Relativistic low angular momentum accretion

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Written by Michal Piróg

I will report on the results of the joint numerical project with J.A. Font and P. Mach. We investigated low angular momentum accretion of inviscid fluids on black holes. The Newtonian calculation in this topic have been already done by D. Proga and M. Begelman. Our work promotes their models to fully relativistic setting. The staring point of those simulations is the Bondi-type accretion solution, perturbed by adding a small amount of angular momentum. The results of simulations will be discussed, emphasizing the similarities and differences with Newtonian models.

 

Causality for nonlocal phenomena

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Written by Tomasz Miller

The talk will be based on a joint work with M.Eckstein [1], in which we propose and study the extension of the causal precedence relation onto the space of Borel probability measures on a given spacetime M. I will present several conditions on what it might mean that one measure causally precedes the other, and show they are all equivalent if a spacetime has a sufficiently robust causal structure. I will also introduce the suitable extension of the Lorentzian distance between measures and discuss possible applications of the developed formalism in the study of causality in quantum theory. [1] M.Eckstein, T.Miller, Causality for nonlocal phenomena, arXiv:1510.06386

 

An extension of the spacetime symmetry group and its relation to SUSY

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Written by Andras Laszlo

In this talk a nontrivial extension of the spacetime symmetry group is presented. It significantly differs from the one applied in SUSY as the generators of the extended part couple to the generators of the conformal Lorentz group rather than to the generators of translations. When acts at spacetime points this group is comprised by a semi-direct product of a nilpotent normal subgroup N and that of the product U(1) x conformal Lorentz group. As a result N combines the U(1) gauge symmetry and the spacetime symmetry action, circuimventing thereby the Coleman-Mandula no-go theorem due to the presence of the nilpotent part N. (Main results are based on arXiv:1512.03328 and arXiv:1507.08039)

 

Relativistic Bondi-Michel accretion: global vs. homoclinic solutions

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Written by Patryk Mach

A spherically symmetric accretion model introduced by Bondi in 1952 belongs to classical textbook models of theoretical astrophysics. Its general relativistic version is due to Michel, who considered spherically symmetric, purely radial, stationary flow of perfect fluid in the Schwarzschild spacetime. Solutions of the Bondi-Michel flow are usually parametrized by fixing asymptotic values of the density and the speed of sound at infinity; they extend smoothly from infinity up to the horizon of the black hole (and below). In contrast to that, local solutions, that cannot be extended to infinity, were recently discovered in the cosmological context. They correspond to homoclinic orbits on phase diagrams of the radial velocity vs. radius (say). More surprisingly, they also appear in the standard Bondi-Michel model for polytropic fluids with polytropic exponents larger than 5/3. In this talk I will discuss recent results on the existence of those local, homoclinic solutions.

 

Non-chaotic vacuum singularities without symmetries

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Written by Paul Klinger

The BKL conjecture proposes a detailed description of the generic asymptotic dynamics of spacetimes as they approach a spacelike singularity. It predicts complicated chaotic behaviour in the generic case, but simpler non-chaotic one in cases with symmetry assumptions or certain kinds of matter fields. Here we use Fuchsian methods to construct a new class of four-dimensional vacuum spacetimes containing spacelike singularities which show non-chaotic behaviour. In contrast with previous constructions, no symmetry assumptions are made. Analogous solutions exist for the case of non-zero cosmological constant and with timelike instead of spacelike singularity.

 

Broken causal lens rigidity: Reconstructing Lorentzian manifolds from geodesic data

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Written by Eric Larsson

Lens rigidity problems concern reconstructing a manifold from data about its geodesics. These and similar problems have been extensively studied for Riemannian manifolds, but there are not as many results for Lorentzian manifolds.
I will speak about the following problem: Suppose that we wish to determine the topology, smooth structure and metric of a Lorentzian manifold with boundary, and suppose that we are allowed to send in probes which follow timelike geodesics and which broadcast their proper time along lightlike geodesics. Can we determine the manifold up to isometry by observing these signals at the boundary? I will discuss a method of solving this and similar problems. One of the features of the method is that it does not require an explicit construction of coordinates.

 

Separability of test fields equations on the C-metric background

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Written by David Kofron

In the Kerr-Newman spacetime the Teukolsky master equation, governing the fundamental test fields, is of great importance. We derive an analogous master equation for the rotating charged C-metric which encompasses a massless Klein-Gordon field, neutrino field, Maxwell field, Rarita-Schwinger field and gravitational perturbations.This equation is shown to be separable in terms of “accelerated spin-weighted spherical harmonics.” It is shown that, contrary to ordinary spin-weighted spherical harmonics, the “accelerated” ones are ;different for different spins. In some cases, the equations for eigenfunctions and eigenvalues are explicitly solved.

