Mathematical Analysis of Single- and Many-black-hole Spacetimes

Einstein’s theory of general relativity is a highly successful and precise description of space and time. One of its most striking and central predictions is the existence of black holes: regions of spacetime where gravity is so strong that nothing, not even light, can escape from them. Another prediction is the existence of gravitational waves, ripples in the geometry of spacetime that propagate at the speed of light. The 2017 Nobel Prize in Physics was awarded for the successful effort to measure the gravitational waves emitted as two black holes as they spiral together and subsequently merge. The merger results in the formation of a single black hole, which rapidly settles down to its equilibrium state through a process called ringdown. The equations at the heart of Einstein’s theory describe these and other phenomena through a complicated system of nonlinear partial differential equations. They can be solved numerically, that is, with the help of computer simulations, but many outstanding foundational problems remain which call for theoretical work and rigorous mathematical analysis. The investigator studies two of these problems related to the aforementioned merger/ringdown scenario. The first problem concerns the construction and detailed description of solutions of Einstein’s field equations that describe large mass ratio mergers or the scattering of two black holes. The second problem asks whether small changes in the initial state of a black hole will compound or decay in time. A rigorous proof of decay will also provide a strong theoretical justification for the ubiquitous use of these black hole models in astrophysics. An important component of this project is the training of the next generation of researchers. Under the investigator’s guidance, they will contribute to this effort through the application and development of cutting-edge mathematical techniques for the study of wave propagation. The project studies the dynamics of spacetimes containing one or more black holes in the context of Einstein’s theory of general relativity. The overarching goal is a detailed description of the late-time dynamics of single or multiple black holes, possibly undergoing merger processes. The investigator plans to address three interdependent objectives: the proof of nonlinear stability of Kerr black holes in the full subextremal range; the stability properties of several classes of many-black-hole spacetimes; and the construction of black hole merger and black hole scattering spacetimes, including a precise description of their global geometry. Nonlinear stability problems have been a driving force behind the development of modern geometric approaches to the study of wave equations. While recent progress suggests that a full resolution of the Kerr stability problem is within reach, the geometric and analytic complexity of subextremal Kerr and its perturbations still calls for additional analytic insights. The second and third objectives provide the framework for a systematic and multi-faceted approach to the construction and analysis of many-black-hole spacetimes. The investigator expects many-black-hole spacetimes to become a topic of central importance in the mathematical theory of general relativity due to the intriguing and delicate dynamical properties they possess, and also due to their clear astrophysical significance. The resolution of the problems considered in this project require the development and sharpening of powerful and flexible tools in the theory of hyperbolic partial differential equations, spectral and scattering theory, microlocal analysis, and geometric singular analysis. The investigator expects these to be of broader applicability and interest in the field of hyperbolic (and related) partial differential equations. 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. NSF Award ID: 2554160 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Peter Hintz | Institution: Pennsylvania State Univ University Park, UNIVERSITY PARK, PA | Award Amount: $300,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554160 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554160.html