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

This talk is about two important trends of scattering theory in general relativity: time dependent analytic scattering and conformal scattering. The former was initiated by Jonathan Dimock and Bernard Kay in the early 1980's and is based on spectral and functional analysis. The latter was proposed by Roger Penrose in 1965 and then constructed for the first time by Gerard Friedlander by putting together Penrose's conformal method and another analytic approach to scattering: the Lax-Phillips theory due to Peter Lax and Ralph Phillips. Professor Nicolas shall review the history of the two approaches and explain their general principles. Professor Nicolas shall also explore an important question: 'can the tools of one approach be used to obtain a complete construction in the other?'.

Professor Jean-Philippe Nicolas, University of Brest, France

Professor Jean-Philippe Nicolas, University of Brest, France

Professor Jean-Philippe Nicolas received their PhD under the supervision of Alain Bachelot in Bordeaux. This was followed by a postdoc under the supervision of Roger Penrose at the University of Oxford. They were appointed as a Lecturer in 1995 in Bordeaux and then as a Professor in Brest in 2007.

Professor Piotr Chruściel will describe a construction, with Erwann Delay and Raphaela Wutte, of families of asymptotically locally hyperbolic time symmetric vacuum general relativistic initial data sets with negative cosmological constant, with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular they obtain new classes of solutions with negative mass.

Professor Piotr Chruściel, University of Vienna, Austria

Professor Piotr Chruściel, University of Vienna, Austria

Professor Piotr Chruściel is Professor of Gravitational Physics at the University of Vienna, Austria. He received his PhD from the Institute for Theoretical Physics of the Polish Academy of Sciences under the supervision of Professor J Kijowski, and has held positions at the Polish Academy of Sciences, Université de Tours and the University of Oxford. He is a Fellow of the Institute of Physics and Clare Hall College, University of Cambridge. He won the Prix Plumey de l'Académie des Sciences de Paris in 2003. He serves on the Editorial Board of the journal Communications in Mathematical Physics.

Photo credit: Barbara Mair

Compactifications of spacetimes in General Relativity have mainly served two purposes until recently. The first purpose is exemplified by Penrose diagrams: they enable one to easily read off the causal structure of the spacetime, and in this manner also guide the analysis of wave equations using energy methods. The second purpose, which plays an important role since Friedrich's work in the 1980s, is to reduce the global-in-time analysis of linear and nonlinear wave equations to the local-in-time analysis of conformally related hyperbolic equations.

In this talk, Professor Hintz describes a third purpose: on a carefully defined compactification of an asymptotically flat spacetime, one can perform precise asymptotic analysis even when the underlying metric is not smooth on the compactification, but merely conormal. This level of generality is crucial for applications to nonlinear stability problems. The process of compactification transforms a partial differential equation (PDE) on a noncompact space into a PDE on a compact space that is typically singular (ie unless the PDE is very special at infinity). The singular behaviour is a structured degeneration at the boundary hypersurfaces at infinity; geometric singular analysis is a general-purpose perspective for the study of such singular PDE. He will explain novel estimates for wave equations on Minkowski space and more general asymptotically flat spacetimes which arise naturally from this perspective, leading in particular to a new proof of the exterior stability of Minkowski space. Parts of this talk are based on joint work with András Vasy.

Professor Peter Hintz, ETH Zürich, Switzerland

Professor Peter Hintz, ETH Zürich, Switzerland

Peter Hintz studies partial differential equations in general relativity. He received his PhD in 2015 from Stanford University under the supervision of Andras Vasy. Following his postdoctoral appointment at the University of California, Berkeley, he was appointed Assistant Professor at the Massachusetts Institute of Technology (MIT) in 2018. Since 2021, he has been an Associate Professor of Mathematics and Physics at ETH Zürich. For his research, Hintz has received a number of awards, including Miller, Clay, and Sloan Research Fellowships. Furthermore, he was an invited speaker at the ICM 2022, and a plenary speaker at the ICMP 2021.

Friedrich's framework of spatial infinity introduces a regular initial value problem at spatial infinity for the conformal Einstein field equations. In this formulation, spatial infinity is represented by a cylinder that connects the endpoints of past and future null infinity, these endpoints are known as the critical sets of null infinity. This representation of spatial infinity can be used to express physical quantities at the critical sets in terms of initial data given on a Cauchy hypersurface. This is essentially the topic of this talk, to show that one can express the asymptotic charges at the critical sets in terms of initial data. The discussion in this talk is heavily inspired by the calculation of the Newman-Penrose constants at the critical sets introduced by Friedrich and Kánnár. Mariem MA Mohamed will start by discussing the calculation of the asymptotic charges for the spin-2 field on Minkowski spacetime, where she shows that the asymptotic charges for the spin-2 field associated with arbitrary functions on 2-spheres are well-defined if and only if the freely specifiable data satisfy certain regularity conditions. She will then discuss the tools and techniques used to extend this to the non-linear gravitational field on Vacuum spacetimes.  

