This workshop **DYNAMICS OF SELF-GRAVITATING MATTER**** **is an event of the Trimester Program taking place at the Institut Henri Poincaré in order to celebrate the 100th Anniversary of General Relativity. It will present recent progress on matter models and other related areas of mathematical general relativity.

*Emile Borel Centre of Henri Poincaré Institute, Paris*

*October 26 to 29, 2015*

**SCHEDULE**

**Monday afternoon Oct. 26**

**Tuesday morning Oct. 27**

**Tuesday afternoon Oct. 27**

**Wednesday morning Oct. 28**

**Wednesday afternoon Oct. 28**

**Buffet Dinner Reception**

**Thursday morning Oct. 29**

**LIST OF SPEAKERS with title and abstracts**

Hakan Andreasson (Gothenburg) Properties of axially symmetric stationary solutions of the Einstein-Vlasov system and the Vlasov-Poisson system

**Abstract.** A large class of stationary axially symmetric solutions to the Einstein-Vlasov system and the Vlasov-Poisson system is constructed numerically using the finite element method implemented in FEniCS. The possible range of solutions that can be obtained is discussed, and the properties of these solutions are investigated. In particular, the rotation curves corresponding to axially symmetric flat solutions of the Vlasov-Poisson system are calculated. The rotation curves are compared with measurements from real galaxies and satisfactory agreement is obtained. This raises the question whether the observed rotation curves for disk galaxies may be explained without introducing dark matter.

Thierry Barbot (Avignon) Fuchsian polygonal surfaces in Minkowski and the decorated Teichmüller space

**Abstract.** In “Fuchsian polyhedra in Lorentzian space-forms” (Mathematische Annalen, 2011), F. Fillastre proved that any singular Euclidean metric with with cone angles larger than 2 pi on a closed surface can be uniquely realized (up to a global isometry) as a convex polygonal surface in a flat and globally hyperbolic spacetime, whose holonomy group is linear. In this talk, I will mention a work in progress by L. Brunswic, aimed at extending this result to the case of flat spacetimes containing massive or BTZ particles. I will also present the connection with the Decorated Teichmüller space (“The Decorated Teichmüller space of punctured surfaces”, Commun. Math. Phys., 1987).

Robert Beig (Vienna) Celestial mechanics of elastic matter

**Abstract. **After a short review of relativistic elasticity, we restrict ourselves to time independent configurations. Our particular interest are configurations corresponding to an elastic body in circular motion in a stationary-axisymmetric gravitational field such as Kerr. We give a rigorous construction of families of such configurations paramterized by a small constant \epsilon, where, for \epsilon = 0, the configuration is stress-free and spacetime Minkowski. The construction is based on an implicit-function argument. Complications arising due to the linearized operator having nontrivial range and kernel are treated by a Liapunoff-Schmidt technique.

David Fajman (Vienna) L2-estimates for geometric transport equations

**Abstract.** We consider the Vlasov equation on spacetimes foliated by compact spatial hypersurfaces. Using the family of Sasaki metrics on tangent bundles of these hypersurfaces, we construct weighted L2-Sobolev norms for distribution functions. We derive energy estimates for solutions to the transport equation, which imply bounds on the corresponding energy momentum tensor. Furthermore, we construct a correction mechanism, which yields improved energy estimates, that imply uniform boundedness of the norm under less restrictive conditions on the space-time. Finally, we present a number of applications to the nonlinear stability problem for the Einstein-Vlasov system.

Marc Mars (Salamanca) Characterization of (asymptotically) Kerr-de Sitter-like spacetimes at null infinity

**Abstract.** The Kerr-de Sitter spacetime, and related metrics such as the Kerr-NUT-de Sitter spacetime, can be characterized, among vacuum spacetimes with positive cosmological constant admitting a Killing vector, in terms of the vanishing of a suitable geometric tensor, in a similar way as the Kerr metric can be identified among Ricci-flat spacetimes. Based on this local characterization, the geometric data at null infinity corresponding to those spacetimes will be identified and analyzed. Along the way, a larger class of data, which we call asymptotically Kerr-de Sitter like, with interesting properties, arises naturally. This is joint work in collaboration with T. Paetz, J.M.M. Senovilla, and W. Simon.

David Maxwell (Fairbanks) Conformal Killing fields and non-CMC solutions of the constraint equations

**Abstract. **It has long been known that conformal method is quite satisfactory for generating CMC solutions of the constraints, but its applicability in the non-CMC regime has not been well-understood. Recently, a picture has been developing pointing to various undesirable non-existence/non-uniqueness results in the non-CMC setting, even for the vacuum conformal method on a compat manifold. Hence there is motivation for finding alternative generalizations of the CMC conformal method. One marked difference between the CMC conformal method and its non-CMC variation is the role played conformal Killing fields. For CMC solutions, conformal Killing fields are essentially harmless, but in the non-CMC setting very little is known. In this talk we show how conformal Killing fields pose a genuine obstruction for the conformal method in the non-CMC regime, and we present alternative modifications of the CMC conformal method that address these issues, at least for near-CMC solutions.

