Abstracts

The aim of this talk is to present recent results related to a notion of energy associated to fourth-order gravitational theories and its relation with \(Q\)-curvature analysis. The \(Q\)-curvature is a fourth order curvature object, which has played a central role in many problems in geometric analysis in recent years, specially within conformal geometry. During this talk, we shall focus on an asymptotic invariant for asymptotically Euclidean manifolds, motivated as a conserved quantity canonically associated to perturbations of solutions of certain fourth-order space-time equations possessing a time-like Killing field. We shall present a positive energy theorem associated with it, which is deeply connected to \(Q\)-curvature analysis, underlying positive mass theorems for the Paneitz operator as well as several rigidity phenomena associated to \(Q\)-curvature. Parallels between the role played by scalar curvature analysis in the usual positive mass theorem of general relativity and \(Q\)-curvature analysis in this higher order case will be highlighted.

I am studying static solutions to the spherically symmetric Einstein-Dirac system. This system couples Einstein's theory of general relativity to Dirac's relativistic description of quantum mechanics. The goal is to study the transition from a quantum mechanical description to a classical description by comparing properties of the solutions to the Einstein-Dirac system to solutions of the Einstein-Vlasov system as the number of particles of the former system increases. In 1999 Finster et al. found for the first-time static solutions to the Einstein-Dirac system in the case of two fermions with opposite spins. Recently this study has been extended to a larger number of particles by Leith et al. In particular, they construct highly relativistic solutions. The structure of the solutions is strikingly similar to the structure of highly relativistic solutions of the Einstein-Vlasov system. In both cases multi-peak solutions are obtained, and moreover, the maximum compactness of the solutions is the same. The compactness is measured by the quantity m/r, where m is the mass and r the areal radius, and in both cases the maximum value is 4/9. In order to compare the solutions, I need to construct solutions numerically to the Einstein-Dirac system in the case of a large number of particles. Which requires a delicate procedure with significant numerical precision when the number of particles in the system grows. This is a joint work with Håkan Andréasson.

It is expected that black hole singularities can be resolved within the framework of quantum gravitational theory. However, the question remains whether it is possible to achieve the same goal using some kind of classical matter fields. In this regard, we consider nonlinear extensions of Maxwell’s Lagrangian, collectively called nonlinear electrodynamic theories. The first systematic approach to the problem was presented by Bronnikov, whose no-go theorems are formulated for spherically symmetric spacetimes sourced by nonlinear electromagnetic Lagrangians depending on one electromagnetic invariant. We generalize Bronnikov’s analysis by taking into account Lagrangians that depend on both electromagnetic invariants. The obtained results significantly narrow down the possibility of regularization using physically plausible Lagrangians. [1] A. Bokulić, T. Jurić, and I. Smolić, Phys.Rev.D 106, 064020 (2022)

I will present the theory of Kerr geodesics. Based on a Weierstrass-Biermann theorem, I will derive two formulas, one for radial motion and one for altitude motion, both describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. These will be a generalisation of last year's method, which describes geodesics in the Schwarzschild metric. Two single expressions work remarkably for an entire geodesic trajectory, whether equatorial or not, even if it passes through turning points. With their help, I will show how to derive expressions for the azimuthal angle and coordinate time along the geodesic.

In this joint work with Cederbaum, Leandro and Dos Santos, we generalize to any dimension n+1 Robinson’s divergence formula used to prove the uniqueness of (3+1)-dimensional static black holes. To this end, we use a tensor first introduced by Cao and Chen for the analysis and classification of Ricci solitons. We thereby prove the uniqueness of black holes and of equipotential photon surfaces in the class of asymptotically flat (n+1)-dimensional static vacuum space-times, provided the total scalar curvature of the horizon is properly bounded from above. In the black hole case, our results recover those of Agostiniani and Mazzieri and partially re-establish the results by Gibbons, Ida, and Shiromizu, and Hwang and finally by Raulot in the case of a spin manifold; in the photon surface case, the results by Cederbaum and Galloway can also be proven. Our proof is not based on the positive mass theorem and avoids the spin assumption.

