A collapse singularity theorem permitting achronologies

A new collapse-type singularity theorem is formulated. The premise of the theorem involves various semi-global geometric conditions whose purpose it is to represent a region of strong gravity that is surrounded by an event horizon. These conditions, though relatively specific, may be argued to be natural from what is known of black holes and the various trapped surfaces associated with them. The restriction on causal structure is that the relevant region be marginally less pathological than non-totally vicious. This is weaker than the chronology assumption of the Hawking-Penrose theorem though comparable to that used in recent strengthened versions thereof. The method of proof is different in that it makes use of recent work on causal structure by Minguzzi, which, in particular, permits bypassing the generic timelike and or null conditions. Another significant feature is that the theorem's conclusion permits locating future incomplete null geodesics within the strong gravity region. An important role is also played by Minguzzi's proposal for the boundary of a chronology violating set.

A simple recipe to generate exact physically acceptable anisotropic solutions in general relativity

By using gravitational decoupling though the minimal geometric deformation approach (MGD-decoupling), we show a simple and powerful recipe to generate physically acceptable analytical solutions for anisotropic stellar distributions in general relativity.

Asymptotics properties of the small data solutions of the Vlasov-Maxwell system in high dimensions

The Vlasov-Maxwell system is a classical model in plasma physics. Glassey-Strauss proved global existence for the small data solutions of this system under a compact support assumption on the initial data. They also established optimal decay rates for these solutions but not on their derivatives.
We present here how vector field methods, developped by Christodoulou-Klainerman ([CK]) for the Maxwell equations (in 3d) and, more recently, by Fajman-Joudioux-Smulevici ([FJS]) for the Vlasov equation, can be applied to revisit this problem. In order to adapt the results of [CK] in high dimensions, and then obtain the optimal pointwise decay estimates on the null components of the electromagnetic field, we study the Vlasov-Maxwell system in the Lorenz gauge. We extend the techniques of [FJS] as we do not use a hyperboloidal foliation (and we then do not need any compact support assumption in space on the initial data) thanks to a new decay estimate for the velocity average of the Vlasov field. It allows us, by making crucial use of the null properties of the system, to remove all compact support assumptions on the initial data and to obtain optimal decay rates for the derivatives of the solutions. The work on the 3d case is in progress.

Bifurcating solutions of Rotating-Bowen-York initial data with a positive cosmological constant

A generalization of the Bowen–York initial data to the case with the positive cosmological constant is investigated numerically.  We follow the construction presented recently by Bizoń, Pletka and Simon, and solve numerically the corresponding Lichnerowicz equation on a compactified domain S1 × S2 . We find and describe new solutions, bifurcating from those discovered by Bizoń et al . We provide numerical arguments suggesting the absence of additional branches of solutions.

Butscher's perturbative method for the construction of initial data sets

The system of Conformal Constraint Equations (CCEs) of H. Friedrich [3] others a promising alternative to the standard Conformal Method for the construction of initial data for the Cauchy problem in GR. As a first step to their analysis, one must first understand the Extended Constraint Equations (ECEs), which can be thought of as a reduction of the CCEs corresponding to a trivial conformal rescaling. While much simpler than the full system, much of the intricate structure of the CCEs is already evident at the level of the ECEs.
In this talk I will outline the pertubative method of A. Butscher [1, 2] for the construction of solutions of the ECEs, with emphasis on its application to the case of closed initial hypersurfaces. Such solutions, constructed as non-linear perturbations of a given \background" initial data set, describe initial data with prescribed extrinsic mean curvature, and prescribed TT parts of the electric and magnetic Weyl curvature. I will give suffcient conditions for the implementation of the method, and some examples of admissible background initial data.
Time permitting, I will describe progress made towards extending the analysis to the full Conformal Constraint Equations.

[1] A. Butscher, Exploring the conformal constraint equations, in The conformal structure of spacetime: Geometry, Analysis, Numerics, edited by J. Frauendiener & H. Friedrich, Lect. Notes. Phys., page 195, 2002.
[2] A. Butscher, Perturbative solutions of the extended constraint equations in General Relativity, Comm. Math. Phys. 272, 1 (2007).
[3] H. Friedrich, Cauchy problems for the conformal vacuum field equations in General Relativity, Comm. Math. Phys. 91, 445 (1983).

