2016
2015
In this talk, I will present a proposal to test directly effects of quantum field theory in the Earth's spacetime using quantum technologies. Under certain circumstances, real spacetime distortions (such as gravitational waves) can produce observable effects in the state of phonons of a Bose-Einstein condensate. The sensitivity of the phononic field to the underlying spacetime can also be used to measure spacetime parameters such as the Schwarzschild radius of the Earth.
Abstract: We turn away from the idea that the Misner spacetime should be Hausdorff as was already discussed by previous authors. In lieu thereof we allow the notion of a non-Hausdorff spacetime and construct an analytic non-Hausdorff extension of Misner space. On this basis we elucidate the global causal structure of the maximally extended Misner spacetime, with the result that there are two fundamentally different maximal extensions and associated covering spaces. From this we can conclude that there exist two versions of Misner space. Furthermore, we wish to shed some new light on the pathologies, e.g. the quasiregular singularities and CTCs. It turns out that the Misner space is related to the pseudo-Schwarzschild spacetime regarding its properties from a chronological and global point of view. According to this result the pseudo-Schwarzschild cylinder can be regarded as a non-flat generalization of the Misner space. This gives rise to a conjecture which says that 4D Misner space and pseudo-Schwarzschild spacetime are isocausal to each other. Furthermore, we create a new chronology violating spacetime that describes a generalization of the two precedent ones: We derive the pseudo-Reissner-Nordstroem spacetime from the well-known Reissner-Nordstreom spacetime and review our main results in this more general setting.
Abstract: We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius α. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter α. We relate this pattern to the Morse index of the static solution with maximal energy. Using a hyperboloidal approach to the initial value problem, we describe the relaxation to the ground state solution for generic initial data and unstable static solutions for initial data of codimension one, two, and three.
Abstract: We introduce a class of overdetermined systems of partial differential equations of on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equations in differential geometry.
Among them there are: the Ricci soliton equations, the vacuum near-horizon geometry equations in general relativity, special cases of Einstein-Weyl equations and their projective counterparts, equations for homotheties and Killing's equation. We provide explicit examples of generalised Ricci solitons in 2 dimensions, some of them obtained using techniques developed by J.Jezierski. This is joint work with Pawel Nurowski available at arXiv:1409.4179.
Note the 10th Vienna Central European Seminar on Particle Physics and Quantum Field Theory - Inflation and Cosmology, Friday November 28 to Saturday November 29, 2014
Abstract: We report on recent progress in the study of spacetimes where the metric is C^{0,1} (locally Lipschitz continuous) or C^{1,1} (first derivatives locally Lipschitz). In particular, we focuss on existence and regularity of geodesics in the first case and discuss the prospects of proving Hawking's singularity theorem in the second case.
Abstract: We explain how fuzzy geometries in extra dimensions can emerge in standard Yang-Mills gauge theory, based on a geometric version of the Higgs effect. In particular, we discuss the 4-and 6-dimensional squashed coadjoint orbits which were recently found in maximally supersymmetric N=4 SYM. The resulting low-energy fluctuation modes lead to 3 generations of chiral fermions coupled to scalar and gauge fields. The discussion is focused on geometrical and group-theoretical aspects. Talk based on arXiv:1409.1440
Abstract: I will introduce the basic concepts of String theory and show how the quantized string unifies gauge theory and gravity. I will further explain why String Theory requires a ten-dimensional space-time and discuss the concept of compactification of extra dimensions.
Abstract: I discuss the recent claim of experimental verification of an analogue of the Hawking effect.
Abstract: Interaction between the components in galaxy clusters - the galaxies and the gas surrounding the galaxies, the so-called intra-cluster medium - have a variety of effects on the cluster. The gas within the galaxies is compressed and sometimes stripped off. Therefore the galaxies change their morphology and their star formation activity. The intra-cluster gas is enriched by the lost gas from the galaxies, hence it changes the metal content and the temperature. All effects are modelled by simulations on galaxy scales as well as clusters scales. Results of the evolution of various properties (metallicity, gas density, star formation rate, temperature, magnetic fields,... ) are presented.
Abstract: Conformal compactification is a well established tool in GR and many related fields. The model for this construction is often taken to be the Poincare ball model of hyperbolic space. There is a refinement of this idea which reveals the Lie groups and Lie group embeddings behind conformal compactification. These structures at once generalise to the curved setting through the conformal Cartan-tractor calculus (i.e. the natural conformally invariant connection and related objects). This provides a conceptual and calculationally effective way to treat many problems linked to conformal compactification.
Abstract: A method for analysing the evolution of the volume of an inhomogeneous irrotational dust universe is presented. In this framework it is possible to go beyond perturbation theory in a numerical analysis. The results of such computations show that the evolution is strongly affected by inhomogeneities, but nevertheless suggest that a cosmological constant is required to account for the observed acceleration of the expansion. Possible loopholes to this conclusion will be discussed.
