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.
TBD (17th May 2022, room SRZ 204)
Renato Velozo (24th May 2022, room SRZ 204)
Melanie Graf (31th May 2022, room SRZ 204)
Holidays and conference break
Christoph Kehle (21th June 2022, room SRZ 204)
Nicolas Besset (28th June 2022, room SRZ 204)
Christopher Straub (5th July 2022, room SRZ 204)
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 L^{2}-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 H^{2} norm of these components can be estimated by the L^{2} 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 L^{2} and second fundamental form of the boundary in H^{1/2} 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.