EPSRC Reference: 
EP/X012417/1 
Title: 
Geometric scattering methods for the conformal Einstein field equations 
Principal Investigator: 
Valiente Kroon, Dr JA 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
Queen Mary University of London 
Scheme: 
Standard Research  NR1 
Starts: 
01 November 2022 
Ends: 
31 October 2023 
Value (£): 
80,353

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Summary on Grant Application Form 
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 largescale 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 themselvescreating 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 socalled Weyl tensor. The Weyl tensor encompasses the gravitational degrees of freedom, and is intimately tied to the conformal structure that is, the lightcone, 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 irregularor singularconformal 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 selfgravitating isolated systems.

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