Kerr度量中的数学问题

发布时间：2018-04-30 15:59

本文选题：渐进平坦时空 + Kerr度量　；参考：《北京邮电大学》2011年硕士论文

【摘要】：当大质量星体发生引力塌缩时,会导致时空的奇点,然而奇点总是被事件视界所遮盖,不能通过任何方式影响事件视界外的观者。这就是说,在事件视界外部,时空具有良好的因果性。同时,从事件视界外的观者的角度来看,大质量星体塌缩后,留下的唯一客体就是黑洞,而事件视界就是它的表面。本文主要可以分为两个部分。第一部分,我们着重强调类光方向及其类光面在时空几何结构中的重要作用。黑洞时空的光锥结构与类光面的几何自然的协调,通过对类光面的分析,我们可以更容易的把握住时空的对称性,同时从数学的角度上讲,类光面的分析也揭示了相对论几何的丰富解析性质。在本文中,我们通过对类光面的分析,给出对引力波及其渐进平坦时空的数学理解。我们讨论在渐进平坦区域的光锥的几何性质,因为这些光锥上面的类光测地线可以到达类光无穷远,这个条件的成立,需要对光锥加一定程度的限制。完成这步后,我们探讨在类光无穷远镶嵌一个标准球面的可能性。为了文章内容的完整性,本文给出了类光面分析背后的物理意义,并推导类光面上几何量的传播方程,及其用数学的语言刻画类光面的几何图像。在文章第二部份,主要考虑Kerr度量。它是由2个参数刻画的一组Einstein场方程的解。黑洞究其本质而言,构成的元素为时间与空间的概念。到目前为止能描述引力塌缩的结果的严格解只有Kerr度量。从这个角度而言,它是最简单的能为物理学准确描述的宏观客体。源于Kerr度量的重要性,历史上有很多的研究针对着怎样从数学上刻画Kerr时空,并且这种数学描述不依赖于坐标。这些研究往往导致一些重要的数学定理的发现。这类定理有一个统一的名字一无毛定理。无毛定理的多种证明使得物理学家坚信：在大质量星体塌缩过程结束后,时空逐渐趋于稳定,事件视界外面的解也逐渐趋于Kerr解。本文用新的方法来证明无毛定理(唯一性定理)的一种类型。我们关注静态具有事件视界的黑洞时空解,这类黑洞无旋转,并且渐进平坦。我们的结论是这种时空完全由Schwarzschild解来描绘。
[Abstract]:When the gravity collapses of the mass stars, it will lead to the singularities of time and space. However, the singularities are always covered by the event horizon and can not affect the viewers outside the event horizon in any way. This means that, outside the event horizon, space-time has a good causality. At the same time, from the point of view of the viewer outside the event horizon, the only object left behind after the collapse of the mass stars is the black hole, and the event horizon is its surface. This paper can be divided into two parts. In the first part, we emphasize the important role of the optical-like direction and its light-like surface in the geometric structure of time and space. The conical structure of black hole space-time is in harmony with the geometric nature of light-like surface. By analyzing the light-like surface, we can grasp the symmetry of space-time more easily, and at the same time, from a mathematical point of view, The analysis of light-like surfaces also reveals the rich analytical properties of relativistic geometry. In this paper, we give a mathematical understanding of gravitational waves and their asymptotically flat spacetime through the analysis of photo-like surfaces. We discuss the geometric properties of optical cones in an asymptotically flat region, because the quasi-optical geodesic lines above these optical cones can reach the quasi-light infinity. This condition needs to be limited to a certain extent. After completing this step, we explore the possibility of embedding a standard sphere in a similar light infinity. In order to complete the content of the paper, this paper gives the physical meaning behind the light-like surface analysis, and deduces the propagation equation of the geometric quantity on the light-like surface, and describes the geometric image of the light-like surface with mathematical language. In the second part of the article, we mainly consider the Kerr metric. It is the solution of a set of Einstein field equations characterized by two parameters. Black holes are essentially the concepts of time and space. The only strict solution to the result of gravitational collapse so far is the Kerr metric. In this sense, it is the simplest macro object that can be accurately described for physics. Due to the importance of Kerr metrics, there have been many studies on how to describe Kerr spacetime mathematically, and this mathematical description does not depend on coordinates. These studies often lead to the discovery of important mathematical theorems. Such theorems have a uniform name, a hairless theorem. Various proofs of the no-Mao theorem make physicists firmly believe that after the collapse process of the massive stars, the space-time becomes stable gradually, and the solutions outside the event horizon tend to the Kerr solution. In this paper, a new method is used to prove a type of wordless theorem (uniqueness theorem). We focus on spatiotemporal solutions of black holes with event horizon, which are non-rotating and asymptotically flat. Our conclusion is that this time and space is completely described by the Schwarzschild solution.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2011
【分类号】：P145.8

【共引文献】