Applications of Riemann-Hilbert Theory to Gravitational WaveInteractions (Physics)

如果引力的线性理论是正确的， 在真空中碰撞的平面引力波 彼此之间没有任何变化， 强度 然而，根据非线性理论， 爱因斯坦的引力，两个碰撞平面的引力 真空中的波相互作用，从而产生散射波 这不是一个平面波，它明显不同于 原始平面波 本研究的主要目的 是找到系统的方法来确定散射波 当给定两个碰撞的平面波时，即，方法 碰撞平面初值问题的求解 引力波 一个相关的目标是寻找答案 关于这种碰撞的一些重要问题。 一 这些问题之一是， 引力波可以告诉我们任何关于真实的 碰撞引力波（不是平面的）。 这项研究首先重新阐述了 一类新Riemann-Hilbert初值问题 问题，涉及这个黎曼-希尔伯特问题的分析， 研究其解的存在性和性质，以及 发现解决问题的实际方法

If a linear theory of gravitation were correct, then colliding plane gravitational waves in a vacuum would pass through one another without any alterations in their shapes or intensities. However, according to the nonlinear theory of gravitation due to Einstein, two colliding plane gravitational waves in a vacuum interact and thereby produce a scattered wave which is not a plane wave and which differs markedly from the original plane waves. The principal objective of this research is to find systematic methods of determining the scattered wave when the two colliding plane waves are given, i.e., methods of solving the initial value problem for colliding plane gravitational waves. A related objective is to seek the answers to certain important questions concerning such collisions. One of these questions is whether the study of colliding plane gravitational waves can tell us anything significant about real colliding gravitational waves (which are not planar). The research, which begins with a reformulation of the initial value problem into a certain new kind of Riemann-Hilbert problem, involves the analysis of this Riemann-Hilbert problem, the study of the existence and properties of its solutions, and the discovery of practical methods of solving the problem