Theoretical Investigations in Classical and Quantum Gravity

最近的观测表明，宇宙中大约70%的能量密度不能归因于物质。对这种“暗能量”最简单的解释来自爱因斯坦最初在他的广义相对论基本方程中引入的宇宙常数。然而，在世纪的大部分时间里，人们普遍认为宇宙学常数为零，因为所有的观测都与这个最简单的可能性相容。因此，描述紧凑天体（如碰撞黑洞和中子星）动力学的详细数学理论以及它们产生的引力波都是基于这一假设。该项目第一部分的目标是系统地扩展该理论，以包括非零宇宙学常数。该扩展将为引力波科学提供更坚实的基础。该项目的第二部分包括强引力和量子物理学界面的研究。它们涉及与黑洞的量子蒸发和我们宇宙最早时期的物理学有关的基本问题，当时引力很强，量子物理学不可或缺。拟议的研究将对广泛的研究领域产生影响，包括广义相对论，几何分析，宇宙学，量子场论和计算科学。因此，通过这项研究，年轻科学家将在跨学科科学方面得到培训。目前，这些研究人员包括代表性不足的群体的成员。通过他的系列纪念广义相对论发现一百周年的编辑，PI将委托专着，旨在传达科学和技术的最新进展，以年轻的研究人员。最后，与以往一样，获得该基金资助的人员将通过视频、半流行文章和公开讲座等方式大力开展外联活动。该项目的第一部分将解决这样一个事实所引起的问题，即即使是宇宙常数的最小值也会对引力波理论产生深远的影响。到目前为止，我们甚至没有一个完整的、非线性的、具有正宇宙学常数的广义相对论中的引力波规范不变概念。我们的目标是构建一个严格的框架来填补这一空白，并利用它以更可靠的方式提取引力波的物理学。这些结果将为研究引力坍缩、紧双星合并、能量、动量和角动量发射等过程以及后牛顿和后闵可夫斯基近似方法等方面的分析和数值工作开辟新的途径。它们也可能通过新的数学结构（与能量的新概念的正性有关）刺激几何分析的研究。该项目的第二部分将解决两类问题：i）概念上的基本问题，即信息是否会在黑洞蒸发中丢失; ii）圈量子引力与早期宇宙观测之间的相互作用。这是可能的，因为圈量子宇宙学现在已经足够成熟，可以根据它在暴胀前的动力学来预测未来的观测，并利用当前的观测来寻求基本原理，这些基本原理可能决定了我们宇宙的初始条件。

Recent observations have shown that about 70% of the energy density in the universe cannot be attributed to matter. The simplest explanation of this `dark energy' comes from the cosmological constant that Einstein had initially introduced in his fundamental equations of general relativity. However, for most of the 20th century, it was widely believed that the cosmological constant is zero because all observations were compatible with this simplest possibility. Therefore, the detailed mathematical theories describing the dynamics of compact astronomical objects such as colliding black holes and neutron stars, and gravitational waves they source, was based on this assumption. The goal of the first part of this project is to systematically extend that theory to include a non-zero cosmological constant. The extension will provide a firmer foundation for gravitational wave science. The second part of the project contains studies at the interface of strong gravity and quantum physics. They address fundamental issues related to the quantum evaporation of black holes and the physics of the earliest epoch of our universe when gravity is strong and quantum physics indispensable. The proposed research will have an impact on a broad range of research areas, including general relativity, geometric analysis, cosmology, quantum field theory, and computational science. As a result, through this research, young scientists will be trained in inter-disciplinary science. Currently, these researchers include members of under-represented groups. Through his editorship of series commemorating the centennial of the discovery of general relativity, the PI will commission monographs aimed at communicating the latest advances in science and technology to young researchers. Finally, as in the past, personnel supported by this grant will contribute vigorously to outreach activities through videos, semi-popular articles and public lectures.The first part of the project will address issues arising from the fact that even a tiniest value if the cosmological constant can cast a long shadow on the theory of gravitational waves. As of now, we do not even have a gauge invariant notion of gravitational waves in full, non-linear general relativity with a positive cosmological constant. The goal is to construct a rigorous framework to fill this gap and use it to extract the physics of gravitational waves in a more reliable fashion. The results will open new avenues for analytic and numerical work in the study of the gravitational collapse, compact binary coalescence, energy, momentum and angular momentum emission in such processes, post-Newtonian and post-Minkowskian approximation methods, etc. They are also likely to stimulate research in geometric analysis through new mathematical conjectures (related to positivity of a new notion of energy). The second part of the project will address two types of issues: i) the conceptually fundamental question of whether information is lost in evaporation of black holes; and, ii) the emerging interplay between loop quantum gravity and observations related to the very early universe. This is possible because loop quantum cosmology is now sufficiently mature to make predictions for future observations based on its pre-inflationary dynamics, and to use current observations to seek fundamental principles that presumably dictate the initial conditions for our universe.