Hyperbolic Geometry and Gravitational Waves

该奖项旨在提高利用双曲几何计算引力波和黑洞扰动的能力。双曲几何在数学、物理、生物和机器学习等多个科学领域发挥着至关重要的作用。通过利用渐近双曲时间曲面（称为双曲曲面）的优点，我们可以有效地表示无限域中的出射波，使我们能够访问远离其源的引力波通量，并避免有限计算网格的外部边界处理。该奖项将有助于更广泛地理解波传播与双曲几何之间的相互作用，并提供应用程序，以改进科学和工程中基本问题的解决方案。此外，它将支持教育和培训一个高度跨学科的学生，并通过外展工作提高公众的科学素养，包括一个关于波传播的暑期学校。双曲紧化技术利用边界固定和Penrose紧化将无限域映射到有限区域，从而将全局问题转化为局部问题。该技术对时空摄动和引力波的数值和解析处理具有重要意义。该奖项活动将分为五个子项目，以不同的重点和风险概况来研究双曲面的应用。前三个子项目侧重于(i)扩大该方法在黑洞摄动理论中的应用范围，（ii）提供详细的数值分析，以及（iii）实现无界域中散射问题的双曲紧化。第四个子项目将尝试弯曲时间面，以提高非线性爱因斯坦方程现有代码的计算效率。第五个子项目将探索双曲面在黑洞时空量子场中的应用。该项目的潜在贡献包括改进时空摄动和引力波的数值和解析处理，为自由参数的选择提供严格的指导方针，并为未来的研究开辟新的方向。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award aims to improve the ability to compute gravitational waves and black hole perturbations using hyperbolic geometry. Hyperbolic geometry plays a crucial role in several scientific fields, including mathematics, physics, biology, and machine learning. By exploiting the benefits of asymptotically hyperbolic time surfaces, called hyperboloidal surfaces, we can efficiently represent outgoing waves in an infinite domain, allowing us to access the gravitational wave flux far away from its sources and avoid the outer boundary treatment of finite computational grids. This award will contribute to the broader understanding of the interactions between wave propagation and hyperbolic geometry and provide applications to improve the solution of fundamental problems in science and engineering. Additionally, it will support the education and training of a highly interdisciplinary student and promote public scientific literacy through outreach efforts, including a summer school on wave propagation.The hyperboloidal compactification technique maps infinite domains to finite regions using scri-fixing and Penrose compactification, thereby translating global problems into local ones. This technique significantly benefits the numerical and analytical treatment of spacetime perturbations and gravitational waves. The award activity will be divided into five subprojects to investigate the applications of hyperboloidal surfaces with different focuses and risk profiles. The first three subprojects focus on (i) broadening the range of applications of the method in black hole perturbation theory, (ii) providing detailed numerical analysis, and (iii) implementing hyperboloidal compactification for scattering problems in unbounded domains. The fourth subproject will experiment with bending up time surfaces to improve the computational efficiency of existing codes for nonlinear Einstein equations. The fifth subproject will explore the application of hyperboloidal surfaces to quantum fields in black hole spacetimes. The potential contributions of the project include improving the numerical and analytical treatment of spacetime perturbations and gravitational waves, providing rigorous guidelines for the choices of free parameters, and opening up new directions for future research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.