Microlocal Analysis in General Relativity

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

This project addresses several problems in the study of black holes in general relativity from a mathematical perspective. At the heart of general relativity lies Einstein's field equation which relates the curvature of spacetime to its energy/matter content. Solutions of this equation describe astrophysical phenomena such as black hole and gravitational waves. The primary goal of this project is a precise description of the dynamics of black holes which are out of equilibrium: as time evolves, they are expected to settle down to a stationary black hole under emission of gravitational waves. Attaining this goal requires the development of novel mathematical tools in microlocal analysis and spectral theory. In a nutshell, the main principle of microlocal analysis is to keep track simultaneously of the position and the momentum (or frequency) of a propagating wave. Spectral theory, in this context, is concerned with the characteristic frequencies and amplitudes of waves surrounding an almost stationary black hole. The problems under investigation are closely connected to recent advances in astrophysics. The gravitational waves emitted by a dynamical black hole were recently detected experimentally; their frequencies and amplitudes contain a large amount of information about the properties of the black hole, such as its mass and angular momentum. Directly motivated by the experimental detection of the merging of binary black holes, the investigator furthermore aims to initiate the mathematical study of the interaction of several black holes. This project provides research training opportunities for graduate students. The nonlinear stability problem of the Kerr family of black hole spacetimes has been a driving force behind the development of the modern theory of wave equations on curved spacetimes. The investigator will study this problem for slowly rotating black holes by combining recent advances in spectral theory and dynamical systems with novel microlocal tools. This will in particular require a significant extension of the set of tools developed for the analysis of elliptic partial differential equations on compactified/singular spaces to the hyperbolic setting. Using similar techniques, the investigator will analyze the sharp asymptotic behavior of solutions of linear wave equations, including the Maxwell and Klein-Gordon equations, on black hole spacetimes. Moreover, the investigator will consider problems related to proving quasinormal mode expansions and ringdown for black holes settling down to their equilibrium state, motivated by recent experimental observations of gravitational waves. The investigator plans to relate the decay rate of spacetime metrics towards equilibrium in the exterior of black holes to their regularity near the inner black hole horizon in the context of Penrose’s Strong Cosmic Censorship conjecture. The second part of this project concerns the construction of multi-black hole spacetimes. In particular, on cosmological spacetimes, the investigator proposes to construct vacuum spacetimes in which the black holes are eventually out of causal contact. The third part concerns inverse problems for passive and active measurements for nonlinear wave equations in settings in which the inverse problem for the corresponding linear equation has not been solved yet. The investigator plans to use the nonlinear interaction of waves deep inside of an object of interest in order to create artificial waves which carry information about the object back to its surface. 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.