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 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.

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.

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.  

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.

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.