Mathematical Problems in General Relativity

【摘要】广义相对论中的数学问题这个建议涉及引力坍缩和黑洞形成问题的数学方面。从解析的角度讲，这是对渐近平坦初始数据的合适爱因斯坦-物质系统的初值问题的研究。这个主题的中心问题是著名的彭罗斯的弱和强宇宙审查猜想。在之前的工作中，作者能够解决在球对称爱因斯坦-麦克斯韦标量场系统的背景下强宇宙审查的问题，证实了一个在物理界一直争论的启发式图像。目前的建议是发展一个完整的理论，引力坍缩在球面对称，同时考虑到角动量的物理。考虑到球对称的约束，通过电荷来模拟角动量。在数学上，对于带电自引力标量场，即爱因斯坦-麦克斯韦带电标量场系统，给出了一个严格的初值问题。该研究大致可分为三个解析上不同的问题族：对捕获面形成的研究，视界上场衰变的研究，以及柯西视界不稳定性的研究。所有问题都需要远远超出拟线性双曲方程理论标准技术的方法。在二维空间中。特别是，黑洞几何特征与非线性波动方程之间的相互作用将发挥重要作用。在球对称的情况下理解这些问题，将有希望指出正确的框架，这些问题最终可以在没有对称的情况下研究。一颗恒星在自身引力的作用下坍缩，随后可能形成黑洞，这是广义相对论中最令人兴奋的预言之一。然而，这个过程并没有被很好地理解。我们目前的大部分课程都来自爱因斯坦方程的所谓“显式解”，即理论的控制方程。然而，这些“明确”的解决方案除了表现出与身体相关的行为外，还表现出被广泛认为是病态的行为。例如，我们对引力坍缩最终状态的直觉在很大程度上基于克尔解，它包含所谓的“封闭类时曲线”。这使得进入黑洞的观察者随后可以进行时间旅行。这是这个解的特殊性质造成的不幸意外，还是黑洞的一般性质？鉴于爱因斯坦方程的强烈非线性，以及在数值或启发式工作的基础上做出准确预测的困难，这似乎是一个严格的数学分析技术可以产生巨大影响的领域。在这个项目中，建议对引力坍缩的现实模型进行严格的数学研究，特别是考虑角动量的模型，角动量是导致克尔解中“因果关系”失效的机制。在作者之前的工作中已经取得了部分进展，特别是，上面提到的图片的稳定性问题已经在一个非常有限的环境中得到了解决。由于这个问题解决了牛顿决定论在完全经典（即非量子）物理理论背景下的显著失败，这项工作可能会影响当前对物理学一些基本原理的看法。

ABSTRACTMATHEMATICAL PROBLEMS IN GENERAL RELATIVITYThis proposal deals with mathematical aspects of the problem ofgravitational collapse and the formation of black holes. Analyticallyspeaking, this is the study of the initial value problem forappropriate Einstein-matter systems for asymptotically flat initialdata. The central questions in the subject are the celebrated weak andstrong cosmic censorship conjectures of Penrose. In previous work,the author was able to resolve the issue of strong cosmic censorship inthe setting of the spherically symmetric Einstein-Maxwell scalar fieldsystem, confirming a heuristic picture that had been the subject of debatein the physics community. The present proposal concerns the developmentof a complete theory for gravitational collapse in spherical symmetry,which at the same time takes into account the physics of angular momentum.In view of the constraint of spherical symmetry, angular momentum issimulated through charge. Mathematically, a rigorous formulation is givenby the initial value problem in the large for a charged self-gravitating scalarfield, i.e. for the Einstein-Maxwell-charged scalar field system. Thestudy can be divided roughly into three analytically distinct familiesof problems: the study of the formation of trapped surfaces,the study of the decay of fields on the event horizon, and the study ofthe instability of the Cauchy horizon.All problems require methods which go well beyond standard techniques ofthe theory of quasilinear hyperbolic p.d.e.'s in two dimensions.In particular, the interaction between geometric features characteristicof black holes with non-linear wave equations will certainly play a bigrole. Understanding these issues in the spherically symmetric case willhopefully point to the correct framework where these issues can eventuallybe studied in the absense of symmetry.The collapse of a star under the force of its own gravity andthe possible subsequent formation of a black hole is one of themost exciting predictions of the general theory of relativity. Thisprocess, however, is not well understood. Most of our currentintuition derives from so-called ``explicit solutions'' of theEinstein equations, the governing equations of the theory.These ``explicit'' solutions, however, in addition to exhibitingbehavior which is hoped to be physically relevant, exhibit alsobehavior widely considered to be pathological. For example, the Kerrsolution, on which our intuition for the final state of gravitationalcollapse is largely based, contains so-called ``closed timelike curves''.These allow observers who enter the black hole to subsequently travel in time.Is this an unhappy accident of the very special nature of thissolution, or is this a general property of black holes? In viewof the strong non-linearity of the Einstein equations, and the difficultyof making accurate predictions on the basis of numerical or heuristicwork, this seems to be an area where the techniques of rigorousmathematical analysis can make a tremendous impact.In this project, it is proposed to undertake a rigorous mathematicalstudy of a realistic model of gravitational collapse, in particular onewhich takes account of angular momentum, the mechanism that leads to thefailure of ``causality'' in the Kerr solution. Partial progress hasalready been made in the author's previous work, and in particular, thequestion of the stability of the picture alluded to above has beenresolved in a very restricted setting. As this problem addresses aspectacular failure of Newtonian determinism in the setting of acompletely classical (i.e. non-quantum)physical theory, this work could impact current viewson some of the fundamental principles of physics.