The "Topics in general relativity" seminar is the seminar of the Holzegel group at Mathematics Münster. It takes place every Tuesday at 12:00 at the Westfälische Wilhelms-Universität. For further details/to receive e-mails concerning this seminar, feel free to contact us.
Matthias Wink (5th April 2022, room SRZ 204)
Title: Vanishing Results for Betti numbers
Abstract: A well known theorem of Bochner says that the first Betti number of compact manifolds with positive Ricci curvature vanishes. More generally, D. Meyer used the Bochner technique to show that manifolds with positive curvature operators are rational homology spheres. In this talk I will explain that this is more generally the case for manifolds with $\lceil \frac{n}{2} \rceil$-positive curvature operators. We will see that this is a consequence of a general vanishing and estimation theorem for the $p$-th Betti number for manifolds with a lower bound on the average of the lowest $(n-p)$ eigenvalues of the curvature operator. This talk is based on joint work with Peter Petersen.
Annegret Burtscher (11th April 2022 at 2pm in room SRZ 204) Caution: unusual time and date
Title: On the volume of generalized tubes
Abstract: Consider a small spherical tube around a compact submanifold M in Euclidean space. In 1939 Weyl showed that the volume of such a tube only depends on the radius of the tube and the intrinsic curvature of M. What happens for tubes with more complicated cross sections D? We will see that under sufficiently strong symmetry assumptions on D the tube volume turns out to be still intrinsic. This gives hope that causal tubes around spacelike submanifolds in Minkowski space also exhibit such nice properties. In the Lorentzian setting, however, the situation is more subtle. Joint work with Gert Heckman.
Arthur Touati (19th April 2022, room SRZ 204)
Title: Construction of high-frequency spacetimes
Abstract: In this talk, I will present recent work on high-frequency solutions to the Einstein vacuum equations. From a physical point of view, these solutions model high-frequency gravitational waves and describe how waves travel on a fixed background metric. There are also interested when studying the Burnett conjecture, which adresses the lack of compactness of the family of vacuum spacetimes. These high-frequency spacetimes are singular and require to work under the regime of well-posedness for the Einstein vacuum equations. I will review the literature on the subject and then show how one can construct them in generalised wave gauge by defining high-frequency ansatz.
Dejan Gajic (26th April 2022, room SRZ 204)
Title: Late-time tails for geometric wave equations with inverse-square potentials
Abstract: I will introduce a new method for obtaining the precise late-time asymptotic profile of solutions to geometric wave equations with inverse-square potentials on asymptotically flat spacetimes. This setting serves as a convenient toy model for understanding novel dynamical properties in the context of Einstein's equations of general relativity that arise in a variety of situations, e.g. when considering the gravitational properties of electromagnetically charged matter, when describing dynamical, rapidly rotating black holes and when considering higher, odd, spacetime dimensions.
Martin Taylor (3rd May 2022, room SRZ 204)
Title: The nonlinear stability of the Schwarzschild family of black holes
Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos--Holzegel--Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.
Fatima-Ezzahra Jabiri (10th May 2022, room SRZ 204)
Title: Stationary axisymmetric Einstein-Vlasov bifurcations of the Kerr spacetime
Abstract: In this talk, I am going to talk about the construction of stationary axisymmetric black hole solutions to the EV system. These solutions have the property that the spatial support of the matter is a finite, axially symmetric shell located away from the black hole. To this end, I will start by reviewing some of the progress made in the context of solutions to the Einstein-Vlasov system. Then, I will explain how the study of trapped timelike geodesics in a perturbed Kerr spacetime allowed us to provide a one-parameter family of solutions.
Canceled (17th May 2022, room SRZ 204)
Renato Velozo (24th May 2022, room SRZ 204)
Title: Stability of Schwarzschild for the spherically symmetric Einstein-massless Vlasov system
Abstract: The Einstein-massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the hyperbolicity of the geodesic flow around the black hole to obtain decay of the stress energy momentum tensor, despite the presence of trapped null geodesics. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.