Mariem Magdy Ali Mohamed, Queen Mary University of London, UK

Mariem Magdy Ali Mohamed, Queen Mary University of London, UK

Mariem MA Mohamed is a PhD student at Queen Mary University of London working under the supervision of Juan A Valiente Kroon and Mahdi Godazgar. After completing a BSc in Astronomy and Physics at Cairo University in 2014, she moved to the UK to pursue postgraduate studies. She completed an MSc in Physics at Imperial College London in 2019. Mariem MA Mohamed has held a faculty position at the Astronomy department of the Faculty of Science at Cairo University since 2016. During her PhD, she published two papers. The first is focused on the investigation of the relation between two different conformal formulations of spatial infinity; namely, Friedrich’s cylinder at spatial infinity and Ashtekar’s hyperboloid at spatial infinity. The second publication is concerned with the evaluation of the asymptotic charges at the so-called critical sets of null infinity for spin-1 and spin-2 fields on Minkowski spacetime.  

Leonhard Kehrberger begins with a brief overview of some of the historically most relevant approaches to modelling the asymptotics of isolated systems. In particular, they discuss the notions of Bondi coordinates, the peeling property and Penrose's asymptotic simplicity (which all model isolated systems by demanding that the spacetime exhibits certain structure near null infinity), as a well as a different notion of asymptotic flatness employed by Christodoulou and Klainerman in their prominent proof of the stability of Minkowski space (which models isolated systems by demanding that spacetime exhibits certain decay near spacelike infinity).

They then present a construction of spacetimes describing the exterior of $N$ massive particles coming from the infinite past. From the physical perspective, this includes a discussion of how Post-Newtonian theory can be used to predict the gravitational radiation emitted by $N$ infalling masses from the infinite past up to some finite advanced time. This already shows that past null infinity $\mathcal{I}^-$ is not smooth. From the mathematical perspective, this prediction, together with the condition that there be no incoming radiation from $\mathcal{I}^-$, is taken as a starting point to set up a scattering problem for the linearised Einstein vacuum equations around Schwarzschild in the infinite past. Leonhard Kehrberger finally outlines how to solve this scattering problem and how to obtain the asymptotic properties of this scattering solution near spacelike infinity $i^0$ as well as future null infinity $\mathcal{I}^+$, and they prove that none of the notions mentioned above are satisfied.

Leonhard Kehrberger, University of Cambridge, UK

Leonhard Kehrberger, University of Cambridge, UK

The path that eventually led Leonhard to the field of analysis in general relativity began in 2014 when they started their Bachelor's studies at the University of Hamburg. Their degree was in physics, but they also took many mathematics courses and found great enjoyment in those, particularly in the analysis courses. After an Erasmus semester in Uppsala, Sweden, and finishing their Bachelor's thesis, they then went to the University of Cambridge for their master's degree. During that time, Leonhard learned much about currently active research fields in mathematical physics and mathematics, and were very happy to discover that general relativity is full of exciting open analytical and physically relevant problems. They thus started doing their PhD under the supervision of Mihalis Dafermos. They were lucky to be able to go on research visits to Münster and Princeton. After their PhD, they will be a postdoctoral researcher at the Max Planck Institute in Leipzig under the supervision of Dejan Gajic.

 

Conformal geodesics represent a conformally invariant generalisation of the notion of standard metric geodesics. The properties of these curves can be used to study the global structure of solutions to the Einstein field equations. In particular, the existence of conformal Gaussian gauge systems associated to these curves together with the extended conformal Einstein field equations can be used to analyse the non-linear stability of vacuum spacetimes with positive cosmological constant. Special focus is given to de Sitter-like spacetimes with spatial sections of negative scalar curvature and to the subextremal Schwarzschild-de Sitter spacetime. The strategy of the proof on the latter case relies on the observation that the cosmological region of this exact solution admits a smooth conformal extension with a spacelike conformal boundary. This region can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain dependence of suitable spacelike hypersurfaces can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows to prove existence and stability results close to the conformal boundary. In particular, small enough perturbations of initial data give rise to a solution to the Einstein field equations which is regular at the conformal boundary.

Marica Minucci, Queen Mary University of London, UK

Marica Minucci, Queen Mary University of London, UK

Marica Minucci is a final year PhD student in Mathematics at Queen Mary, University of London under the supervision of Dr Juan Antonio Valiente Kroon. Her research is devoted to understanding the mathematical aspects of Einstein’s theory of General Relativity. In particular, she works on devising new strategies to prove the non-linear stability of vacuum spacetimes with positive cosmological constant. The main strategy relies on conformal methods along with hyperbolic reduction procedures. She is also interested in problems arising from the conformal structure of spacetime such as the problem of spatial infinity.

Before starting her PhD, Marica received a BSc and an MSc in Theoretical and Mathematical Physics at University of Naples "Federico II".