Abstract: We construct a class of vacuum space-times without Killing vectors and with "asymptotically velocity dominated" singularities.
Abstract: In 2008 Sergio Dain proved that the ADM mass of axially symmetric, AF initial data is greater or equal than the root of the angular momentum, and equality holds for extreme Kerr (only). We describe recent, stronger inequalities which also contain higher "momenta", focusing on the special case where the data are close to extreme Kerr in a suitable sense.
Abstract: I will describe bifurcation phenomena in thevacuum Lichnerowicz equation with positive cosmological constant on $S^1\times S^2$ with $U(1)\times SO(3)$-invariant seed data.
Abstract: I describe the construction of certain classes of axially symmetric initial data with positive cosmological constant via the conformal method.
Abstract: I will explain how the well-known vector field method, which was one of the most important tool to understand the asymptotic behavior of the wave equation, can also be applied to the Vlasov fields.
Abstract: I will describe ongoing work on the construction of solutions to the collisionless Boltzmann equation on a Kerr black hole background.
Abstract: I will briefly describe my research project on acoustic perturbations of radial accretion flows.
Abstract: We continue our discussion of the geometry and topology of asymptotically flat initial data sets, including discussion of a different approach based on solutions of Jang’s equation.
Abstract: With a distance of about 8 kpc, the center of the Milky Way is the closest galactic nucleus to us. Hence, it provides us with a unique opportunity to study a galactic nucleus up close. Longterm observations of stellar kinematics of the Nuclear Star Cluster point to the existence of a super-massive black hole (SMBH) at the position of Sagittarius A* (SgrA*), with a mass of 4 million suns. SgrA* shows flare emission from the millimeter to the X-ray domain. A detailed analysis of the infrared light curves allows us to address the accretion phenomenon in a statistical way. The analysis shows that the near-infrared flare amplitudes are dominated by a single state power law, with the low states in SgrA* limited by confusion through the unresolved stellar background. There are several dusty objects in the immediate vicinity of SgrA*. The source G2/DSO is one of them. Its nature is unclear. It may be comparable to similar stellar dusty sources in the region or may consist predominantly of gas and dust. In this case a particularly enhanced accretion activity onto SgrA* may be expected in the near future.
A relativistic model that could explain the flaring nature of SgrA* are hotspots, overdense compact emitting regions, moving inside an accretion flow. To model compact sources orbiting in the immediate vicinity of SgrA*, it is necessary to use the metric for a rotating black hole, the Kerr-metric. There are a couple of relativistic effects on the emission that need to be taken into account, most importantly the gravitational Doppler-shift and gravitational lensing.
Abstract: We consider the static Einstein-Vlasov system in spherical symmetry. Existence of different types of solutions to this system for zero cosmological constant has been shown for the isotropic and anisotropic case by Rein-Rendall, Rein and Wolansky. In this talk I review the results on static solutions for the Einstein-Vlasov system and eventually describe a method to prove existence of static solutions to the Einstein-Vlasov system with positive cosmological constant. The energy density and the pressure of these solutions have compact support and outside a finite ball these solutions are identical to a Schwarzschild deSitter spacetime. The results presented in the talk are joint work with H. Andréasson and D. Fajman.
Abstract: The dynamical gravitational collapse of a complex scalar field coupled with Maxwell field in dilaton gravity, allowing a phantom coupling to gravity, will be described.
Abstract: Known theorems and work in progress establishing the existence of solutions describing isolated bodies will be discussed. There are results for fluids as well as for elastic matter, with and without gravity in Newtonian and Einstein's theory.
Abstract:We give an elementary proof of positivity of the Trautman-Bondi mass of light-cones with complete generators in asymptotically flat space-times.
Abstract: We present a geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems (Huisken-Yau, Metzger, Huang). We then explain the pseudo-Newtonian character of these formulas and show that they converge to Newtonian mass and center of mass in the Newtonian limit, respectively, using Ehlers' frame theory. Moreover, we present a novel physical interpretation of the level sets of the canonical lapse function and apply it to prove uniqueness results.
Abstract: We discuss the initial-boundary value problem which arises when formulating the Cauchy problem in general relativity on a finite domain with an artificial outer boundary, like is usually the case in numerical relativity simulations. First, the restrictions on the boundary data that result from the requirement of constraint propagation and the attenuation of spurious reflections will be analyzed. Then, we will introduce the important concept of strong well-posedness and explain it first in the simple example of the wave equation on the half-plane. For systems of wave equations, strong well-posedness allows to treat a certain class of boundary conditions which is general enough to cover many evolution systems in physics, including Einstein’s equations in harmonic coordinates. Finally, open issues related to a geometric formulation of the initial-boundary value problem will be mentioned.