 

Mode stability of Kerr revisited

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Written by Claudio Paganini

Whitings proof of mode stability plays a central role in the black hole stability problem. We give a simplified proof of mode stability for complex frequencies which applies to fields of arbitrary spin, in particular half integer spin.

Joint work with: Siyuan Ma

 

Null geodesics in Kerr

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Written by Blazej Ruba

Joint work with: Claudio Paganini

We consider null geodesics in the domain of outer communication of a sub-extremal Kerr spacetimes. We will show, that most fundamental properties of null geodesic motion can be read off from a plot of the radial pseudo potential. In particular one can see immediately that the ergoregion and trapping are separated in phase space.

 

Hamiltonian dynamics in asymptotically Kerr spacetimes. Quasilocal mass in Kerr spacetime.

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Written by Tomasz Smolka

We present a variational formula which lead to the Hamiltonian dynamics in the space of general-relativistic initial data sets with asymptotically Kerr ends. In Kerr spacetime, we examine particular choice of surfaces: rigid spheres (with constant external curvature) and round spheres (with constant intrinsic two-dimensional curvature).

 

Killing spinors as a characterisation of rotating black hole spacetimes

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Written by Michael J. Cole

We investigate the implications of the existence of Killing spinors in spacetimes. We show that in vacuum and electrovacuum a Killing spinor, along with some assumptions on the associated Killing vector in an asymptotic region, guarantee that the spacetime is locally Kerr or Kerr-Newman. We show that the characterisation of these spacetimes in terms of Killing spinors is a alternative expression of characterisation results of Mars (Kerr) and Wong (Kerr-Newman) involving restrictions on the Weyl curvature and matter content.

 

Algorithmic characterization of the Kerr-NUT-(A)dS family

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Written by Tim Paetz

One of the most important families of exact solutions to Einstein's vacuum field equations is provided by the Kerr space-time. Usually the Kerr space-time, or, more general, the Kerr-NUT-(A)dS space-time, is given in Boyer-Lindquist-type coordinates, which are adapted to the stationary and axial symmetries of this family. In some situations, though, one may have to deal with coordinate systems which do not reflect these symmetries. In such cases it is convenient to have a gauge-invariant characterization at hand.

Such a characterization has been given by Mars and Senovilla, based on the so-called Mars-Simon tensor. This approach requires the existence of a Killing vector field. To check whether a space-time, given in arbitrary coordinates, belongs to the Kerr-NUT-(A)dS family, one therefore needs to solve the Killing equation first.

In this talk we modify their approach to obtain an algorithmic characterization of the Kerr-NUT-(A)dS family, which avoids the need of solving PDEs. We will do that on a space-time level, and on the level of a Cauchy surface.

 

Weak cosmic censorship, dyonic Kerr-Newman black holes and Dirac fields

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Written by Gabor Zsolt Toth

It was investigated recently, with the aim of testing the weak cosmic censorship conjecture, whether an extremal Kerr black hole can be converted into a naked singularity by interaction with a massless classical Dirac test field, and it was found that this is possible. We generalize this result to electrically and magnetically charged rotating extremal black holes (i.e. extremal dyonic Kerr-Newman black holes) and massive Dirac test fields. We show that the possibility of the conversion is a direct consequence of the fact that the Einstein-Hilbert energy-momentum tensor of the classical Dirac field does not satisfy the null energy condition, and is therefore not in contradiction with the weak cosmic censorship conjecture. We give a derivation of the absence of superradiance of the Dirac field in dyonic Kerr-Newman background without making use of the complete separability of the Dirac equation, and we determine the range of superradiant frequencies of the scalar field. The range of frequencies of the Dirac field that can be used to convert a black hole into a naked singularity partially coincides with the superradiant range of the scalar field.

reference: arXiv:1509.02878 [gr-qc]

 

The Yang-Mills fields on curved space-times

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Written by Sari Ghanem

I will present the proof of the non-blow up of the Yang-Mills curvature on arbitrary curved space-times using a Kirchoff-Sobolev type representation formula derived by Klainerman and Rodnianski, combined with suitable Grönwall type inequalities. While the argument of Chruściel and Shatah requires a control on two derivatives of the Yang-Mills curvature, we can get away by controlling only one derivative instead, and write a new gauge independent proof of the non-blow up of the Yang-Mills curvature on arbitrary, fixed, sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds.