In this talk, I will discuss the Lie group structure of asymptotic symmetries in General Relativity. To this end, I will first introduce different approaches to asymptotically flat spacetimes and then present the notion of an infinite-dimensional Lie group. Next, I will introduce two prominent such symmetry groups, the Bondi--Metzner--Sachs (BMS) group and the Newman--Unti (NU) group. In particular, I will highlight the following new results from a recent collaboration with Alexander Schmeding: The BMS group is regular in the sense of Milnor, satisfies the Trotter property as well as the commutator property, but is not real analytic. This motivates us to conjecture that it is not locally exponential. The corresponding situation for the NU group is much more subtle: In a natural coarse topology it becomes only a topological group, lacking a manifold structure. However, in a finer Whitney-type topology the unit component can be turned into an infinite-dimensional Lie group. Interestingly, this implies that the BMS group cannot be embedded into it, contrary to the situation of their Lie algebras. Based on: - Class. Quantum Grav. 39 (2022) 065004; arXiv:2106.12513 [gr-qc] - Class. Quantum Grav. 39 (2022) 155005; arXiv:2109.11476 [gr-qc]

I will begin this short talk by discussing the explicitly-solvable toy-model for the problem of waves propagating outside of an extremal black hole. Next I will show how one can apply the methods developed there to the case of the Reissner-Nordström-anti-de Sitter spacetime. This is a joint work with Claude Warnick.

We analyze asymptotics of silent equations, which are a class of linear wave equations on cosmological spacetimes. Specifically, we obtain asymptotic estimates of all orders for solutions, and show that solutions are uniquely determined by the asymptotic data in these estimates. As an application, we analyze the behavior of solutions to source free Maxwell's equations on Kasner backgrounds near the initial singularity.

I will explain a result obtained in collaboration with P. Hintz and A. Vasy on the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural generalized wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed finite dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory. The restriction to small angular momentum mainly comes from the analysis of mode solutions and I will explain at the end of the talk how this analysis can be carried out also in the case of large angular momentum of the black hole (this last part is based on joint work with L. Andersson and B. Whiting).

Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici '13), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain ‘close’ and dissipate sufficiently fast.

In the traditional setting, spectral asymptotics explores the interplay between the spectral data of geometric differential operators and the underlying Riemannian geometry. In this talk, a general relativistic generalisation of this notion will be presented. In particular, I shall show the Weyl asymptotic on the manifold of lightlike geodesics by generalising the Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact spacetime. (Based on J. Geom. Anal. 33, 57 (2023))

Astrophysical black holes, while usually modelled by the Kerr(-Newmann) metric, are never isolated. In fact, the interaction with surrounding matter is the only way how we observe them. As matter accretes onto the black hole, it often results in the formation of disc-like structures. Despite these structures being of high importance, obtaining their common gravitational field is a challenging task, particularly when searching for analytical solutions. Nonetheless, under assumptions of stationarity (or staticity) and axial symmetry, it can be done – albeit not many explicit solutions are known. Our recent contributions to this area will be presented, including closed-form potentials and the complete (exact) metrics of a black hole encircled by a self-gravitating static disc, along with their basic physical properties. We will also address some challenges of the stationary (i.e., rotating) case, and provide a brief overview of the ongoing work in this direction.

We seek to describe the generic dynamics of Bianchi type VIII and IX models, which are spatially homogeneous and anisotropic. We will see how a specific perturbation (within a modified gravity theory) of such models unravels well-known and brand new dynamical features. In particular, we will show that general relativity is associated with a bifurcation where chaos becomes generic. Moreover, we shall discern when such perturbations yield good or bad approximating schemes of general relativity. These results were fruit of collaborations with K.E. Church (U Montreal), V.H. Daniel (Columbia U), J. Hell (FU Berlin), O. Hénot (McGill U), J.P. Lessard (McGill U), H. Sprink (FU Berlin) and C. Uggla (Karlstads U).

An invariant characterization of a spacetime is a longstanding problem in general relativity. One of approaches is the so-called IDEAL characterization, which consists of a set of tensorial equations built from the Riemann tensor and its concomitants, which are satisfied if and only if the metric is locally isomorphic to a given reference one. In the case of the 4 dimensional Kerr metric, IDEAL characterization was found by Ferrando and Saez. In our work, which is still in progress, we aim to extend their result to the whole family of higher-dimensional Kerr-NUT-(A)dS spacetimes by exploiting the relation between curvature and the special closed-conformal Killing-Yano form. Namely, we have investigated various constraints, and their consequences for the curvature tensor, including an integrability condition of the conformal Killing-Yano equation and the second Bianchi identity. As an intermediate result we have shown how the resulting algebraic conditions bring the Riemann tensor to Kerr-NUT-(A)dS form in 5 dimensions. We are working on extending this finding to higher dimensions.