Cauchy characteristic matching in numerical relativity

An important aspect of a simulation in numerical relativity is the extraction of the emitted gravitational waves, which is non-trivial since the simulation is on a physical domain of finite extent but gravitational waves are unambiguously defined only at future null infinity. There are a number of methods for waveform estimation, but only in characteristic extraction is the waveform calculated at scri+.

We present a new algorithm and implementation of characteristic extraction. It has the key feature of being simply extendible to characteristic matching, in which the characteristic evolution provides outer boundary data for the "3+1" simulation. The key advantage of characteristic matching is that it would lead to a significant speed-up in the time required to complete a numerical simulation.

Colliding circular polarised plane waves

I will introduce a generalisation of the Szekeres class of colliding plane wave solutions obtained by means of an inverse scattering method for the hyperbolic Ernst equation. Instead of the collinear polarisation of the waves within the Szekeres class, the incoming waves feature a monotonically increasing phase angle in their characteristic Weyl tensor component and thus can be regarded as "circular polarised". These waves can occur with two different helicities, and interestingly the interaction properties differ dramatically for collisions of waves with the same or respectively opposite helicities. The class seams also to admit a limit of "circularly polarised impulsive waves".

Electrically charged black hole on AdS3: scale invariance and the Smarr formula

The Einstein-Maxwell theory with negative cosmological constant in three spacetime dimensions is considered. It is shown that the Smarr relation for the electrically charged BTZ black hole emerges from two different approaches based on the scaling symmetry of the asymptotic behaviour of the fields at infinity. In the first approach, we prove that the conservation law associated to the scale invariance of the action for a class of stationary and circularly symmetric configurations, allows to obtain the Smarr formula as long as a special set of holographic boundary conditions is satisfied. This particular set is singled out making the integrability conditions for the energy compatible with the scale invariance of the reduced action. In the second approach, it is explicitly shown that the Smarr formula is recovered through the Euler theorem for homogeneous functions, provided the same set of holographic boundary conditions is fulfilled.

Electromagnetic and gravitational Hopfions<

Hopfions are a family of `solitonary’ field solutions which have non-trivial topological structure. I will focus on two physical applications of Hopfions: electromagnetism and linear gravitation. I will show that electromagnetic ( or linearized gravity) field can be quasi-locally described in terms of complex scalar field. New definition of topological charge for linearized gravity will be presented. Using Hopfion solution, I will discuss problem of energy in linearized gravitation.

Energy in higher-dimensional Spacetimes

Based on the geometric Hamiltonian formalism we derive new expressions for the total Hamiltonian energy of gravitating systems in higher dimensions in terms of the Riemann tensor, allowing a cosmological constant. In this way, we also obtain the ADM and the Komar expressions of the corresponding spacetimes. We then apply this analysis to a large class of higher dimensional spacetimes with various asymptotics known to us, which satisfy certain conditions. In particular, our analysis covers asymptotically flat spacetimes, Kaluza-Klein asymptotically flat spacetimes, as well as asymptotically (anti-)de Sitter spacetimes. As it turns out, the Komar mass equals the ADM mass in stationary asymptotically flat spacetimes in all dimensions, generalizing the four-dimensional result of Beig. We show that in general, the Hamiltonian mass, the ADM mass and the Komar mass do not coincide with each other in the non-asymptotically flat setting.
This talk is based on my master's thesis under supervision of Prof. P. T. Chrusciel.

Energy- and angular momentum-like Noether currents for the Teukolsky Master Equation

Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime
are constructed for the Teukolsky Master Equation (TME). Three partly different approaches are discussed,
in which three variants of Noether's theorem are applied. The three approaches give essentially the same results, nevertheless. The first approach provides an example of the application of an extension of Noether's theorem to nonvariational differential equations. The variant of Noether's theorem applied in the third approach is a generalization of the standard construction of conserved currents associated with spacetime symmetries in general relativity, in which the currents are obtained as the contraction of the symmetric energy-momentum tensor with the relevant Killing vector fields. The constructed currents involve two independent solutions of the TME with opposite spin weights.

