为光学焦散的数学理论指明了一个新的、有前途的方向。此外，它们的严格处理也可能导致对引力透镜类星体的通量增强及其在前景星系周围的超密度，以及活动星系核中x射线的快速变化有更深的理解。同样重要的是，这些研究可能会揭示理解和组织引力透镜中基本物理概念的新的有力方法，并可以为数值模拟的结果提供理论检验。引力透镜效应是指来自遥远光源（如类星体、扩展星系）的光由于中间物质分布（如星系、星系团）而发生的偏转。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”。这就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一特定量的概率。对这一问题的严谨研究直接关系到“放大偏差”问题。“另一个重要的问题是通过动态透镜（例如，旋转黑洞）来研究引力透镜。这在物理上是相关的，因为大多数星系的中心似乎都有黑洞。此外，旋转黑洞的引力透镜效应可能解释了活动星系核中x射线的快速变化。上述问题将使我与天体物理学家和天文学家进行物理讨论和最新观察，与计算机科学家进行在高性能计算机上运行的复杂数值代码，与数学家进行奇点理论和微分几何问题的互动。此外，这个总体目标隐含着分离和研究由引力透镜引起的潜在数学结构。一旦确定，这些结构就可以作为潜在的数学工具，用于可能需要它们的其他物理系统（可能是流体中的激波形成）。

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.