The generalized Jang equation was introduced by Bray and Khuri in an attempt to prove the Penrose inequality in the setting of asymptotically Euclidean initial data sets for the Einstein equations. Since then it has appeared in a number of arguments allowing to prove geometric inequalities for initial data sets by reducing them to known inequalities for Riemannian manifolds provided that a certain geometrically motivated system of equations can be solved. We will present a novel argument along these lines that could potentially lead to a proof of the positive mass theorem for asymptotically hyperbolic initial data sets modeling constant time slices of asymptotically anti-de Sitter spacetimes. Furthermore, we will show how to construct a geometric solution of the generalized Jang equation in this setting, in the case when the dimension is less than 8 and for very general asymptotics, using methods from geometric measure theory.

To study the interplay of general relativity and quantum theory, experiments to measure gravitationally induced phase shifts for single photons and entangled photon pairs are currently under development. Previous theoretical descriptions of such proposals commonly used ad-hoc insertions of gravitational phase shifts into the formalism of quantum optics in flat space-time. This talk presents a theoretical framework of quantum optics in curved space-times, which allows for a comprehensive study of such setups from first principles.

The study of wave propagation on black hole spacetimes has been an intense field of research in the last few decades. This interest has been driven by the stability problem for black holes and by scattering questions. For Maxwell equations and the equations of linearized gravity on a Kerr background, it is possible to base the analysis on the study of the Teukolsky equation, which has the advantage of being scalar in nature. This linear second-order equation involves a half-integer parameter \(s\) whose value depends on the field under examination: \(\pm 1\) for Maxwell and \(\pm 2\) for linearized gravity. In my presentation, I will present a result that provides the large time leading-order term for initially localized and regular solutions. The strength of this result is that it is valid for the whole subextremal range of black hole parameters and for all values of the parameter \(s\). It generalizes a previous result obtained by Ma-Zhang in the case of a slowly rotating Kerr black hole for \(s=\pm 1\), \(s=\pm2\) and a previous result by Hintz in the case of general subextremal Kerr black holes for \(s=0\). The proof relies on microlocal and spectral methods recently introduced in the context of general relativity. In particular, it makes use of Vasy's non elliptic Fredholm theory, an analysis of trapping due to Dyatlov and Wunsch-Zworski, and a mode stability result for the Teukolsky equation due to Whiting.

A spacetime singularity is called ``strong" if any object hitting it is crushed to zero volume. It is called at least ``locally naked" if it is a past endpoint of some causal curve in the spacetime manifold. We show the existence of a nonzero measured set of initial data that gives rise to such strong, at least locally naked singularity formed as an end state of an unhindered gravitational collapse of a spherically symmetric inhomogeneous perfect fluid. (Phys. Rev. D 101, 044052, 2020).

For gravitational instantons of Petrov type D, we study Ricci-flat perturbations of the metric. Through a Riemannian analog of the Newman–Penrose formalism, we see that the linearized vacuum Einstein equations give rise to equations for the various components of the linearized Weyl curvature. In particular, we see that the equations for the components \(\dot{\Psi}_0^\pm\) decouple, giving rise to a Riemannian version of the Teukolsky equation. This equation is studied in the particular case of the Riemannian Kerr instanton.

Matter structures are among the key observations on large scales. Hence, their formation, or lack thereof, in certain mathematical models provides insight into whether these models are physically valid. In this talk we explore structure formation utilizing the tools of modern PDE analysis. Due to work by D. Christodoulou, it is known that the relativistic Euler equations are unstable for a rather comprehensive class of equations of state on Minkowski space. However, solutions exist globally for exponentially expanding FLRW-type models or even power law expansion, as was explored in the previous decade by J. Speck among many others. In our paper on dust as well as our followup on massive fluids, we prove that the homogeneous fluid solutions coupled to the Milne-universe are fully nonlineary stable solutions to the coupled Euler-Einstein-system with a linear equation of state. In particular, this shows that even linear expansion is sufficient for regularizing dust and massive fluids. This is joint work with David Fajman, Todd Oliynyk and Zoe Wyatt.

We present Hamilton’s equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism necessary to compute Hamilton’s equations in both Weitzenböck gauge and covariant formulation, where for the latter we must introduce new fields: Lorentz matrices and their associated momenta. We also derive explicit relations between the conjugate momenta of the tetrad and the conjugate momenta for the metric that are traditionally defined in GR, which are important to compare both formalisms.

The Hawking energy is one of the most famous local energies in general relativity, by using a Lyapunov-Schmidt reduction procedure we construct unique local foliations of critical surfaces of the Hawking energy on initial data sets. Any quasilocal energy should satisfy the so-called small sphere limit, therefore we also discuss the relation between these surfaces and the small sphere limit. In particular, we discuss some discrepancies on the small sphere limit, so when approaching a point with these foliations and when approaching as in the small sphere limit.

After an overview of the literature concerning the dynamics of big bang formation, the talk will provide a geometric perspective on the underlying mechanisms.