Melanie Graf (31th May 2022, room SRZ 204)
Title: Coordinates are messy
Abstract: In General Relativity, an “isolated system at a given instant of time” is modeled as an asymptotically Euclidean initial data set (M,g,K). Such asymptotically Euclidean initial data sets (M,g,K) are characterized by the existence of asymptotic coordinates in which the Riemannian metric g and second fundamental form K decay to the Euclidean metric delta and to 0 suitably fast, respectively. Using harmonic coordinates Bartnik showed that (under suitable integrability conditions on their matter densities) the (ADM-)energy, (ADM-)linear momentum and (ADM-)mass of an asymptotically Euclidean initial data set are well-defined. To study the (ADM-)angular momentum and (BORT-)center of mass, however, one usually assumes the existence of Regge-Teitelboim coordinates on the initial data set (M,g,K) in question, i.e. the existence of asymptotically Euclidean coordinates satisfying additional decay assumptions on the odd part of g and the even part of K. We will show that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge-Teitelboim coordinates. This allows us to easily give examples of (vacuum) asymptotically Euclidean initial data sets which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.
Holidays and conference break
Christoph Kehle (21th June 2022, room SRZ 204)
Title: Strong Cosmic Censorship for Λ < 0
Abstract: The statement that general relativity is deterministic finds its mathematical formulation in the celebrated "Strong Cosmic Censorship Conjecture" due to Roger Penrose. I will present my results on the linear analog of this conjecture in the case of negative cosmological constant. It turns out that this is intimately tied to Diophantine properties of a suitable ratio of mass and angular momentum of the black hole due to the presence of slowly decaying quasinormal modes on the black hole exterior. If time permits, I will further discuss a related result (joint with M. Van de Moortel) on Strong Cosmic Censorship for spherical symmetry dynamical black holes in the presence of slow decay.
Nicolas Besset (28th June 2022, room SRZ 204)
Title: Parametrix construction and Fredholm theory for totally characteristic wave type operators
Abstract: Totally characteristic wave type operators naturally appear in cosmological black hole type spacetimes in General Relativity. We propose a parametrix construction for such operators using Mellin type quantization. We then deduce the index 0 Fredholm property which allows us to define resonances of these operators as the zeros of a Fredholm determinant. If time allows it, we will discuss how to use this method to numerically approximate resonances with control of the error.
Christopher Straub (5th July 2022, room SRZ 204)
Title: Linearly Stable Shells of Collisionless Matter Surrounding a Schwarzschild Black Hole
Abstract: The asymptotically flat Einstein-Vlasov system is used to describe the evolution of an isolated globular cluster or galaxy in a relativistic setting. We consider this system in Schwarzschild coordinates with a fixed central black hole as well as without singularity. After constructing compactly supported equilibria in these cases, we discuss their stability properties. We derive a sharp criterion for linear stability by developing a Birman-Schwinger type principle originating from quantum mechanics. We then conclude by discussing applications of our criterion, including the linear stability of "small" Vlasov matter shells surrounding a black hole. The talk is based on joint work with Sebastian Günther and Gerhard Rein.
Christopher Kauffman (12th October 2021, room MA 503 (5th floor, main building))
Title: Global Stability of Minkowski for the Einstein-Maxwell-Klein-Gordon system
Abstract: We discuss the global stability of Minkowski space for the Einstein-Maxwell-Klein-Gordon system, based on joint work with Hans Lindblad. This proof uses modified wave coordinates which are adapted to the behavior of the light cones of the Schwarzschild metric. This in particular provides improved energy bounds for the scalar and electromagnetic fields, as well as asymptotics for the metric. We conclude with a modular result which is applicable to similar problems.
Sam Collingbourne (19th October 2021, room M5, Einsteinstrasse 64)
Title: The Gregory—Laflamme Instability of the 5D Schwarzschild Black String Exterior
Abstract: In this talk, I will discuss my work on a direct proof of the Gregory—Laflamme instability for the 5D Schwarzschild black string (https://aip.scitation.org/doi/full/10.1063/5.0043059). In particular, I will discuss how one proves the existence of a regular, exponentially growing, low frequency, mode solution of the linearised vacuum Einstein equation in harmonic/transverse-traceless gauge on the Schwarzschild black string exterior.
Leonhard Kehrberger (26th October 2021, room SRZ 102 CIP, Orléans-Ring 12)
Title: On the Relation Between Conservation Laws, Late-Time Asymptotics and the Failure of Peeling
Abstract: I will discuss certain generalisations of the conservation laws associated to the "Newman-Penrose constants". I will then explain how one can use these to read off late-time asymptotics for solutions to the wave equation (or, more generally, to the Teukolsky equations) from their peeling behaviour near future null infinity, i.e. from how regular they are in the conformal variable $1/r$. Finally, I will show that solutions generically fail to be conformally regular near future null infinity, going back to arguments by Christodoulou and Damour. This failure of peeling then leads to notable differences in the late-time asymptotics compared to the usual Price's law asymptotics.
