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.

一般相对论是最佳的重力理论。它将重力相互作用描述为由物质引起的时空曲率的表现，这种关系在爱因斯坦磁场方程中编码。通过爱因斯坦磁场方程，一般相对论既描述了整个宇宙的大规模结构，也描述了较小的隔离系统（如恒星和黑洞）的动力学。孤立系统的概念是一个特别有用的数学理想化，因为它允许人们将最重要的宇宙学效应的影响分开，例如宇宙的扩展与现象，人们希望将其归因于所研究的特定系统 - 例如产生的辐射。爱因斯坦场方程本身创造了一种近来的实验确认的重力波现象，即相同的方程式描述了（动力学）背景，而引力波则在其上传播。这是爱因斯坦场方程在数学上特别复杂的一种方式。在总体相对论中，一个长期存在的问题是了解引力场与孤立系统的大距离之间的行为，也就是说，远离产生或分散它的来源。编码此信息的数学对象是所谓的Weyl张量。 Weyl张量包括重力自由度，并与保形结构密切相关 - 即背景时空的轻锥或因果关系。罗杰·彭罗斯爵士（Sir Roger Penrose）在1965年的著名见解是为了确定一个事实，即在某些空间上，魏尔张量的各个组成部分应以层次结构的方式衰减（一种称为脱皮的一种腐烂）。这种认识对我们对重力领域渐近行为的理解产生了深远的影响。导致重力浪潮发现的科学努力，这一发现获得了2017年诺贝尔物理价格的发现，最终可以追溯到彭罗斯（Penrose）的剥离定理。遵守剥离定理的空间具有定期的形式结构。当时，尤其是在一个世纪的最后一个世纪中，数学界已经进行了系统的努力，以了解对爱因斯坦字段方程的全球属性的全球特性，通过对偏微分方程理论工具的有方法使用来理解Einstein场方程的全球属性。这些事态发展导致人们意识到，彭罗斯的剥离图片实际上不是通用的，而且大多数孤立的系统的weyl张量会以一种更为复杂的方式腐烂。在整形图中，这意味着通用的空间具有不规则或奇异的符合条件结构。尽管研究人员在过去几十年中一直持续努力，但迄今为止，没有令人满意的数学理论，这将允许严格研究重力领域的渐近衰变，而无需做出临时假设。该项目的目的是结合有关此问题的两种方法的想法，迄今为止的相互作用有限。一方面，一种方法有利于问题的几何方面（共形程序），另一方面是基于硬数学分析（几何散射）的一组工具。可以预期，这两种方法的合并将提供一套坚实但通用的数学工具集，这些工具将允许实现Penrose的开创性视觉，尽管以修改形式，但描述了相对论自我植物隔离系统的描述。

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.