Analysis of black hole stability.

The black hole stability conjecture is a challenging problem in the analysis of nonlinear partial differential equations and is one of the major open problems in mathematical relativity. Because of work by me and others, there has been great progress on this problem in the last decade. The proposed grant would partially fund a post-doctoral research assistant (PDRA), who will be essential to maintaining the recent pace of progress.

General relativity is a geometric theory of gravity, in which the universe is described by a four-dimensional set of space-time points and by an indefinite inner-product, which must satisfy the Einstein equations.

Physicists believe that black holes will play a crucial role in our understanding of theoretical physics, will absorb all matter in the late stages of the universe, and are the enormously massive objects known to exist at the centre of most galaxies. Kerr's family of explicit solutions to the Einstein equation are parametrised by mass and angular momentum, and they describe black holes when the angular momentum is small relative to the mass. For zero mass, the solution reduces to the Minkowski solution, and for angular momentum zero to the Schwarzschild solution.

For solutions having the appropriate asymptotic behaviour, the Kerr family is the unique family of stationary, black hole solutions of the Einstein equations. Although physicists believe that there can be no reasonable doubt that all black holes will asymptotically approach a Kerr solution, as with the Navier-Stokes equation, there is an enormous gap between what is expected on physical grounds and what can be proved. Hence, there is great interest in proving the asymptotic
stability of the Kerr solutions:

Conjecture K: If a set of initial data is very close to one that generates a Kerr solution, then the corresponding solution will eventually approach a Kerr solution.

It is unlikely that that this conjecture can be proved without estimates on the rate of decay to the Kerr solution. For a nonlinear wave equation, which can serve as a model for the Einstein equation, it is known that for sufficiently small initial data and a sufficiently weak nonlinearity, the smallness of the initial data guarantees that the influence of the nonlinearity is small up to intermediate times, allowing the solution to decay at the same rate as a solution to the linear wave equation. Then, from intermediate to late times, since the nonlinear term is smaller than the linear terms when they are small (but larger than the linear terms, when the linear terms are large -this being the nature of the relevant nonlinear terms), the influence of the nonlinearity remains small and diminishing, so that the solution to the nonlinear equation behaves like solutions to the linear equation.

Analysis of the Einstein equation is challenging because it is nonlinear and geometric and because it has both solutions for which the curvature diverges in finite time and globally smooth solutions. A thorough investigation of divergent solutions has been possible only when solutions have a high degree of symmetry. One of the landmark results in the study of mathematical relativity was the proof that the flat space (known as Minkowski space) is stable. This built on decay estimates for the wave, Maxwell, and linearised Einstein equations.

In the Schwarzschild case, these equations have also been studied. In the general Kerr case, decay estimates for the wave equation have been proved, and I anticipate that my collaborators and I will have completed our analysis of the Maxwell equation by the start of this proposed period of the grant. The purpose of this grant is to continue this program, and to employ a post-doctoral researcher to investigate the linearised Einstein equation in the Kerr context. This should provide important progress that will help the mathematical relativity community resolve the Kerr stability conjecture.

The primary impact of this project will be in developing and maintaining expertise in Britain in the field of nonlinear partial differential equations (PDEs). Nonlinear partial differential equations occur very frequently in engineering, and the physical, life, and social sciences and can be used to describe an enormously diverse range of phenomena. Funding for this project will help train a post-doctoral research assistant (PDRA), who will become more of an expert in the subject and whose presence will strengthen the recently established centre for analysis and nonlinear partial differential equations (CANPDE).

Because of the diversity and complexity of the wide range of phenomena described by PDEs, it is not commonly possible to find explicit solutions, and, instead, there is a need to develop techniques which can be used to get qualitative and quantitative estimates on the solutions. I believe an effective way of tackling these problems is by establishing groups of people who are experts in PDEs and are willing to work on problems arising from outside pure mathematics. Expertise in this area is and will remain crucial to maintaining and developing a technologically advanced society.

In the last year, I have worked with a biologist on a project on an epidemiological model for estimating disease infection in society. In 2008, the H1N1 flu virus and fear about the problems it could cause cost the UK alone more than a billion pounds. While there is a real risk of a severe and gravely dangerous pandemic, there is a wide-spread belief that in this case, the scientific community over estimated the risks. This diverted medical resources from other necessary tasks, cost vast sums of money, and ran the risk of making the public complacent about future risks. These concerns motivated my collaborator, N. Fefferman, from the department of Ecology, Evolution, and Natural Resources at Rutgers University. Because of my expertise in differential equations, N. Fefferman asked me to help with this problem.

We investigated how a change in the rate at which healthy people are tested for a disease after an outbreak, coupled with a rate of false positives, could affect the estimated virulence of the disease and number of people who will be affected in the future. I was interested in this problem because it allowed me to put into practice my claim that developing mathematical techniques allows for advances in other fields and has a broader impact, because I hoped to disseminate these techniques more broadly, and because applications have historically been a source of interesting mathematical problems. In this case, the techniques required were quite basic mathematically but not commonly used in biology, demonstrating the need for maintaining a broad expertise in mathematics.

A central part of the proposed grant will be training a post-doctoral researcher. I will also be open to other collaborations outside mathematics and will encourage the post-doctoral researcher to do the same.