Ernst Mach and Machian effects in general relativity

I shall start with some remarks on Ernst Mach (1838-1916), who spent many years at the University in Prague and at the Vienna University. I briefly recall his and Einstein’s ideas on the origin of inertia and their influence on the construction of general relativity. I mention the direct experiment verifying relativistic dragging/gravitomagnetic effects - the Gravity Probe B; the results were summarized only recently. I shall then turn to several specific general-relativistic problems illustrating the gravitomagnetic effects: the dragging of particles and fields around a rotating black hole, dragging inside a collapsing slowly rotating spherical shell of dust, linear dragging in a static situation, and the way how Mach’s principle can be formulated in cosmology. A more detailed discussion will be devoted to the dragging effects by rotating gravitational waves.

Expanding Singularities and C^0-(in)extendibiltiy

We introduce the concept of an "expanding singularity" in a globally hyperbolic spacetime, characterised by asymptotic blowup of the (Riemannian) diameter of certain subsets of Cauchy hypersurfaces. Such singularities are present in Kasner and Gowdy spacetimes for example. We show that C^0 extensions across expanding singularities must have a boundary that is non-compact and null almost everywhere.

Gauge invariant description of weak gravitational field on a spherically symmetric background

We construct a formalism for analysis of the true degrees of freedom in the linearized Cauchy problem for the Einstein equation on a spherically symmetric vacuum background with cosmological constant. Our approach does not require an a priori splitting of the data into spherical harmonics, but is compatible with it. Some motivaton for our method and several simple applications will also be presented.

Gödel, Brno, time, universe

The lecture reminds Gödel’s youth spent in Brno. Then the lecture will focus on his contribution to physics. I will show how his interest in solving Einstein’s equations naturally stemmed from his interest in understanding the nature of time. It will also show how the results enriched general relativity and cosmology.

Gravitational multipole moments from Noether charges

We define the mass and current multipole moments for an arbitrary theory of gravity in terms of canonical Noether charges associated with specific residual transformations in canonical harmonic gauge, which we call multipole symmetries. We show that our definition exactly matches Thorne's mass and current multipole moments in Einstein gravity. For radiative configurations, the total multipole charges -- including the contributions from the source and the radiation -- are given by surface charges at spatial infinity, while the source multipole moments are naturally identified by surface integrals in the near-zone or, alternatively, from a regularization of the Noether charges at null infinity. The conservation of total multipole charges is used to derive the variation of source multipole moments in terms of the radiative multipole fluxes.

Killing initial data on the timelike conformal boundary on anti-de Sitter-like spacetimes

By making use of the conformal Einstein field equations, we analyse initial-boundary value problems on anti de Sitter-like spacetimes ensuring the existence of Killing vectors via a homogeneous system of wave equations. In this problem, both the conformal boundary and the initial hypersurface are analysed with the help of the confromal constraint equations via a 3+1 formalism; this leads to an analogous problem on the boundary. Adopting a particular gauge on the conformal boundary, an obstruction to the existence of a Killing vector is found. The boundary data is made consistent with those at the initial hypersurface by means of appropriate corner conditions.

Lorentzian length spaces

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The role of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity. This is joint work with Michael Kunzinger. Preprint: https://arxiv.org/abs/1711.08990

Non-linear perturbations of Schwarzschild with the conformal field equations

The aim of this talk is to introduce an initial boundary value problem (IBVP) for the generalised conformal field equations (GCFEs), and as an application study gravitational perturbations of a black hole space-time. 

The main ideas of the GCFEs will be summarised and issues associated with forming a well posed (at least numerically) IBVP will be discussed. A framework for the IBVP will be presented and numerical evidence of its success will be given. As an application I will discuss a gravitational wave impinging on a Schwarzschild black hole. In particular, I will discuss how the IBVP is formulated for this situation and how to calculate global properties directly at future null infinity.

