Applications of Riemann-Hilbert Theory to Gravitational Wave Interactions

如果线性引力理论是正确的，那么在真空中相互碰撞的平面引力波就会相互穿过，而不会改变它们的形状或强度。然而，根据爱因斯坦的非线性引力波理论，真空中两个相互碰撞的平面引力波相互作用，从而产生一种散射波，这种散射波不是平面波，与原来的平面波有明显的不同。本研究的主要目的是寻找在给定两个碰撞平面波时确定散射波的系统方法，即求解碰撞平面引力波初值问题的方法。一个相关的目标是寻求有关这种碰撞的某些重要问题的答案。其中一个问题是，对碰撞平面引力波的研究是否能告诉我们关于真实碰撞引力波（不是平面的）的任何重要信息。本研究从将初值问题重新表述为一类新的黎曼-希尔伯特问题开始，包括对该黎曼-希尔伯特问题的分析，研究其解的存在性和性质，并发现求解该问题的实用方法。

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.