Hyperbolic Geometry and Gravitational Waves

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.