Non-minimal Einstein-Maxwell theory: the Fresnel equation and the Petrov classification of a trace-free susceptibility tensor

Title: Non-minimal Einstein-Maxwell theory: the Fresnel equation and the Petrov classification of a trace-free susceptibility tensor

We construct a classification of dispersion relations for the electromagnetic waves non-minimally coupled to the space-time curvature, based on the analysis of the susceptibility tensor, which appears in the non-minimal Einstein-Maxwell theory. We classify solutions to the Fresnel equation for the model with a trace-free non-minimal susceptibility tensor according to the Petrov scheme. For all Petrov types we discuss specific features of the dispersion relations and plot the corresponding wave surfaces.

Numerical studies of a hyperbolic formulation of the constraint equations

In the framework of General Relativity, the formulation of the initial value problem is given in terms of two systems of PDEs, namely evolution and constraints equations, which have to be solved simultaneously during the whole evolution. Initial data sets are given as solutions of the second system. However, because of the non-linear nature of this set of PDEs, their numerical and analytical treatment is, in general, a complex task.

In this talk we will consider a hyperbolic formulation of the constraints equations. Using a pseudo-spectral approach based on spin-weighted spherical harmonics, we will construct initial data sets which can be interpreted as nonlinear perturbations of a Schwarzschild initial data in Kerr-Schild coordinates. Our numerical results suggest that generic initial data sets obtained with this method may violate fundamental asymptotic conditions.

On Lie derivation of spinors against arbitrary tangent vector fields

It is well-known since the work of Lichnerowicz and Kosmann in the '70-s, that the notion of Lie derivation along conformal Killing vector fields can be uniquely lifted to the two-spinor bundle over a four dimensional spacetime manifold, via requiring compatibility with the Infeld-Van der Waerden symbol and the preservation of the complex phase of the spinor maximal form. That formula, also called Kosmann Lie derivation, is known to provide a faithful Lie algebra representation of the conformal Killing vector fields within the Lie algebra of the derivation operators on spinor fields. This naturally motivated the question: can this notion be generalized to arbitrary tangent vector fields of the spacetime, which are not necessarily conformal Killing? Lot of discussion appeared in the literature on this and related questions, and are expanded in a large set of papers of Forgach, Manton, Kolar, Michor, Slovak, Godina, Jhangiani, Hurley, Vandyck etc throughout the years. Even nowadays papers appear on the matter. Unfortunately, it seems that the confusion in the literature about the possible generalization of the spinor Lie derivation to arbitrary tangent vector field still persists. Motivated by this, we managed to find a compact answer via the more generic approach of Lie derivations over arbitrary vector bundles. The key finding, usually not mentioned in the literature, is that the Kosmann formula fails to be a Lie algebra homomorphism for tangent vector fields which are not conformal Killing. It shall be shown, this does not mean an interpretational problem, but rather signifies that one needs to allow also a vertical part of the perninent Lie derivation. It will be shown that such notion of spinor Lie derivation is uniquely characterized by preserving the following structures: the vector bundle structrure of the spinor bundle and the solderability of the spinor bundle to the tangent bundle, but not preserving the soldering form, for instance. It also shall be shown that it can serve as a natural unified framework to encode spacetime and internal (gauge) symmetries.

On the Bartnik mass of CMC Bartnik data

We study the Bartnik mass in asymptotically flat Riemannian 3-manifolds with inner boundary and non-negative scalar curvature. The Bartnik mass is an important notion of "local mass" of the inner boundary, although it is notoriously difficult to compute. The problem of computing it can be rephrased as an extension problem for Bartnik data, i.e., Riemannian 2-surfaces with mean curvature H.

Recently, C. Mantoulidis and R. Schoen constructed asymptotically flat extensions of Bartnik data with H=0 allowing them to compute their Bartnik mass. We will describe how to adapt their ideas to construct extensions and obtain estimates for the Bartnik mass of Bartnik data with H a positive constant. In addition, we will discuss a Bartnik mass analog in the context of asymptotically hyperbolic manifolds, construct extensions and prove the corresponding estimates. This talk is based on joint projects with C. Cederbaum, S. McCormick, and P. Miao.

