GLOBAL STUDIES OF EINSTEIN SPACETIMES

在爱因斯坦的广义相对论中，一个主要的开放性理论问题是，在场方程的许多其他合理解中，时空奇点的出现，在那里曲率可以变得无限大，或者“爆炸”，或者在那里时空几何中的一些更微妙的崩溃可以出现。 根本的问题是，这些奇点是否代表爱因斯坦引力理论的经典决定论的真正失败，从而成为广义相对论本身的潜在致命缺陷，或者它们实际上是否以某种微妙的方式与决定论相一致。 罗杰·彭罗斯的一个猜想，称为“宇宙审查猜想”，如果它能被证明是正确的，将解决这个基本问题，并确保爱因斯坦理论在经典（即，非量子）水平。 为了阐明这一难题，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）如何聚焦和（有时）爆炸以产生物理奇点。该项目的主要目的之一是发展理论技术，用于精确分析这种传播、聚焦和曲率爆破，以期为彭罗斯猜想的最终证明提供数学工具。该项目的另一个方面涉及一个迷人的联系，这是最近才曝光的，纯粹黎曼几何学家研究的“爱因斯坦空间”和数学广义相对论学家研究的“爱因斯坦时空”之间的区别。 从动力学的观点来看，时空被看作是空间通过时间的演化，某些爱因斯坦空间为这种动力学演化提供了自然吸引子。 这不仅发生在经典维的时空中，也发生在高于或低于4维的时空中。虽然高维时空的物理意义在很大程度上仍然是推测性的，但毫无疑问，爱因斯坦理论的上述性质（在各种可能的维度上）具有真正的数学意义。该项目的最后一个方面涉及改进技术的发展，用于对两个黑洞碰撞等奇异过程中发射的引力波进行数值研究。 这种波几乎肯定会被当前或拟议中的实验搜索最终检测到，但可以想象，这些努力的最终成功可能取决于先前为所讨论的假设波形开发出准确的理论模板。 该项目正在进行的工作是研究计算波信号并分析其属性的可行性，一直到零无穷大，理论演变的自然边界。

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.