The "Topics in general relativity" seminar is the seminar of the Holzegel group at Mathematics Münster. It takes place Tuesdays at 12:00 at the Westfälische Wilhelms-Universität, in Room 503 of the department building (Einsteinstr. 62). For further details/to receive e-mails concerning this seminar, please contact us. The page for previous semesters may be found here
Note: Seminars on 25 October and 6 December will be a part of the PDE Colloquium. There will be no GR seminar on 20 December, but we encourage people to attend the Geometry seminar on 19 December, where Anna Sakovich is scheduled to speak.
Christopher Kauffman (WWU Münster) (11 October, Room 503)
Title: Stability for perturbed wave equations on subextremal Kerr spacetimes
Abstract: We prove stability results for small first-order perturbations of the wave equation on the exterior region of the Kerr spacetime. This builds on techniques and results for the covariant wave equation developed by Dafermos, Rodnianski, and Shlapentokh-Rothman, and we expect our result to be extendible into certain nonlinear wave equations. Our approach relies on a carefully constructed microlocal commutator W, which gives a nondegenerate ILED estimate for trapped frequencies. We will combine these local results with a weak Coifman-Meyer-type pseudodifferential bound to globally bound perturbative terms. This talk is based on joint work with Gustav Holzegel
Athanasios Chatzikaleas (WWU Münster) (18th October, Room 503)
Title: Non-linear periodic waves in AdS and their stability
In 2006, Dafermos-Holzegel conjectured that the Anti-de Sitter spacetime is unstable solution to the Einstein equations under reflective boundary conditions at the conformal infinity for generic initial data. Recently, Moschidis established a rigorous instability proof. Moreover, Rostworowski-Maliborski enhanced this conjecture by providing strong numerical evidence indicating the existence of "special" initial data leading to time-periodic solutions for the Einstein-Klein-Gordon system in spherical symmetry which are in fact stable. Motivated by these, we construct families of arbitrary small time-periodic solutions to several toy models providing a rigorous proof of the numerical constructions above in a simpler setting. The models we consider include the conformal cubic wave equation and the spherically-symmetric Yang-Mills equation on the Einstein cylinder and our proof relies on modifications of a theorem of Bambursi-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. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is non-resonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to back-reaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting. Finally, we also consider the massive wave equation in the fixed Anti-de Sitter with a cubic non-linearity, construct families of arbitrary small time-periodic solutions from "special" initial data and show that these are non-linear stable for exponentially long times.
Sebastian Herr (Bielefeld) (25 October, 2:15 PM PDE Colloquium, Room TBA)
Title: Global wellposedness of the Zakharov System below the ground state
Abstract: The Zakharov system is a quadratically coupled system of a Schrödinger and a wave equation, which is related to the focussing cubic Schrödinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schrödinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate. This is joint work with Timothy Candy and Kenji Nakanishi.
Grigorios Fournodavlos (University of Crete) (8 November, Room 503)
Title: On the nature of the Big Bang singularity
Abstract: Abstract: 100 years ago, Friedmann and Kasner discovered the first exact cosmological solutions to Einstein’s equations, revealing the presence of a striking new phenomenon, namely, the Big Bang singularity. Since then, it has been the object of study in a great deal of research on general relativity. However, the nature of the ‘generic’ Big Bang singularity remains a mystery. Rivaling scenarios are abound (monotonicity, chaos, spikes) that make the classification of all solutions a very intricate problem. I will give a historic overview of the subject and describe recent progress that confirms a small part of the conjectural picture.
Max Weissenbacher (Imperial College London) (15 November, Room 503)
Title: (Non-)decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström
Abstract: We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner-Nordström spacetimes. We prove that moments of solutions decay at an exponential rate in the subextremal case and obtain a sharp inverse polynomial decay rate along the event horizon for the extremal case. Furthermore in the extremal case we derive a condition under which transversal derivatives of certain components of the energy--momentum tensor do not decay along the horizon. Our results may be viewed as analogous to the Aretakis instability. Our proof does not rely on the existence of a conservation law.
Klaus Kröncke (KTH Stockholm) (22 November, Room M3)
Title: Stable cosmological Kaluza-Klein Spacetimes
Abstract: We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a wave map type and a Maxwell type equation by the Kaluza-Klein reduction. The Milne universe solves those field equations when the additional parts arising from the toroidal dimensions are chosen constant. We prove future stability of the Milne universe within this class of spacetimes, which establishes stability of a large class of cosmological Kaluza-Klein vacua. A crucial part of the proof is the implementation of a new gauge for Maxwell-type equations in the cosmological context, which we refer to as slice-adapted gauge. This is joint work with Volker Branding and David Fajman.
Georgios Moschidis (EPFL) (29 November, Room 503)
Title: Weak turbulence in general relativity.
Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. One way to introduce confinement to the equations is by imposing a negative value for the cosmological constant. In this setting, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will discuss a number of problems in connection with the turbulent dynamics of the equations in the asymptotically AdS setting. I will also present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system.