Abstract: Recent results concerning derivation of the conservative equations of motion of compact binary systems up to the 4th post-Newtonian approximation of general relativity will be presented. The derivation is made within the ADM canonical formalism. It employs Dirac delta distributions to model the compact bodies what leads to divergencies which are regularized by a combination of Riesz-implemented Hadamard's partie finie approach and dimensional regularization. It also requires taking into account tail-transported nonlocal-in-time interaction between the bodies.
Abstract: I will discuss an ongoing project on flat steady states for the Vlasov-Poisson system, which in astrophysics are used as models of disk-like galaxies. We construct solutions numerically and study in particular the shape of the rotation curves. It is often claimed that a system obeying Newton's law of gravity should have a rotation curve which declines in a Keplerian manner far out in the galaxy. However, observations indicate that the rotation curves are approximately flat and this discrepancy is one of the reasons for introducing dark matter. In our numerical study we find a large class of solutions for which the rotation curves are flat all the way out to the boundary of the steady state. This is a joint work with Gerhard Rein.
Abstract: We discuss some results concerning the geometry and topology of asymptotically flat initial data sets in three and higher dimensions, with and without horizons. More specifically, we explore the relationship between the topology of such initial data sets and the occurrence of marginally outer trapped surfaces in the initial data. We shall discuss the rationale for this and present relevant background material. This involves work with several collaborators, L. Andersson, K. Baker, M. Dahl, M. Eichmair and D. Pollack.
Abstract: I will give an introduction to the linearization stability problem for the Einstein equations. Furthermore I will introduce two criterions for linearization stability (established by Vincent Moncrief [1][2]) and sketch the corresponding proofs from those references.
[1] V. Moncrief, Spacetime symmetries and linearization stability of the Einstein equations. I ,
J. Math. Phys. 16, 493 (1975); dx.doi.org/10.1063/1.522572
[2] V. Moncrief, Spacetime symmetries and linearization stability of the Einstein equations. II ,
J. Math. Phys. 17, 1893 (1976); dx.doi.org/10.1063/1.522814
Abstract: I will describe a configuration space of two surfaces rolling on each other without sleeping or twisting. A relation between this space and totally null planes in 4-dimensional conformal geometry of signature (2,2) will be established and used to construct new surfaces that roll on each other without sleeping or twisting and exhibit the symmetry of the exceptional simple Lie group G2.
Abstract: Some results on the mechanism of interactions among fermion fields and cosmic strings in curved spacetime, as well as on the influence of spinor fields on Yang-Mills black holes, will be presented.
Abstract:
1. A short introduction to convenient calculus in infinite dimensions.
2. Manifolds of mappings (with compact source) and diffeomorphism groups as convenient manifolds
3. A diagram of actions of diffeomorphism groups
4. Riemannian geometries of spaces of immersions, diffeomorphism groups, and shape spaces, their geodesic equations with well posedness results and vanishing geodesic distance.
5. Riemannian geometries on spaces of Riemannian metrics and pulling them back to diffeomorphism groups.
6. Robust Infinite Dimensional Riemannian manifolds, and Riemannian homogeneous spaces of diffeomorphism groups.
We will discuss geodesic equations of many different metrics on these spaces and make contact to many well known equations (Cammassa-Holm, KdV, Hunter-Saxton, Euler for ideal fluids), if time permits.
Abstract: I will review the status of a conformal constrained ADM-like formulation of the Einstein (+matter) equations on hypersurfaces of constant mean curvature, developed with V. Moncrief. This has been adapted and implemented numerically for several applications: a gravitationally perturbed Schwarzschild black hole in axisymmetry, late-time tails of massless scalar and Yang-Mills fields in spherical symmetry, critical phenomena in the Einstein-Yang-Mills system, and massive scalar fields / evolution of (mini) boson stars.
Abstract: Newton’s Law of Gravity is considered valid from sub-millimetre distances up to inter-galactic space, but fails to describe important features of cosmology like the accelerating expansion component of our universe. While the most straightforward candidate for such a component is Einstein’s cosmological constant, a plausible alternative is dynamical vacuum energy, or ”quintessence”, changing over time. Although it is traditional to neglect (or set to zero) the couplings of this light scalar to the standard model, it is natural for a scalar quintessence field to evolve on cosmological time scales today while having couplings to matter, as expected from string theory. Hence the presence of such a field would provide energy changes to Newton’s gravity potential of the earth at short distances invisible to electromagnetic interactions.
We present a novel direct search strategy with neutrons based on Rabi spectroscopy of quantum transitions in the gravity potential of the earth. The sensitivity for deviations on Newton’s gravity law is right now E = 10^{-15} eV, providing a severe restriction on quintessence fields and on any possible new interactions on that level of accuracy. If some undiscovered dark matter or dark energy particles interact with a neutron, this should result in a measurable energy shift of the observed quantum states. In the case of some dark energy scenarios with a coupling to matter, the experiment has the potential to find or exclude these hypothetical particles in full parameter space.