 

Asymptotic behaviour of solutions to the massive Vlasov-Nordstrom system

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Written by Jeremie Joudioux

I will present  a global existence result for the massive Vlasov-Nordstrom system, in dimension 4+1, for small initial data, and will describe the asymptotic behavior of solutions of this system. The method is based on the contruction of commutators for the linear massive transport equation. This techniques of commutators is widely inspired by similar techniques developed by Klainerman for the wave equation, known as the vector fields methods.
After explaining the decay of solutions of the linear massive transport equation, I will explain how this result applies to the massive Vlasov-Nordstrom system in dimension 4+1. I will finally explain how these techniques can be adapted to deal with the 3+1 Vlasov-Nordstrom system.
This work is a collaboration with D. Fajman (Vienna), and J. Smulevici (Orsay).

 

Stability of Anti-de Sitter spacetime for Robin boundary conditions

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Written by Dominika Hunik

We consider the four-dimensional spherically symmetric Einstein-Klein-Gordon equations with cosmological constant Λ and mass square m2=2/3Λ. These equations are well-behaved at the conformal boundary $\scri$ which makes them a good toy model for studying how the stability properties of anti-de Sitter spacetime depend on the boundary conditions at $\scri$. We find that for the Dirichlet and Neumann boundary conditions the dynamics is similar to the massless case, i.e. for small perturbations of AdS of size ε we observe instability against black hole formation on the time scale 1/ε2. However, for the Robin boundary conditions sufficiently small perturbations do not grow which indicates that there is a threshold for instability. We conjecture that the existence of the threshold is due to the fact that the Robin spectrum of linear perturbations of AdS, in contrast to the Dirichlet and Neumann spectra, is not fully resonant.

 

Resonant dynamics and the instability of anti-de Sitter spacetime

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Written by Maciej Maliborski

The resonant approximation for the small perturbations of AdS will be discussed with an emphasis on Einstein-massless-scalar field system in five dimensions. It will be demonstrated that for large class of initial data solution of the resonant system develops an oscillatory singularity in finite time. The quasi-periodic solutions obtained within this approximation will be presented. Finally, comparison of resonant system with fully nonlinear evolution will be given. (Based on arXiv:1506.03519.)

Conformal properties of the Schwarzschild-de Sitter spacetime

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Written by Edgar Gasperin-Garcia

This talk is based on a work in collaboration with Juan Valiente Kroon.

Conformal methods constitute a powerful tool for the global analysis of spacetimes, e.g., in the proof of the global non-linear stability of de Sitter and the semiglobal non-linear stability of the Minkowski spacetime. In this talk we will briefly discuss the conformal Einstein field equations and conformal Gaussian systems. In addition, we will discuss how to use this formalism to pose an asymptotic initial value problem (the initial hypersurface is the conformal boundary) and analyse non-linear perturbations close to the Schwarzschild-de Sitter spacetime in the asymptotic region. In particular, it will be shown that small enough perturbations of asymptotic initial data for the Schwarzschild de-Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.

 

Helical symmetry in general relativity

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Written by Martin Scholtz

In the flat spacetime, helically symmetric solutions represent particles moving periodically along the closed orbits, e.g. two particles orbiting about their common centre of mass. Such solutions exist in Maxwell's theory or in the scalar gravity, where the energy loss by radiation is compensated by introducing the advanced potentials. However, there are fundamental obstacles in defining the notion of helical symmetry in the context of full general relativity, although such solutions are believed to play an important role in numerical simulations of the binary inspiral. In this talk, we present 2-particle helically symmetric solution of linearized Einstein's equations and discuss some of its properties: conditions of equilibrium, asymptotic behaviour and peeling properties of the Weyl tensor and geodesics in helically symmetric spacetimes.

 

Universal spacetimes

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Written by Tomáš Málek

Universal metrics solve, by definition, vacuum equations of all theories of gravitation with the Lagrangian described by any polynomial curvature invariant constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order. Thus, all quantum corrections vanish for these metrics. We will study Weyl type II, III and N universal spacetimes of Lorentzian signature in arbitrary dimension and provide examples of such metrics. We will also discuss metrics of neutral signature. It turns out, in contrast to the Lorentzian case, that universal metrics of neutral signature need not to belong to the Kundt class.