Todd Oliynyk (Monash) Future stability of the FLRW fluid solutions with a positive cosmological constant

**Abstract.**We will outline a new method for establishing the future non-linear stability of perturbations of FLRW solutions to the Einstein-Euler equations with a positive cosmological constant and a linear equation of state. The method we present is based on a conformal transformation of the Einstein-Euler equations that compactifies the time domain thereby reducing the global problem to a local one, albeit singular. We will discuss how a variation of the usual approach to establishing the local-in-time existence and uniqueness of solutions to symmetric hyperbolic systems can be used to obtain the existence and uniqueness of solutions to our local singular problem and obtain the asymptotic behaviour in the far future. Some advantages of our method over previous approaches are the conciseness of the resulting stability proof and the fact that the equation of state parameter values can be handled in a uniform manner. Due to the compactified time domain, we expect this method will also be useful for numerically simulating solutions globally to the future.

Volker Schlue (Toronto) Non-existence of time-periodic vacuum spacetimes

**Abstract.** In general relativity, we expect that the dynamics of self-gravitating matter cannot display time-periodic behavior, due to the emission of gravitational waves. We show that any asymptotically flat solution to the vacuum equations, which is assumed to be periodic in time, is necessarily stationary near infinity. Thus genuinely time-periodic solutions to the Einstein equations do not exist, at least far away from the sources. This is joint work with S. Alexakis.

Bernd Schmidt (Potsdam) Transition conditions for isolated self-gravitating bodies

**Abstract.** For freely floating self-gravitating bodies the boundary conditions on physical grounds are: the vanishing of the normal stress at the boundary for all times. We expect that these conditions together with initial data determine a unique solution of the evolution equations. However, if the density of the matter at the surface of the body is positive, further “transition conditions” are needed to imply sufficient differentiability of the solution inside and outside the body. I will discuss the origin of these conditions first for a simple model problem and the for self-gravitating bodies in Newton’s and Einstein’s theory of gravity.

Jared Speck (Cambridge, USA) Stable big crunches in general relativity

**Abstract.** I will discuss the global Cauchy problem for perturbations of the Big Crunch solution to the Einstein-scalar field system.The Big Crunch is a member of the well-known Friedmann-Lemaître-Robertson-Walker family that features topology RxS3, positive spatial curvature, and spacelike singularities at its past and future boundaries due to spacetime curvature blow-up. The main new result is that the Big Crunch solution is globally stable. More precisely, if one slightly perturbs (in a suitable Sobolev space) its data at the “present,” then the perturbed solution exhibits precisely the same kind of blow-up, namely past and future curvature blow-up along spacelike hypersurfaces. The key to the proof is an L2-type approximate monotonicity identity, which is visible in well-constructed gauges. This kind of identity first appeared in my recent work with I. Rodnianski, which addressed the case of T3 spatial topology. Because the geometry is very different in the S3 case, the proof of the stability of the Big Crunch solution relies on a new geometric framework tailored to the approximate symmetries of nearly round metrics.

Shadi Tahvildar-Zadeh (Rutgers) Spin and topo-spin: Topological aspects of Kerr-Newman spacetimes and their consequences for the Dirac equation

**Abstract.** We study the topology of the maximal analytical extension of Kerr-Newman spacetimes and show that their ring singularity and their double-sheetedness survive in the zero-gravity limit, which is when G, Newton’s constant of universal gravitation, tends to zero. We show how the ring singularity of this limiting spacetime displays features of a single spin-half particle which is simultaneously an electron and a positron, thereby solving one of the long-standing mysteries about the Dirac equation. This is a joint project with Michael Kiessling.

Eric Woolgar (Alberta) Lorentzian N-Bakry-Emery cosmological singularity and splitting theorems

**Abstract.** Lorentzian Bakry-Emery geometry is a natural language for scalar-tensor gravitation. It can directly yield singularity theorems in the so-called Jordan frame with conditions on energy-momentum that are more natural than those obtained by translating standard theorems from the Einstein frame. Furthermore, recent developments in Riemannian Bakry-Emery comparison geometry show the theory to be of mathematical interest in its own right. In this talk, I will treat the Lorentzian N-Bakry-Emery theory, with N taking any real value lying outside the interval (1,n], where n is the spacetime dimension. I also treat N=infinity. This theory yields cosmological singularity theorems similar to those of general relativity. In addition to the usual assumptions of an energy condition and a condition on mean curvature of the initial hypersurface, when N=infinity or when N is less than or equal to 1 (including negative N) an additional condition allowing for mild control of the weight function is required. When the condition on the initial hypersurface is borderline-violated, complete spacetimes arise but the geometry must split off a timelike line. This is a product splitting except that when N=1 a warped product occurs. This talk is based on joint work with G.J. Galloway and joint work with W. Wylie.

**ORGANIZERS**

*Lars Andersson (Potsdam)*

*Sergiu Klainerman (Princeton) *

Philippe G. LeFloch *(Paris)*

This conference is part of the **Three-Month Program on MATHEMATICAL GENERAL RELATIVITY** — Institut Henri Poincaré, Paris