GLOBAL STUDIES OF EINSTEIN SPACETIMES

爱因斯坦广义相对论中的一个主要悬而未决的理论问题是，在对场方程的许多其他合理的解中，出现了时空奇点，在这个奇点上，曲率可以变成无穷大，或者说，在这个奇点上，时空几何中可能会出现一些更微妙的崩溃。根本问题是，这些奇点是否代表了爱因斯坦引力理论经典决定论的真正失败，从而成为广义相对论本身的一个潜在的致命缺陷，或者相反，它们实际上是否在某种微妙的方式上与决定论一致。一个由罗杰·彭罗斯提出的猜想，被称为“宇宙审查猜想”，如果它被证明是正确的，将解决这个根本问题，并确保爱因斯坦理论在经典(即非量子)水平上的逻辑一致性。要阐明这一难题，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)如何聚焦并(有时)爆炸产生物理奇点。这个项目的主要目的之一是开发理论技术来精确分析这种传播、聚焦和曲率膨胀，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最近才曝光的一个有趣的联系，即纯黎曼几何学家研究的“爱因斯坦空间”和数学广义相对主义者研究的“爱因斯坦时空”之间的联系。从动力学的观点来看，时空被视为空间通过时间的演化，某些爱因斯坦空间为这种动态演化提供了自然吸引子。这不仅发生在经典维度的时空中，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(在各种可能的维度中)具有真正的数学兴趣。该项目的最后一个方面涉及开发改进的技术，用于数值研究在两个黑洞碰撞等奇异过程中发出的引力波。这种波最终几乎肯定会被当前或计划中的实验搜索所探测到，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。这个项目正在进行的一项工作是研究计算波信号并分析其特性的可行性，直到零无穷远，这是理论发展的自然边界。

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.