Geometric scattering methods for the conformal Einstein field equations

共形爱因斯坦场方程的几何散射方法

• 批准号: EP/X012417/1
• 负责人: Juan Antonio Valiente Kroon
• 金额: $ 10.24万
• 依托单位: Queen Mary University of London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX012417%2F1
• 关键词: Geometric; scattering; methods; conformal; Einstein

General Relativity is the best available theory of gravity. It describes gravitational interaction as a manifestation of the curvature of spacetime caused by matter, a relationship encoded in the Einstein field equations. Through the Einstein field equations, General Relativity describes both the large-scale structure of the whole Universe as well as the dynamics of smaller isolated systems like stars and black holes. The notion of an isolated system is a particularly useful mathematical idealisation, as it allows one to separate the effects of the most important cosmological effects like the expansion of the Universe from phenomena which one would like to ascribe to the particular system under study -like the radiation produced. The recently experimentally confirmed phenomenon of gravitational waves is described by the Einstein field equations themselves-creating a situation whereby the same system of equations describes the (dynamical) background, and the gravitational waves propagating on it. This is one way in which the Einstein field equations are particularly complex mathematically.A longstanding question in General Relativity has been to understand how the gravitational field behaves at large distances from isolated systems -that is, far away from the sources that produce or scatter it. The mathematical object that encodes this information is the so-called Weyl tensor. The Weyl tensor encompasses the gravitational degrees of freedom, and is intimately tied to the conformal structure -that is, the light-cone, or causality, structure- of the background spacetime. A famous insight by Sir Roger Penrose in 1965 was to identify the fact that, for certain spacetimes, the various components of the Weyl tensor should decay in a hierarchical manner (a type of decay known as peeling). This realisation has had a deep influence on our understanding of the asymptotic behaviour of the gravitational field. The scientific endeavour that led to the detection of gravitational waves, a discovery that was awarded the 2017 Nobel price in Physics, can ultimately be traced back to Penrose's peeling theorem. Spacetimes that obey the peeling theorem have regular conformal structures.Since then, especially in the last quarter of a century, the mathematical community has engaged in a systematic effort to understand the global properties of generic solutions to the Einstein field equations by making a methodical use of the tools of the theory of partial differential equations. These developments have led to realise that Penrose's peeling picture is in fact not generic, and that most isolated systems have Weyl tensors that decay in a much more complicated way. In the conformal picture, this means that generic spacetimes possess irregular-or singular-conformal structures. Despite the sustained effort of researchers in the last decades, there is to date, no satisfactory mathematical theory which would allow to rigorously study the asymptotic decay of the gravitational field in fine detail and without having to make ad hoc assumptions. The aim of this project is to combine ideas of two approaches to this problem which, hitherto, have had limited interaction. On the one hand one has an approach which favours the geometric aspects of the problem (the conformal programme) and on the other hand a set of tools based on hard mathematical analysis (geometric scattering). It is expected that the merge of these two approaches will provide a solid, yet versatile, set of mathematical tools which will allow to fulfil Penrose's seminal vision, albeit in a modified form, of the description of relativistic self-gravitating isolated systems.

广义相对论是目前最好的引力理论。它将引力相互作用描述为物质引起的时空弯曲的表现，这是爱因斯坦场方程中编码的关系。通过爱因斯坦场方程，广义相对论描述了整个宇宙的大尺度结构以及恒星和黑洞等较小孤立系统的动力学。孤立系统的概念是一种特别有用的数学理想化，因为它允许人们将最重要的宇宙效应（例如宇宙的膨胀）与人们想要归因于所研究的特定系统的现象（例如产生的辐射）分开。最近通过实验证实的引力波现象是由爱因斯坦场方程本身描述的，这创造了一种情况，即用同一方程组描述（动态）背景以及在其上传播的引力波。这是爱因斯坦场方程在数学上特别复杂的一种方式。广义相对论中一个长期存在的问题是理解引力场在远离孤立系统（即远离产生或散射它的源头）时如何表现。编码该信息的数学对象是所谓的韦尔张量。韦尔张量包含引力自由度，并且与背景时空的共形结构（即光锥或因果结构）密切相关。罗杰·彭罗斯爵士 (Sir Roger Penrose) 于 1965 年提出的一个著名见解是，他发现了这样一个事实：对于某些时空，韦尔张量的各个组成部分应该以分层方式衰变（一种称为剥离的衰变类型）。这种认识对我们对引力场渐近行为的理解产生了深远的影响。导致引力波探测的科学努力最终可以追溯到彭罗斯的剥离定理，这一发现荣获 2017 年诺贝尔物理学奖。服从剥离定理的时空具有规则的共角结构。从那时起，特别是在过去的四分之一个世纪里，数学界一直致力于系统地努力，通过有条不紊地使用偏微分方程理论的工具来理解爱因斯坦场方程泛型解的全局性质。这些发展使我们认识到彭罗斯的剥离图像实际上并不通用，并且大多数孤立系统的韦尔张量以更复杂的方式衰减。在共形图中，这意味着通用时空具有不规则或奇异共形结构。尽管研究人员在过去几十年中不断努力，但迄今为止，还没有令人满意的数学理论可以允许详细地严格研究引力场的渐近衰变，而不必做出临时假设。该项目的目的是将解决该问题的两种方法的想法结合起来，迄今为止，这两种方法的相互作用有限。一方面有一种有利于问题的几何方面的方法（共形程序），另一方面有一套基于硬数学分析（几何散射）的工具。预计这两种方法的合并将提供一套可靠且通用的数学工具，这些工具将允许实现彭罗斯描述相对论自引力孤立系统的开创性愿景（尽管形式有所修改）。

项目成果

At the interface of asymptotics, conformal methods and analysis in general relativity.

• DOI: 10.1098/rsta.2023.0048
• 发表时间: 2024-03-04
• 期刊: PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
• 影响因子: 5
• 作者: Taujanskas, G.;Valiente Kroon, J. A.
• 通讯作者: Valiente Kroon, J. A.