 

Hyperboloidal evolution of the Einstein equations applied to black hole superradiance of a charged scalar field

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Written by Olaf Baake

We study the superradiant behavior of a complex scalar field coupled to the Einstein-Maxwell equations. We write the system in an ADM-like formulation on constant mean curvature slices. The equations are derived in a conformally rescaled spacetime before spherical symmetry is imposed. We show that these equations are regular at infinity and solve them numerically in the massless case. The solutions we produce show superradiant as well as non-superradiant behavior.

 

Constructing highly deformed non-uniform black string solutions

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Written by Michael Kalisch

We construct numerically static non-uniform black string solutions in five and six dimensions by using pseudo-spectral methods. An appropriately designed adaptation of the methods in regard of the specific behaviour of the field quantities in the vicinity of our numerical boundaries provides us with extremely accurate results, that allows us to get solutions with an unprecedented deformation of the black string horizon. Consequently, we are able to investigate in detail a critical regime within a suitable parameter diagram. In particular, we observe three clearly pronounced turning points in the curves of thermodynamic quantities, resulting in a spiral curve in the black string’s phase diagram.

 

Static self-forces in arbitrary dimensions

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Written by Abraham Harte

This talk discusses forces and torques acting on static extended bodies in static spacetimes with any number of dimensions. Non-perturbatively, these results have the same form in all dimensions. Meaningful point particle limits are quite different, however. Such limits are defined and evaluated, resulting in simple "regularization algorithms" which can be used in concrete calculations. In them, self-interaction is shown to be progressively less important in higher numbers of dimensions; it generically competes in magnitude with increasingly high-order extended-body effects. Conversely, we show that self-interaction effects can be relatively large in 1+1 and 2+1 dimensions. The static self-force problem in arbitrary dimensions provides a useful testbed with which to continue the development of general, non-perturbative methods in the theory of motion. Several new insights are obtained in this direction, including a significantly improved understanding of the renormalization process. Much of this also generalizes to the dynamical regime.

 

The integrability issue of the Mathisson-Papapetrou equations

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Written by Georgios Lukes-Gerakopoulos

The motion of a stellar compact object around a supermassive black hole can be approximated by the motion of a spinning test particle in a Kerr spacetime background. The equations of motion of a spinning particle are given by the Mathisson-Papapetrou (MP) equations. The spin of the particle in the MP equations has to be defined with respect to a centroid. A centroid defines the wordline of the particle, and it is set by a spin supplementary condition (SSC). We shall discuss the issues arising from the existence of various SSCs, and focus on the impact they have on the integrability issue of the MP equations.

 

A no-hair theorem for non-isolated black holes

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Written by Norman Gurlebeck

The no-hair theorem states that black holes are entirely characterized by their mass, angular momentum and charge alone. For this result to hold, the black hole must be isolated, i.e., there should be no additional sources of the gravitational field in their neighborhood like accretion disks. However, measurements of the angular momentum of the black hole rely heavily on the existence of such an accretion disk. Naturally, the question arises if the additional matter, say, an accretion disk impedes the suggested tests of the no-hair theorem. I will give a possible formulation of the no-hair theorem for such astrophysical black holes surrounded by matter alongside with a proof for static black holes. The proof employs the source integral formalism, which I review shortly. In the end of my talk, I will elude to some appliciations of this no-hair theorem to certain existence and uniqueness questions in mathematical relativity .

 

Models for Self-Gravitating Photon Shells and Geons

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Written by Maximilian Thaller

In this presentation the massless Einstein-Vlasov system in spherical symmetry will be considered and the existence of static solutions whose matter quantities have bounded support will be proved. To this end we apply a method yielding very detailed information about the solutions at hand, in particular we will see That the matter is arranged in highly relativistic shells of massless particles. Finally the relation of these solutions to geons -a concept introduced in 1955 by John Wheeler as an alternative field-theoretic description of single particles- will be discussed. The results emerged from a collaboration with Håkan Andréasson (Chalmers, Gothenburg) and David Fajman (Univ. of Vienna), and are available on the ArXiV: http://arxiv.org/abs/1511.01290

 

The Positive Mass Conjecture for compact Riemannian manifolds

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Written by Andreas Hermann

The Positive Mass Conjecture for the ADM mass of an asymptotically flat Riemannian manifold has been proved in some special cases (e.g. for manifolds of dimension at most 7 or for spin manifolds) but the general case is still subject to current research. In this talk we introduce a more general notion of mass on compact Riemannian manifolds without boundary. We discuss some properties of this mass which might be useful for a proof of the Positive Mass Conjecture. This is joint work with Emmanuel Humbert.