Principal Investigator: Gregory J. Galloway
Recent developments in gravitational theory, both theoretical and empirical, have lead to an increased interest in the study of asymptotically anti-de Sitter and asymptotically de Sitter spacetimes. In this project, techniques of Lorentzian and Riemannian geometry will be used to attack certain problems pertaining to such spacetimes, as well as some problems concerning new developments in the theory of black holes. One of the specific aims of this project is to continue development of an approach to establishing positivity of mass for asymptotically hyperbolic Riemannian manifolds (with spherical conformal infinity) which does not require spin assumption. This approach, which makes use of the brane action introduced by Witten and Yau in their work on the AdS/CFT correspondence, may also be useful in settling some positive mass conjectures for asymptotically anti-de Sitter spacetimes with toroidal topology at infinity. The PI also plans to continue investigations into uniqueness and rigidity phenomena, and the development of singularities in asymptotically de Sitter spacetimes. The classical definition of a black hole, though mathematically elegant, is problematic in practice (especially for numerical studies) in that it requires knowing the full future evolution of spacetime all the way to infinity. As an alternative, Ashtekar and Krishnan have introduced a new, fully nonlinear and dynamical, approach to the study of black holes, based on the quasi-local notion of a dynamical horizon. Another aim of this project is to investigate some open problems concerning the occurence, i.e., existence, uniqueness and position of dynamical horizons.
Modern theories of gravity are geometrical in nature. The gravitational field and other fields, black holes and related objects, may be described and analyzed using geometric methods. In more general terms, this project is concerned with the study of certain features of gravity of current scientific interest from this geometric point of view, utilizing the tools of Riemannian geometry, a mathematical theory of space, and Lorentzian geometry, a mathematical theory of spacetime. These theories provide a method for studying the relationship among three fundamental aspects of the spacetime universe: curvature (i.e., the bending of space or spacetime), topology (i.e., the global shape and complexity of space or spacetime) and causal structure (i.e., the large scale behavior of light rays and light cones).
