GLOBAL STUDIES OF EINSTEIN SPACETIMES

在爱因斯坦的广义相对论中，一个主要的开放理论问题是，在许多其他合理的场方程解中，时空奇点的出现，曲率可以变得无限，或者“爆炸”，或者在时空几何中出现一些更微妙的破坏。最根本的问题是，这些奇点是否代表了爱因斯坦引力理论的经典决定论的真正失败，从而是广义相对论本身的潜在致命缺陷，或者它们是否实际上以某种微妙的方式与决定论相一致。罗杰·彭罗斯（Roger Penrose）提出的一个猜想，称为“宇宙审查猜想”（cosmic censorship conjecture），如果它被证明是正确的，将解决这个基本问题，并确保爱因斯坦的理论在经典（即非量子）水平上的逻辑一致性。要阐明这个难题，就需要研究时空几何中扰动的演化，以及这些扰动（以光速传播）如何聚焦，（有时）如何爆发，从而产生物理奇点。该项目的主要目的之一是发展精确分析曲率传播、聚焦和膨胀的理论技术，以期为彭罗斯猜想的最终证明提供数学工具。这个项目的另一个方面涉及到最近才被发现的、由纯粹黎曼几何学家研究的“爱因斯坦空间”和由数学广义相对论家研究的“爱因斯坦时空”之间的奇妙联系。从动力学的观点来看，时空被看作是空间随时间的演化，某些爱因斯坦空间为这种动态演化提供了自然的吸引子。这不仅适用于经典维度的时空，也适用于高于或低于4维的时空。虽然高维时空的物理意义在很大程度上仍是推测性的，但毫无疑问，爱因斯坦理论（在各种可能的）维度上的上述性质具有真正的数学意义。这个项目的最后一个方面涉及发展改进的技术，用于在像两个黑洞碰撞这样的奇异过程中发射的引力波的数值研究。这样的波几乎肯定最终会被当前的或提议的实验搜索探测到，但可以想象，这些努力的最终成功可能取决于对所讨论的假设波形的精确理论模板的预先发展。这个项目正在进行的一项工作是研究计算波信号的可行性，并分析它们的性质，一直到零无穷，这是理论演化的自然边界。

One of the main open theoretical problems in Einstein's theory of general relativity is the occurrence, in many otherwise reasonable solutions to the field equations, of spacetime singularities at which the curvature can become infinite, or "blow up", or at which some more subtle breakdown in the spacetime geometry can arise instead. The fundamental issue is whether these singularities represent a genuine failure of classical determinism for Einstein's theory of gravitation, and thus a potentially fatal flaw of general relativity itself, or whether instead they are in fact consistent with determinism in some subtle way. A conjecture due to Roger Penrose, called the "cosmic censorship conjecture" would, if it could be proven true, resolve this fundamental question and assure the logical consistency of Einstein's theory at the classical (i.e., non-quantum) level. To shed light on this difficult problem requires a study of the evolution of disturbances in the spacetime geometry and of how these disturbances (which propagate at the speed of light) can focus and (sometimes) blow up to yield physical singularities. One of the main aims of this project is to develop theoretical techniques for the precise analysis of such propagation, focusing and blowup of curvature with a view towards providing mathematical tools for an ultimate proof of Penrose's conjecture.Another aspect of this project deals with a fascinating connection, that has come to light only recently, between the "Einstein spaces" studied by purely Riemannian geometers and the "Einstein spacetimes" studied by mathematical general relativists. From the dynamical point of view, in whichspacetime is seen as the evolution of space through time, certain Einstein spaces provide natural attractors for this dynamical evolution. This occurs not only for spacetimes of the classical dimension, but also for those of higher or lower than 4 dimensions as well. While the physical significance of higher dimensional spacetimes is still largely speculative there is little doubt thatthe aforementioned property of Einstein's theory (in various possible) dimensions) is of genuine mathematical interest.A final aspect of this project involves the development of improved techniques for the numerical study of gravitational waves emitted in exotic processes like the collision of two black holes. Such waves will almost surely be detected eventually by current or proposed experimental searches but it is conceivable that the ultimate success of these efforts may hinge on the prior development of accurate theoretical templates for the hypothetical waveforms in question. An ongoing effort of this project is to study the feasibility of computing the wave signals and analyzing their properties all the way out to null infinity, the natural boundary for the theoretical evolution.