On the propagation of gravitational waves in a \LambdaCDM universe

We study how the presence of non-zero matter density and a cosmological constant could affect the observation of gravitational waves in Pulsar Timing Arrays. Conventionally, the effect of matter and cosmological constant is included by considering the redshift in frequency due to the expansion. However, there is an additional effect due to the change of coordinate systems from the natural ones in the region where waves are produced to the ones used to measure the pulsar timing residuals. This change is unavoidable as the strong gravitational field in a black hole merger distorts clocks and rules. Harmonic waves produced in such a merger become anharmonic when detected by a cosmological observer. The effect is small but appears to be observable for the type of gravitational waves to which PTA are sensitive and for the favoured values of the cosmological parameters.

Quasinormal modes of black branes

I review linear perturbations o five dimensional Schwarzschild-AdS black branes. I present an approach to this subject based on work by Rostworowski (10.1103/PhysRevD.96.124026) and compare it with known results.

Some results on geometric inequalities in spherically symmetric spacetimes

Recent papers [1,2,3,4] on geometric inequalities present results not only on black holes but on normal bodies too. In spherical symmetry there is a highly accepted quasi-local notion of mass: the Misner-Sharp mass. Utilizing this notion it is possible to examine geometric inequalities on any SO(3) invariant surface on generic spherically symmetric spacetimes. My talk is to present results on trapped, marginally trapped and untrapped surfaces and based on the paper [5].

The talk was supported by OTKA grant K115434.

[1] Dain, S. and Jaramillo, J. L. and Reiris, M.,Area-charge inequality for black holes,
Classical and Quantum Gravity, vol. 29, 2012, 035013

[2] Khuri, M. A., Inequalities Between Size and Charge for Bodies and the Existence of Black Holes Due to Concentration of Charge,
Journal of Mathematical Physics, vol. 56, 2015, 11, 112503

[3] Reiris, Martin, On the shape of bodies in General Relativistic regimes,
Gen.Rel. Grav., vol. 46, 2014, 1777

[4] Anglada, Pablo and Dain, Sergio and Ortiz, Omar E., Inequality between size and charge in spherical symmetry,
Phys. Rev., vol. D93, 2016, 4, 044055

[5] Csukás, Károly Zoltán, Geometric inequalities in spherically symmetric space-times,
Gen. Rel. Grav., vol. 49, 2017, 7, 94

The Fingerprints of Black Holes - Shadows and their Degeneracies

First I will introduce the concept of the shadow of a black hole and what it means for the shadows of two observers to be degenerate. I will then present preliminary results showing that no continuous degenerations exist between the shadows of observers at any point in the exterior region of any Kerr-Newman black hole spacetime of unit mass. Therefore an observer can, by measuring the black holes shadow, in principle determine the angular momentum and the charge of the black hole under observation, as well as his radial distance from the black hole and his angle of elevation above the equatorial plane.

The weak-null condition and Kaluza-Klein spacetimes

In string theory, our most developed theory of quantum gravity to date, one is interested in manifolds of the form $R^{1+3} \times K$ where $K$ is some $n-$dimensional compact Ricci-flat manifold. In the first and simplest case considered by Kaluza and later Klein, $K$ is the $n-$torus with the flat metric. An interesting question to ask is whether this system, viewed as an initial value problem, is stable to small perturbations of the initial data. Motivated by this problem, I will outline the proof of stability in a restricted class of perturbations, and discuss the physical justification behind this restriction. Furthermore the resulting PDE system exhibits the weak-null condition, and so I will discuss how it can be treated by generalising the proof of the non-linear stability of Minkowski spacetime given by Lindblad and Rodnianski.

Weak field dynamics in the Schroedinger-Poisson system with a harmonic potential

I will discuss the resonant approximation for the Schroedinger-Poisson system with a harmonic potential (SPh) which models  weak field dynamics of a self-gravitating Bose-Einstein condensate in a harmonic trap. The SPh system can be viewed as a nonrelativistic limit of the Einstein-Klein-Gordon equations with negative cosmological constant (EKG-AdS) so it is natural to ask which features observed for the  EKG-AdS system (most notably, weakly turbulent instability of AdS) persist in the limit. This is joint work with Piotr Bizon.