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

09 - 10 May 2023 09:00 - 17:00 The Royal Society Watch online
The Watcher Over the Sea of Fog. Credit Paul Tod

Scientific discussion meeting organised by Dr Juan Valiente Kroon and Dr Grigalius Taujanskas.

In the sixty years since Penrose’s original insight that conformal geometry could be used to study the global structure of fields and spacetimes, these ideas have become central to many parts of mathematical and theoretical physics. Today many inspired methods and frameworks exist for asymptotic analysis in general relativity. This meeting aimed to bring together researchers working on these and related topics.

The schedule of talks and speaker biographies and abstracts are available below. Meeting papers will be published in a future issue of Philosophical Transactions of the Royal Society A.

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Organisers

  • Juan Valiente Kroon

    Dr Juan A Valiente Kroon, Queen Mary University of London, UK

    Juan A Valiente Kroon is a Reader in Applied Mathematics at the School of Mathematical Sciences, Queen Mary University of London. He was a Lise Meitner fellow of the Austrian Science Fund (FWF), an Engineering and Physical Sciences (EPSRC) Advanced Research fellow and he specialises in various aspects of mathematical general relativity including the application of methods of conformal geometry to study global properties of the solutions to the Einstein field equations. He is author of over 70 research articles and a research monograph (Conformal Methods in General Relativity, Cambridge University Press 2016). He has supervised eight PhD students.

  • Grigalius Taujanskas

    Dr Grigalius Taujanskas, University of Cambridge, UK

    Grigalius Taujanskas is currently a College Assistant Professor, Fellow and Director of Studies in Mathematics at Trinity Hall, University of Cambridge. His research interests are in the analysis of equations arising in general relativity, and especially global questions about fields and spacetimes. Previously he was a Stipendiary Lecturer at Hertford College, University of Oxford and a mathematical consultant with the Smith Institute in Oxford. He completed his PhD at the University of Oxford in 2019 with Lionel Mason.

    Photo credit: Phil Mynott

Schedule

Chair

Zoe Wyatt

Dr Zoe Wyatt, King's College London, UK

09:00-09:05 Introduction
09:05-09:55 The relations between conformal and spectral analytic scattering theories

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

09:55-10:00 Discussion
10:00-10:50 Conformally compact Riemannian manifolds with negative mass

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

10:50-10:55 Discussion
10:55-11:20 Break
11:20-12:10 Analysis on compactifications of asymptotically flat spacetimes

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

12:10-12:15 Discussion

Chair

Chris Stevens

Dr Chris Stevens, University of Canterbury, New Zealand

Malcolm MacCallum

Professor Malcolm MacCallum, Queen Mary University of London, UK

13:15-13:55 The calculation of the asymptotic charges at the critical sets of null infinity

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

13:55-14:00 Discussion
14:00-14:40 The case against Smooth Null Infinity

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

14:40-14:45 Discussion
14:45-15:15 Break
15:15-15:55 Conformal geodesics and the evolution of spacetimes with positive cosmological constant

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

15:55-16:00 Dicussion
16:00-17:00 Plenary talk: Conformal Infinity - from the origins of the idea to Cyclic Cosmology
Sir Roger Penrose OM FRS, Mathematical Institute, University of Oxford, UK

Sir Roger Penrose OM FRS, Mathematical Institute, University of Oxford, UK

17:00-18:00 Networking

Chair

Katy Clough

Dr Katy Clough, Queen Mary University of London, UK

09:00-09:50 Asymptotics in General Relativity from the perspective of Cartan geometry

There are several classical notions of asymptotic flatness in General Relativity: The exact procedures and geometries are however very different depending on whether one considers an asymptote in null, time-like, or space-like directions. Dr Herfray will show that these can nonetheless all be derived from one unique concept, curved obit decomposition of Cartan geometry and that there is therefore a profound unity in these notions of asymptotics.

Dr Yannick Herfray, University of Tours, France

Dr Yannick Herfray, University of Tours, France

09:50-09:55 Discussion
09:55-10:45 Conformal methods in mathematical cosmology

When he first introduced the notion of a conformal boundary into the study of asymptotically empty space-times Penrose was already aware that the boundary would be null, space-like or time-like according as the cosmological constant Λ was zero, positive or negative. While most applications of the idea have been to the zero-Λ, asymptotically-Minkowskian case, there also has been work on the nonzero cases. Here Professor Tod concentrates on positive Λ, which is the appropriate case for cosmology of the universe in which we live.

With a space-like future conformal boundary, we may follow Friedrich and contemplate using the boundary as a Cauchy surface with data for a conformal version of the Einstein equations. Assuming a suitable form of Penrose's Weyl Curvature Hypothesis, we may also contemplate rescaling an initial 'Big Bang' singularity for use as a Cauchy surface for a similar system. Finally we may follow Penrose's 'outrageous suggestion' and contemplate a universe of successive aeons, each of which is an expansion from a regularised Big Bang surface to a regularised future conformal boundary that provides the regularised Big Bang of the next aeon. Now there is a regular conformal metric common to all aeons, while the (conformally-related) physical metric runs from singularity to singularity.