Within the framework of Einstein-Maxwell theory geometric properties of charged rotating discs of dust, using a post-Newtonian expansion up to tenth order, are discussed. Investigating the disc’s proper radius and the proper circumference allows us to address questions related to the Ehrenfest paradox. In the Newtonian limit there is an agreement with a rotating disc from special relativity. The charged rotating disc of dust also possesses material-like properties. A fundamental geometric property of the disc is its Gaussian curvature. The result obtained for the charged rotating disc of dust is checked by additionally calculating the Gaussian curvature of the analytic limiting cases (charged rotating) Maclaurin disc, electrically counterpoised dust-disc and uncharged rotating disc of dust. We find that by increasing the disc’s specific charge there occurs a transition from negative to positive curvature.

A result by Choquet-Bruhat and Geroch (Y. Choquet-Bruhat, R. Geroch, Global Aspects of the Cauchy Problem in General Relativity, Commun. math. Phys. 14, 329335 (1969)) allows us to study the Einstein Equation as a Cauchy problem, describing the evolution of suitably defined initial data. As observed by Chruściel and Tod (P.T. Chruściel, P. Tod, An angular momentum bound at null infinity, Advances in Mathematical and Theoretical Physics 13 (2009) 1317–1334), there exists a simple algebraic transformation mapping initial data with constant mean curvature (CMC) and \(\Lambda<0\) to CMC initial data with \(\Lambda=0\). This correspondence in particular links the hyperbolic slice in Anti-de Sitter (AdS) to the hyperboloidal slice in Minkowski, and it has been used to heuristically transfer knowledge between asymptotically flat and asymptotically AdS Spacetimes. In this talk, I will present results on the effect of this correspondence on the symmetries of two spacetimes originated by a pair of such initial data sets. This is done in vacuum by studying the Killing initial data (KIDs) of two corresponding CMC initial data sets. This work is partly in collaboration with Cederbaum.

Addressing cosmological questions exclusively based on observations requires a formulation on the past lightcone of a cosmic observer. In this talk, the question of how to define gravitational energy associated with the past lightcone of a cosmic observer is studied by introducing Hawking’s quasi-local energy as a tentative energy measure of the observable Universe. I will introduce the Hawking Energy on lightcones, discuss its properties in cosmological spacetimes, and explore its applications.

To understand to which extent cosmological models make sense as an approximation of our universe, it is of particular interest to study their stability within the Einstein equations. In this talk, I will present the global nonlinear stability of Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes with hyperbolic spatial metric \(\gamma\) and in presence of homogeneous scalar field matter. In the contracting direction, we have shown that these spacetimes exhibit stable collapse into a Big Bang curvature singularity. Under a mild additional spectral assumption on \(\Delta_\gamma\), the perturbed solutions also asymptotically approach Milne spacetime in the expanding direction. As a consequence, the Strong Cosmic Censorship conjecture holds for this class of near-FLRW solutions in the \(C^2\)-sense. This is based on joint work with David Fajman.

Identifying any conformally round metric on the \(2\)-sphere with a unique cross section on the standard lightcone in the \(3+1\)-Minkowski spacetime, we gain a new perspective on \(2d\)-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch--Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from \(2d\)-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we obtain a new proof of Hamilton's classical result using only the maximum principle.

The (in-)extendibility of a spacetime is a recurrent topic well known to play a central role in General Relativity. In general, it focusses on proving the inextendibility of a spacetime in a certain regularity (and/or possibly symmetry) class or to explicitly construct an extension of this spacetime. However, uniqueness of these extensions still remained uncharted territory. Recently, Sbierski tackled the issue of (local) uniqueness of low-regularity spacetime extensions at the boundary. Nevertheless, is it possible to retain any notion of uniqueness on a more global scale? I will present joint work with Melanie Graf on the existence of a unique maximal future boundary for \(C^{2}\)-extensions. Furthermore, this can be seen as a first step in order to obtain a global uniqueness result for low regularity spacetime extensions. Our techniques make use of ideas that stem from the maximal Cauchy development proof by Choquet-Bruhat and Geroch using Zorn’s lemma as well as from the work on maximal conformal boundaries by Chrusciel and are reminiscent of the g-boundary construction going back to Geroch.

We consider the Yang-Mills field propagating on the Einstein Universe and study the long-time behaviour of small solutions using weakly non-linear perturbation theory. We derive the resonant system, find an explicit formula for the interaction coefficients and study its properties, including conserved quantities. Furthermore, we find an invariant manifold of the resonant system and analyze motion on this manifold. Finally, we conclude that, for generic data, solutions have bounded higher Sobolev norms.