Hyperbolic Geometry and Gravitational Waves

This page collects research results, educational resources, and community activities supported by NSF Award #2309084: Hyperbolic Geometry and Gravitational Waves.

Award period: May 1, 2023–April 30, 2027. Principal investigator: Anıl Zenginoğlu, University of Maryland.

Last updated: September 14, 2026.

Goal

The award investigates the role of hyperbolic geometry in relativity and related areas, with a focus on computing gravitational waves from astrophysical sources.

The common thread is the use of hyperboloidal surfaces: spacelike surfaces that approach the paths of outgoing light rays at large distances. These surfaces allow us to represent radiation all the way to null infinity, where an idealized distant observer measures gravitational waves. Hyperboloidal compactification brings this infinite domain onto a finite computational grid and avoids imposing an artificial outer boundary.

I combine mathematical analysis, numerical experiments, and physical insight to study black-hole perturbations, nonlinear Einstein equations, and scattering problems in physics and engineering.

Research highlights

• Late-time tails in merging black holes: evidence for late-time gravitational-wave tails in fully nonlinear simulations, published in Physical Review Letters in 2025.
• Bridging time across null horizons: a unified treatment of time coordinates across event horizons, null infinity, and cosmological horizons, published in General Relativity and Gravitation in 2025.
• A null infinity layer for wave scattering: a method for computing scattering and far-field patterns on compactified domains, published in SIAM Journal on Scientific Computing in 2026.

See the award publications for the full list of associated papers.

Activities

The following pages describe the research program, freely available learning materials—including our online book Hyperboloidal Compactification in NGSolve—and the workshops, seminars, and publications of the Hyperboloidal Research Network.