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”最简单的解释来自于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20世纪的大部分时间里，人们普遍认为宇宙常数是零，因为所有的观测都符合这种最简单的可能性。因此，描述紧凑天体动力学的详细数学理论，如黑洞和中子星的碰撞，以及它们产生的引力波，都是基于这一假设。这个项目第一部分的目标是系统地扩展这个理论，使其包括一个非零的宇宙常数。这一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界面的研究。它们解决了与黑洞的量子蒸发有关的基本问题，以及我们宇宙早期的物理学，当时引力很强，量子物理学必不可少。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，青年科学家将在跨学科的科学训练。目前，这些研究人员包括代表性不足的群体的成员。通过编辑纪念广义相对论发现100周年的系列丛书，PI将委托撰写旨在向年轻研究人员传达最新科学技术进展的专著。最后，同过去一样，这项赠款支助的人员将通过录象、半流行文章和公开演讲大力促进外联活动。该项目的第一部分将解决这样一个问题，即即使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，我们甚至没有一个规范不变的引力波的概念，在完整的，具有正宇宙常数的非线性广义相对论中。我们的目标是建立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特性。这一结果将为研究引力坍缩、紧二元聚并过程中的能量、动量和角动量发射、后牛顿和后闵可夫斯基近似方法等的解析和数值工作开辟新的途径。它们也可能通过新的数学猜想（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，因为环量子宇宙学现在已经足够成熟，可以根据它的前暴胀动力学来预测未来的观测，并利用当前的观测来寻找可能决定我们宇宙初始条件的基本原理。

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.