This Seminar is part of the Trimester Program “Mathematical General Relativity” taking place at the Institut Henri Poincaré in order to celebrate the 100th Anniversary of General Relativity.

*Emile Borel Centre of Henri Poincaré Institute, Paris*

*September to December, 2015*

### Wednesday December 9, 2015

2:00pm **Hans Lindblad** (Baltimore) ** A sharp counter example to local existence for Einstein’s equations in wave coordinates **

** Abstract. **We are concerned with how regular initial data have to be to ensure local existence for Einstein’s equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in Sobolev spaces Hs for s>2. We give an example of data in H2 for which there is no local solution to the vacuum Einstein equations in H2. This is joint work with Boris Ettinger.

3:30pm **Florian Beyer (Dunedin) Self-gravitating Gowdy-symmetric fluids near the big bang singularity
**

** Abstract. **We present new results (in a joint work with P.G. LeFloch) about the construction and analysis of singular Gowdy-symmetric self-gravitating fluid solutions by means of the Fuchsian method. A crucial role is played by a critical phenomenon which is induced by the competition between the (by definition) isotropic internal fluid forces and the highly anisotropic gravitational forces. Einstein’s equations are written in some particular (generalized) wave gauge which both renders the evolution equations hyperbolic and allows us to handle the constraints. For this we make significant use of further new results regarding the vacuum Einstein equations in generalized wave gauges by E. Ames, F.B., J. Isenberg and P.G. LeFloch.

### Wednesday November 25, 2015

2:00pm **Anna Sakovich** (Vienna) **On the center of mass of asymptotically hyperbolic initial data sets**

** Abstract. **There are two main approaches which can be pursued to define the center of mass of an asymptotically Euclidean / hyperbolic initial data set. The first definition is derived from the Hamiltonian charges approach, while the second definition is purely geometric. After reviewing the situation in the asymptotically Euclidean setting, we will discuss how to unite these approaches in the asymptotically hyperbolic case.

3:30pm **Sung-Jin Oh** (Berkeley)** ****Linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations**

**Abstract: **Consider the linear scalar wave equation on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge. In this talk, I will present a proof that generic smooth and compactly supported initial data on a Cauchy hypersurface give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole, through combination of the infinite blue shift effect along the Cauchy horizon and sharp polynomial decay rate of the scalar field in the exterior. This is a joint work with J. Luk.

### Wednesday November 4, 2015

2:00pm **Jared Wunsch** (Evanston)** Asymptotics of scalar waves near null infinity**

**Abstract.** I will discuss joint work with Baskin and Vasy that addresses the asymptotics of scalar waves on backgrounds that admit compactifications analogous to the radial compactification of Minkowski space. For a wide class of such spacetimes, we find complete asymptotic expansions of the solution in various regimes near null infinity, and give a prescription for reading off the powers arising in the asymptotic expansion of the Friedlander radiation field (the rescaled solution at null infinity) in terms of solutions to an auxiliary elliptic problem on an asymptotically hyperbolic space.

3:30pm **Anne Franzen** (Utrecht) **Boundedness of massless scalar waves on Reissner-Nordström interior backgrounds**

**Abstract.** We consider solutions of the massless scalar wave equation, without symmetry, on fixed sub-extremal Reissner-Nordström backgrounds with non-vanishing charge. Previously, it has been shown that for solutions arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon . Using this, we show here that the solutions are in fact uniformly bounded in the black hole interior up to and including the bifurcate Cauchy horizon . The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates.

### Wednesday October 21, 2015

2:00pm **Jesus Oliver** (San Diego)** ****A vector field method for radiating blackhole spacetimes**

Abstract. We study the global decay properties of solutions to the wave equation in 3+1 dimensions on non-stationary, weakly asymptotically flat black hole space-times without symmetries. Assuming local energy decay estimates, we prove that sufficiently regular, well-localized solutions to this equation have bounded conformal energy with any number of scalings, rotations, and translation derivatives applied to the solution. We will also discuss a non-linear application of this method.

### Wednesday October 14, 2015

2:00pm **Lars Andersson** (Potsdam) **Spin geometry of the Kerr spacetime**

**Abstract.** I will give an introduction to the 2-spinor formalism and discuss some applications in the special case of the Kerr black hole geometry. A covariant variational formalism for spinors will be introduced which can be used to study the equations of linearized gravity. The special geometry of the Kerr spacetime is expressed via the existence of a Killing spinor. Important consequences of this fact are symmetries and conservation laws for fields on the Kerr spacetime, including Maxwell and linearized gravity.

3:30pm **Steffen Aksteiner **(Potsdam) **Symbolic computer algebra and applications in linearized gravity**

**Abstract.** The symbolic computer algebra package xAct for Mathematica is an efficient tool for abstract differential geometric calculations. I will review basic features of xAct and the use of spinors and Newman-Penrose formalism with examples. Starting from the linearized gravity field equations a covariant form of the Teukolsky master equation and Teukolsky-Starobinski identities will be presented.

### Wednesday October 7, 2015

2:00pm **Martin Taylor** (Cambridge, UK) **Global stability of Minkowski space for the massless Einstein–Vlasov system **

**Abstract.** Massless collision-less matter is described in general relativity by the massless Einstein–Vlasov system. I will present a proof that for smooth asymptotically flat Cauchy data for this system which is sufficiently close, in a suitable sense, to the trivial solution, Minkowski space, the resulting maximal development exists globally in time and asymptotically decays appropriately. By appealing to the corresponding result for the vacuum Einstein equations, a monumental result first obtained by Christodoulou–Klainerman in the early ’90s, the proof reduces to a semi-global problem. A key step is to estimate certain Jacobi fields on the mass shell, a submanifold of the tangent bundle of the spacetime endowed with the Sasaki metric.

3:30pm **Bruno Premoselli** (Cergy-Pontoise) **Stability and instability phenomena for the equations of the conformal method**

**Abstract.** The conformal method allows one to reformulate the constraint equations for the initial-value problem of the Einstein equations in terms of a nonlinear, critical, coupled system of elliptic equations. In this talk we shall investigate stability and instability properties for such a system in a scalar-field setting. Here the stability of the system is understood as the continuous dependence of the set of its positive solutions in the choice of the background physics data of the conformal method. When a non-trivial scalar-field is allowed, these equations might develop concentration phenomena eventually leading to instability results. We will investigate the conditions ensuring stability and discuss their optimality by constructing blowing-up counterexamples when they are not satisfied. Some of these results have been obtained in collaboration with O. Druet (UCBL, Lyon) and J. Wei (UBC, Vancouver).