Approximate Symmetries and the Dynamics of Black Holes

该项目的主要目标是开发数学工具，用于研究非微扰状态下的动态黑洞。这将通过解决以下研究问题来完成：1。目前的近似Killing矢量结构不适用于含黑洞的爱因斯坦方程的初始条件。为此，有必要为近似Killing方程建立一个边值问题。这反过来又需要确定边界条件：（a）编码黑洞的存在;（B）使相关的边值问题椭圆，以便可以应用所需的PDE理论。要保证这两个条件，就需要对近似基灵方程的结构与黑洞及其视界几何之间的相互作用进行细致的分析。近似的Killing矢量为解决静止黑洞的唯一性问题提供了一个诱人的新策略。2.近似扭量方程和GR中质量的正性之间的关系提供了一种新的旋量方法来构造几何不等式-即与黑洞的各种物理和/或几何性质有关的不等式。这些不等式的例子是黑洞的质量在其面积或角动量方面的界限。几何不等式在非常一般的假设下成立，因此，为动态黑洞的性质提供了独特的定性见解。在这个项目的这一部分中，我们将研究基于近似对称概念的几何不等式的构造。近似对称性的概念通常是在时空的单个超曲面上计算的。然而，没有理由不能在一个叶状体的每一片叶子上考虑它们。这自然提出了每个叶子上相关不变量之间的关系问题：它是保持不变还是满足某种类型的单调行为？这种行为对渐近边界条件的依赖性是什么（例如，渐近平坦与双曲面）？一个相关的问题涉及近似的基灵矢量衰减到真正的基灵矢量的条件-从而提供了对黑洞如何稳定到稳态的见解。

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.