Problems in numerical relativity

爱因斯坦的相对论自提出以来的一百多年里，深刻地影响了我们对引力在宇宙中作用的理解。该理论做出了绝对惊人的预测，包括引力波和黑洞的存在，以及宇宙的膨胀。虽然它已经通过了它所受到的每一个实验和观测挑战，但关于该理论对物理学的影响，仍然存在许多未解的问题，特别是当引力非常强时。在理论层面上，我的主要研究领域-数值相对论-使用强大的计算机模拟来回答其中的一些问题。拟议的研究将解决强场引力中的各种问题。其中包括：（1）所谓的临界引力坍缩的性质，黑洞形成的过程被认为表现出相变的许多特征（2）一种特殊类型的宇宙的稳定性，称为反德西特时空，这是弦理论家特别感兴趣的（3）一个新发现的具有强引力场的恒星状物体模型的性质，以及（4）使用机器学习和其他人工智能技术来有效求解我们用来模拟引力相互作用的方程的可能性。所有这些研究都将由计算驱动。就像现代科学的许多其他领域一样，如气象学、气候学和流体动力学，爱因斯坦引力场方程的复杂性意味着，只有借助强大的计算机能力和先进的算法，我们才能在准确推导这些方程对许多最感兴趣的情况的描述方面取得进展。因此，拟议研究活动的大部分日常工作将涉及设计和执行计算机代码，以模拟上述物理情景。研究结果将有助于我们对引力相互作用的基本理解。这项工作对那些在引力、弦理论、天体物理学、宇宙学和应用数学领域工作的人来说将是重要的。在本科和研究生水平的学生，以及博士后研究员的培训将是研究计划的一个重要方面。学员将成为专家，制定和解决复杂的问题，使用先进的信息技术，使他们成为加拿大的日益知识型经济的关键贡献者。

In the hundred plus years since it was formulated, Einstein's theory of relativity has profoundly impacted our understanding of the role of gravity in the universe. The theory makes absolutely spectacular predictions including the existence of gravitational waves and black holes, and the expansion of the universe. Although it has passed every experimental and observational challenge to which it has been subjected, many unanswered questions about the theory's implications for physics remain, especially when gravity is very strong. At a theoretical level, my primary research field---numerical relativity---uses powerful computer simulations to provide answers to some of these questions. The proposed research will address a variety of issues in strong field gravity. These include: (1) the nature of so-called critical gravitational collapse, where the process of black hole formation is seen to exhibit many features of a phase transition (2) the stability of a particular type of universe, known as anti-de Sitter spacetime, which is of particular interest to string theorists (3) the properties of a newly discovered model for a star-like object with strong gravitational fields and (4) the possibility of using machine learning and other artificial intelligence techniques to efficiently solve the equations we use to model the gravitational interaction. All of this research will be computationally driven. As is the case in many other fields of modern science, such as meteorology, climatology and fluid dynamics, the complexity of Einstein's equations for the gravitational field means that it is only with significant computer power and advanced algorithms that we can make progress in accurately deducing what those equations have to say about many of the situations of most interest. Thus, much of the day-to-day work of the proposed research activities will involve the design and implementation of computer codes to simulate the physical scenarios sketched above. Results of this research will contribute to our fundamental understanding of the gravitational interaction. The work will be important to those working in the fields of gravity, string theory, astrophysics, cosmology and applied mathematics. Training of students at undergraduate and graduate levels, as well as postdoctoral fellows will be a crucial aspect of the research program. Trainees will become expert in formulating and solving complex problems using advanced information technology, allowing them to become key contributors to Canada's increasingly knowledge-based economy.