Approximate Symmetries and the Dynamics of Black Holes

该项目的主要目标是开发用于研究非摄动状态下动态黑洞的数学工具。这将通过解决以下研究问题来完成：1。目前的近似杀戮向量构造不适用于具有黑洞的爱因斯坦方程的初始条件。为此，有必要为近似的基宁方程制定一个边值问题。反过来，这需要确定边界条件：(a)编码黑洞的存在；(b)使相关边值问题椭圆化，以便应用所需的偏微分方程理论。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的几何形状之间的相互作用进行细致的分析。近似杀伤向量为解决静止黑洞的唯一性问题提供了一种诱人的新策略。2. 在GR中，近似扭扭方程和质量正数之间的关系为构造几何不等式（即与黑洞的各种物理和/或几何性质有关的不等式）提供了一种新的旋旋方法。这些不等式的例子是黑洞的质量边界，根据它的面积或角动量。几何不等式在非常普遍的假设下成立，因此，为动态黑洞的性质提供了独特的定性见解。在项目的这一部分，我们将研究基于近似对称概念的几何不等式的构造。近似对称的概念通常是在单一时空超曲面上计算的。然而，没有理由认为它们不能出现在叶子的每一片叶子上。这自然提出了每个叶子上相关不变量之间关系的问题：它是保持常数还是满足某种单调行为？这种行为对渐近边界条件（如渐近平坦与双曲）的依赖是什么？一个相关的问题涉及到在什么条件下近似的杀伤向量衰减为真正的杀伤向量——从而提供了对黑洞稳定到静止状态的方式的洞察。

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.