Conference break
Zoe Wyatt (9th November 2021, room SRZ 102 CIP, Orléans-Ring 12)
Title: Stabilising relativistic fluids on slowly expanding cosmological spacetimes
Abstract: On a background Minkowski spacetime, the relativistic Euler equations are known, for a relatively general equation of state, to admit unstable homogeneous solutions with finite-time shock formation. By contrast, such shock formation can be suppressed on background cosmological spacetimes whose spatial slices expand at an accelerated rate. The critical case of linear, ie zero-accelerated, spatial expansion, is not as well understood. In this talk, I will present two recent works concerning the relativistic Euler and the Einstein-Dust equations for geometries expanding at a linear rate. This is based on joint works with David Fajman, Todd Oliynyk and Max Ofner.
Claude Warnick (16th November 2021, room SRZ 102 CIP, Orléans-Ring 12)
Title: Defining quasinormal modes
Abstract: Quasinormal modes — the characteristic decaying oscillations by which a perturbed black hole rings-down to its ground state — are now an observable part of the gravitational wave signals being measured at interferometers. By considering a model problem, I will show how finding the quasinormal modes for a given black hole can be reduced to a spectral problem for a non-self adjoint operator, and compare this approach with others in the literature. Joint work with Dejan Gajic.
Fatima-Ezzahra Jabiri (23th November 2021, CANCELED DUE TO COVID)
Title: Stationary axisymmetric Einstein-Vlasov bifurcations of the Kerr spacetime
Abstract: In this talk, I am going to talk about the construction of stationary axisymmetric black hole solutions to the EV system. These solutions have the property that the spatial support of the matter is a finite, axially symmetric shell located away from the black hole. To this end, I will start by reviewing some of the progress made in the context of solutions to the Einstein-Vlasov system. Then, I will explain how the study of trapped timelike geodesics in a perturbed Kerr spacetime allowed us to provide a one-parameter family of solutions.
Sakis Chatzikaleas (30th November 2021, room SRZ 102 CIP, Orléans-Ring 12)
Title: Non-linear periodic waves on the Einstein cylinder
Abstract: Motivated by the study of small amplitudes non-linear waves in the AdS spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to various toy models that mimic certain properties of nonlinear waves in the AdS spacetime. These include the conformal cubic wave equation and the spherically-symmetric Yang–Mills equations on the Einstein cylinder. Our proof relies on modifications of a theorem of Bambusi–Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. This is a joint work with Jacques Smulevici.
Olivier Graf (7th December 2021, room SRZ 102 CIP, Orléans-Ring 12)
Title: An L2-curvature pinching result for the Euclidean 3-disk
Abstract: In harmonic coordinates the principal terms of the Ricci curvature tensor of a Riemannian manifold are the Laplace-Beltrami operators of the metric components. By elliptic regularity, we expect that the H2 norm of these components can be estimated by the L2 norm of the Ricci tensor. In this talk, I will make this idea concrete in the case of Riemannian 3-manifolds with Ricci curvature in L2 and second fundamental form of the boundary in H1/2 both close to their respective Euclidean unit 3-disk values. The key idea is a refined Bochner identity with boundary for harmonic functions. This talk is based on a result that I obtained in [Global nonlinear stability of Minkowski space for spacelike-characteristic initial data, Appendix A].
Annegret Burtscher (14th December 2021, CANCELED DUE TO COVID)
Title: TBA
Abstract: TBA
Christmas break
Max Weissenbacher (11th January 2022, CANCELED)
Title: TBA
Abstract: TBA
Gustav Holzegel (18th January 2022, room SRZ 102 CIP, Orléans-Ring 12)
Title: Unique Continuation in asymptotically anti-de Sitter spacetimes
Abstract: TBA
Allen Fang (25th January 2022, meeting in front of the lecture building)
Title: Nonlinear stability of the slowly-rotating Kerr-de Sitter family
Abstract: The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof relies on using the vectorfield method to uncover a spectral gap and corresponding exponential decay of the solution at the level of the linearized equations, and then using the exponential decay at the linearized level in a bootstrap proof to conclude nonlinear stability.