Professor Paul Tod, University of Oxford, UK

Professor Paul Tod, University of Oxford, UK

10:45-10:50 Discussion
10:50-11:20 Break
11:20-12:10 Cosmological models: hierarchies of asymptotic behaviour

De Sitter space-time is a time-like and null geodesically complete solution to the Einstein-λ vacuum field equations with cosmological constant λ > 0 and compact Cauchy hypersurfaces that admits smooth conformal boundaries at future and past time-like infinity. Consider general smooth Cauchy data for the same equations on a Cauchy hypersurface S in this space-time. If these data are sufficiently close to the de Sitter data induced on S the conformal Einstein equations imply that they develop into solutions that also admit smooth conformal boundaries and even extend, as solutions to the conformal Einstein equations, beyond these boundaries. In this talk Professor Friedrich discusses their interest in and some answers to the question to what extent the existence of smooth conformal future extensions generalise to the Einstein-λ equations with matter fields. Simple FLRW solutions show that this may require quite restrictive conditions on the matter fields. Examples suggest that in the case of the Einstein-λ equations coupled to conformally covariant matter fields with trace-free energy-momentum tensor the existence of smooth conformal extensions is a general feature of such systems. He discusses cases of solutions with smooth conformal extensions where the matter fields are conformally non-covariant but still conformally privileged in some sense and cases in which such a property seems to be given only asymptotically and smooth conformal extensions exist or their existence remains undecided so far.

Professor Helmut Friedrich, University of Potsdam, Germany

Professor Helmut Friedrich, University of Potsdam, Germany

12:10-12:15 Discussion

Chair

Shabnam Beheshti

Dr Shabnam Beheshti, Queen Mary University of London, UK

13:15-14:05 The good-bad-ugly system near spatial infinity on flat spacetime

A model system of equations that serves as a model for the Einstein field equation in generalised harmonic gauge called the good-bad-ugly system is studied in the region close to null and spatial infinity in Minkowski spacetime. This analysis is performed using H Friedrich’s cylinder construction at spatial infinity and defining suitable conformally rescaled fields. The results are translated to the physical set up to investigate the relation between the polyhomogeneous expansions arising from the analysis of linear fields using the spatial-infinity-cylinder framework and those obtained through a heuristic method based on Hormander’s asymptotic system.

Dr Edgar Gasperin, Instituto Superior Técnico, Lisbon, Portugal

Dr Edgar Gasperin, Instituto Superior Técnico, Lisbon, Portugal

14:05-14:10 Discussion
14:10-15:00 Hyperboloidal foliations and the gravitational self-force programme

Direct observation of gravitational waves (GWs) marks a new era in astrophysics. The arrival of highly sensitive ground- and space-based detectors shall make precision gravitational wave astronomy routine. Critical sources of GWs for the LISA Mission are the so-called extreme mass ratio inspirals (EMRI), ie relatively light objects orbiting a supermassive black hole. Among the methods to tackle the two-body problem in General Relativity (GR), the gravitational self-force approach is the best option to describe EMRIs within the accuracy demanded by LISA. On the theoretical side, it is essential to model highly accurate waveform to optimise the scientific gain of future detections. For this purpose, there has been diagnosed a need for a robust and systematic framework adapting to the geometrical structure of the spatial scales near the black hole and the (infinitely) far radiation zone. On the mathematical side, the conformal approach to GR, implemented in practice using hyperboloidal coordinates, has been identified as the best strategy to tackle the problem. Complementary, spectral methods provide a robust tool to solve the underlying equations accurately. In this talk, Dr Panosso Macedo will discuss the recent developments within this research programme.

Dr Rodrigo Panosso Macedo, Nils Bohr Institute, University of Copenhagen, Denmark

Dr Rodrigo Panosso Macedo, Nils Bohr Institute, University of Copenhagen, Denmark

15:00-15:05 Discussion
15:05-15:35 Break
15:35-16:25 Characterisations of Kerr-de Sitter in arbitrary dimension from null infinity

The Kerr metric has remarkable local geometric properties that are shared its generalisation to positive cosmological constant, namely the Kerr-de Sitter metric. These geometric properties essentially characterise these metrics among all Λ-vacuum solutions of the Einstein field equations admitting a Killing vector. Gibbons et al found a generalisation of the Kerr-de Sitter metric to higher dimensions. However, the geometric characterisation above requires four spacetime dimensions, so characterising geometrically the Kerr-de Sitter metric in higher dimensions requires an alternative method. One viable approach is to understand the behaviour of these metrics at future null infinity. After reviewing the Fefferman-Graham expansion and well-posedness results of the Λ-vacuum field equations in arbitrary dimensions, as well as the necessary and sufficient condition for the existence of a Killing vector, we shall introduce a canonical TT tensor at null infinity constructed from a conformal Killing vector. Conformal flatness and this canonical TT tensor provide initial data to a family of Λ-vacuum spacetimes to which the Kerr-de Sitter metric belongs. This can be used to characterise geometrically these metrics in a way that extends the four-dimensional case. We shall also present various uniqueness properties of the Kerr-de Sitter-like metrics in dimensions bigger than four.

Dr Marc Mars, University of Salamanca, Spain

Dr Marc Mars, University of Salamanca, Spain

16:25-16:30 Discussion
16:30-16:35 Closing remarks