The project concerns the study of the local and global behavior of solutions to the Einstein equations of general relativity, a non-linear system of hyperbolic partial differential equations at the intersection of analysis, geometry, theoretical physics, cosmology and astrophysics. An outstanding problem is to establish the non-linear stability of the Kerr family of black hole solutions, and it is precisely here where the project intends to make a contribution. The past ten years have seen tremendous progress in understanding the boundedness and decay properties of (linear and non-linear) wave equations on black hole backgrounds, a problem which is generally thought of as a primer to the full non-linear stability problem, which would take the full tensorial structure and quasi-linearity of the Einstein equations into account. Previous work of the principal investigator has generalized some of these techniques available for the wave equation to the setting of the Bianchi equations establishing in particular a conditional result on the propagation of regularity and decay in the setting of metrics converging to Schwarzschild. Future research is directed towards generalizing the work to the Kerr setting, as well as to isolate and study an appropriate linear problem for gravitational perturbations in this context. This work will be carried out in collaboration with M. Dafermos and I. Rodnianski. A second research direction is to continue the work on stability and instability of asymptotically anti-de Sitter spacetimes initiated by the principal investigator and J. Smulevici in 2011. In particular, in collaboration with J. Luk and J. Smulevici, the principal investigator plans to study the conjectured non-linear instability of anti-de Sitter space within the model of the spherically symmetric scalar field.

General relativity is the currently accepted theory of spacetime dynamics. While the field equations were already written down by Einstein in 1915, the full mathematical content of the theory is still far from being understood. One of the most exciting predictions of the theory is the existence of so-called black hole solutions of the field equations. Some of these can be written down explicitly and one two-parameter family of solutions (the Kerr family) in particular is conjectured to play a fundamental role as the final state of any vacuum gravitational collapse. While there is now sufficient astrophysical evidence that black holes indeed exist in the observable universe,the simple mathematical question of stability of these solutions is unknown. The difficulty of this problem roots in the strong non-linearities in the Einstein equations and their tensorial structure. In this way studying the Einstein equations in the context of black hole stability has broader impact for the entire field of non-linear partial differential equations (a mathematical technique to understand non-linearities may generalize to other equations) and geometry, as well as astrophysical observations.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Bruce P. Palka
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Princeton University
United States
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