Approximate Symmetries and the Dynamics of Black Holes

The Primary objective of the project is the development of mathematical tools for the study of dynamical black holes in the non- perturbative regime. This will be done by addressing the following research questions:
1. The current approximate Killing vector construction does not apply to initial conditions for the Einstein equations with black holes. For this, it is necessary to formulate a boundary value problem for the approximate Killing equation. This, in turn, requires identifying boundary conditions which: (a) encode the existence of a black hole; (b) make the associated boundary value problem elliptic, so that the required PDE theory can be applied. Guaranteeing these two conditions requires a delicate analysis of the interplay between the structure of the approximate Killing equation and the geometry of the black hole and of its horizon. Approximate Killing vectors offer a tantalising novel strategy to the question of the uniqueness of stationary black holes.
2. The relation between the approximate twistor equation and the positivity of the mass in GR provides a novel spinorial approach to the construction of geometric inequalities -that is, inequalities relating various physical and/or geometric properties of a black hole. Examples of these inequalities are bounds of the mass of a black hole in terms of its area or its angular momentum. Geometric inequalities hold under very general assumptions and, thus, provide unique qualitative insights into the properties of dynamic black holes. In this part of the project, we will investigate the construction of geometric inequalities based on the notion approximate symmetries.
3. The notions of approximate symmetries are usually computed on a single hypersurface of spacetime. There is, however, no reason why they cannot be considered on each of the leaves of a foliation. This naturally raises the question of the relation between the associated invariants on each of the leaves: does it remain constant or does it satisfy some type of monotonic behaviour? What is the dependence of this behaviour on the asymptotic boundary conditions (e.g. asymptotically flat versus hyperboloidal)? A related question concerns the conditions under which an approximate Killing vector decays to a true Killing vector -thus providing an insight into the way a black hole settles to a stationary state.