A Look at Some Mathematics Research Problems in General Relativity

Michael Holst
Department of Mathematics, University of California, San Diego

In 1915, Einstein (and Hilbert) formulated the equations of general relativity, the Einstein Equations, as a geometric description of what we experience as gravity, displacing Newton's theory of gravity. Already by 1916, it was realized that the equations predict that gravitationally interacting bodies will emit radiation at the speed of light that will carry energy away from the interaction. Unlike other types of radiation, this radiation is actually a "ripple" in space (and time). Massive interactions that occur at relatively close ranges, such as in-spiraling and colliding black holes, emit enough radiation that they could potentially be detected by a device that we could build using late 20th/early 21st century technology. Such a device was built,called LIGO (Laser Interferometer Gravitational-Wave Observatory). At more than $600M, it is the most ambitious and expensive NSF project ever undertaken.On 11 February 2016, it was announed that LIGO detected a clear, unambiguous,loud and violent inspiral, collision, merger, and ringdown of a binary black hole pair, each of which had a solar mass in the range 10-50, with roughly the equivalent of three solar masses in energy released as gravitational radiation.This radiation traveled outward from the collision at the speed of light, reaching the LIGO detectors on earth roughly 1.3 billion years later. Three additional detections were made between February 2016 and September 2017,the most recent of which was confirmed by the new VIRGO detector in Europe.Last week, the Nobel Prize in Physics was awarded to three of the key researchers that made LIGO possible.
How do LIGO/VIRGO scientists know what they are detecting? The answer is that the signals detected by the devices were shown, after extensive data analysis and computer simulations of the Einstein evolution and constraint equations, to be a very close match to simulations of wave emission from very particular types of binary collisions. In this lecture, we will examine some fundamental mathematics research questions involving the Einstein constraint equations. We begin with an overview of the most useful mathematical formulation of the constraint equations, and then summarize the known existence, uniqueness, and multiplicity results through 2008. We then present a number of new existence and multiplicity results developed since 2008 that substantially change the solution theory for the constraint equations. We then shift gears and consider Petrov-Galerkin type approximation methods for developing "provably good" numerical methods for solving this type of system. We examine how one proves rigorous error estimates for particular classes of numerical methods, including both classical finite element methods and newer methods from the finite element exterior calculus.