Stefan Czimek (Universität Leipzig) (POSTPONED)
Title: TBA
Abstract: TBA
Jared Speck (Vanderbilt) (2:15 PM 6 December, PDE Colloquium, Room TBA)
Title: Advances in the theory of multi-dimensional shock waves
Abstract: A shock singularity in a quasilinear hyperbolic PDE solution is a mild singularity such that one of the solution’s derivatives blows up, though the solution itself remains bounded. Importantly, the mild nature of the singularity opens the door to the possibility that the solution might be continued uniquely as a weak solution past the singularity, under suitable selection criteria. While the rigorous 1D theory is in a mature stage due to the availability of well-posedness results for BV initial data, multi-dimensional hyperbolic PDEs are typically ill-posed in BV. Consequently, the theory of multi-dimensional shocks is permeated with fundamental open problems, many with deep ties to geometry. Despite the challenges in higher dimensions, for specific systems, including the compressible Euler equations and relativistic Euler equations in 3D, there has been dramatic progress in the last 15 years, starting with Christodoulou’s 2007 monograph on shock formation in irrotational solutions. In this talk, after providing an introduction to the 1D problem, I will give a non-technical description of recent advances in multi-dimensions, with a focus on the multi- dimensional compressible Euler equations with vorticity and entropy. In particular, I will describe my recent series of works on the 3D compressible Euler equations with vorticity and entropy, which, for open sets of initial data, reveal the full structure of the maximal classical development, including the full structure of the singular set as well the emergence of a Cauchy horizon from the singularity. The proof relies on nonlinear geometric optics in conjunction with a new formulation of compressible Euler flow exhibiting miraculous geo-analytic structures and regularity properties. Finally, time permitting, I will discuss some of the many open problems in the field. Various aspects of this program are joint with L. Abbrescia, J. Luk, and M. Disconzi.
Alessandra Tullini (13 December, Room 503)
Title: Energy blowup of linear waves near the Cauchy horizon of Reissner—Nordström—AdS
Abstract: Reissner—Nordström—Anti-de Sitter spacetimes are black hole solutions to the Einstein-Maxwell system of equations under the assumption of a negative cosmological constant. They present with a Cauchy horizon and are thus of interest in the context of Strong Cosmic Censorship Conjecture. In a series of works, Kehle addressed the linear formulation of the conjecture in RN-AdS, which involves the study of the initial value problem for Klein-Gordon’s equation. In 2019, he proved uniform boundedness and continuity of solutions emanating from a spacelike hypersurface and satisfying Dirichlet boundary conditions, thus disproving the C0 formulation. In 2021, under the same boundary conditions, he recovered the H1 formulation by identifying a class of initial data whose resulting solution presents with unbounded local energy near the Cauchy horizon. We focus on the latter work and shine a light onto the identification of such a class and on the notion of genericity thus associated. The argument rests on the possibility of finding a quasi normal mode solution with unbounded local energy near the Cauchy horizon. Then, because of the linear nature of the problem, we may perturb any initial data (among those for which we have well-posedness) with initial data for the QNM exhibiting energy blowup and still obtain (local) energy blowup. The blowup behaviour is generic in the following sense: the set of initial data such that the resulting solution has bounded local energy near the Cauchy horizon, is at most a codimension 1 subset of the set of all admissible initial data. In conclusion, under this notion of genericity, the linear H1 formulation of the conjecture is restored in RN-AdS.
Anna Sakovich (19 December, Geometry Seminar, Time and Room TBA)
Title: TBA
Abstract: TBA
Zoe Wyatt (2:15 PM 20 December, PDE Colloquium, Room TBA)
Title: Global stability of product spacetimes
Abstract: Spacetimes formed from the cartesian product of Minkowski space and a compact Ricci flat space with special holonomy play an important role in supergravity and string theory. In this talk I will discuss two results concerning the global, nonlinear stability of such spacetimes: one is work in collaboration with Andersson, Blue and Yau, the other is upcoming work with Huneau and Stingo. Both results are related to claims of Penrose and Witten concerning the validity of supergravity and string theory.
Oliver Lindblad Petersen (17 January, Room 503)
Title: Wave equations in subextremal Kerr-de Sitter spacetimes
Abstract: In 2013, Vasy proved that solutions to linear wave equations in Kerr-de Sitter spacetimes have asymptotic expansions in quasinormal modes up to an exponentially decaying term, assuming the angular momentum of the black hole satisfies certain bounds. This was the first step towards the proof of non-linear stability for slowly rotating Kerr-de Sitter black holes by Hintz and Vasy in 2018. In this talk, we extend Vasy’s result to the full subextremal range of Kerr-de Sitter spacetimes, by removing the restrictions on the angular momentum of the black hole. The proof is based on a new Fredholm setup and a new analysis of the trapping of photons around a Kerr-de Sitter black hole. This is joint work with Andras Vasy.
Fabienne Klatt (24 January, Room 503)
Title: The Formation of Black Holes and Singularities in Spherical Symmetric Gravitational Collapse
Abstract: This talk will be about the paper with the title stated above that Demetrios Christodoulou published in 1991. In this work, Christodoulou studies solutions to the Einstein Vacuum Equations in spherical symmetry und proved that under certain assumptions on the mass and radii of spheres in an annular region of the initial future cone, a trapped sphere will form in the future. After Penrose's theorem, a trapped sphere is an indicator for geodesic incompleteness or closed timelike curves. With the spherical symmetry and the curvature assumptions, we will derive a metric and a system of equations containing information about mass, radii and curvature and use this in order to derive the desired result with some estimates from monotonicity arguments.