Abstract: I discuss the recently announced discovery of a B-mode signal in the cosmic microwave background and its significance for cosmology.
Abstract: In this talk I will make use of a representation of the Einstein Cosmos based on the properties of conformal geodesics to discuss the global evolution in time of massless spin-2 fields. In view of the conformal properties of the massless spin-2 equation, the constructed solutions can be reinterpreted as global solutions in the anti de Sitter space-time. I will discuss how this analysis can be generalized to the case of the conformal field equations.
Abstract. Joint Work with Philippe G. LeFloch. We consider vacuum spacetimes with two spatial Killing vectors and with initial data prescribed on T^3. The main results that we will present concern the future asymptotic behaviour of the so-called polarized solutions. Under a smallness assumption, we derive a full set of asymptotics for these solutions. Within this symetry class, the Einstein equations reduce to a system of wave equations coupled to a system of ordinary differential equations. The main difficulty, not present in previous study of similar systems, is that, even in the limit of large times, the two systems do not directly decouple. We overcome this problem by the introduction of a new system of ordinary differential equations, whose unknown are renormalized variables with renormalization depending on the solution of the non-linear wave equations.
Abstract: We exhibit a class of theories, with the relativistic fluid a special case, which naturally take the form of a symmetric hyperbolic system. The 'reason' for this is that they possess a convex extension, with the role of convex entropy being played by the particle number density. This is joint work with Philippe LeFloch.
Abstract: In this talk, I discuss a peculiar black hole instability that arises in the presence of short distance dispersion. Its origin is to be found in the spectral properties of the wave equation on a background geometry containing two horizons. I will start by qualitatively describing this effect. In a second part, I will show that the presence of complex eigen-frequencies in the spectrum encodes this instability. Such eigen-frequencies are allowed only because the conserved scalar product is non positive definite. I will then compute the spectrum through a WKB approximation. In a last part, I will present an abstract toy model to discuss general feature of the appearance complex eigen-frequencies. This model is directly inspired from the ``Friedrich model'' of resonances. This will allow to make contact with quasi-normal modes of black holes and other known black hole instabilities.
Abstract: I will describe results of my joint work with Piotr Bizoń on instability of three-dimensional asymptotically AdS spacetime coupled to a massless scalar field. As in higher dimensions, for a large class of perturbations we observe a turbulent cascade of energy to high frequencies, However, in contrast to higher dimensions, small perturbations cannot evolve into a black hole, because their energy is below the threshold for a black hole formation. To determine the long-time evolution we use the analyticity strip method, well known in fluid dynamics, which provides a powerful numerical tool.
Abstract: We review recent and older work on impulsive gravitational waves. These space-times have become textbook examples modelling short but intense gravitational wave impulses. Mathematically they have been described by a distributional - the so-called Brinkmann - metric as well as by a continuous metric - referred to as Rosen form. Our main focus will be on geodesics in these geometries. First we will discuss the behaviour and regularity of geodesics in the distributional form and the notion of geodesic completeness in an even wider class of impulsive wave-type spacetimes. Then we will turn to the Rosen form, and examine the regularity of geodesics in the various subclasses of impulsive wave spacetimes.
Abstract: This talk will explore issues related to the motion of extended bodies in curved spacetimes. Non-perturbative notions of linear and angular momentum will be introduced and some of their properties discussed. Most important among these properties is that forces and torques are “almost” preserved by a certain class of deformations which may be applied to the relevant field (electromagnetic, gravitational, or otherwise). Here, the “almost” refers to terms which can be interpreted purely as finite shifts in an object’s apparent multipole moments. The freedom to choose different fields can be used to dramatically simplify problems where self-interaction affects the motion. Usual results on the self-force emerge as a simple special case of this formalism. In another special case, full multipole expansions for the forces and torques acting on extended test bodies are recovered as well.
Abstract: The behaviour of particles, both from a classical as well as a quantum mechanical perspective, with respect to impulsive background fields is investigated. Due to the singular nature of the problem, which requires the definition of products of distributional objects, a generalized framework like Colombeau's new generalized functions has to be used.
Abstract: The study of the asymptotic behavior of the Maxwell and gravitational fields is a key point in the understanding of the stability properties of solutions of the Einstein equations. Penrose introduced in the beginning of the 60s a method based on the construction of Hertz potentials satisfying a wave equation to determine the asymptotic behavior of massless free fields of arbitrary spin from a decay Ansatz on solutions of the scalar wave equation. The purpose of this talk is to adapt this idea in the context of a Cauchy problem: consider a Cauchy problem for the Maxwell and gravitational fields on the Minkowski space-time with initial data in weighted Sobolev spaces; in the framework of this Cauchy problem, the existence of a Hertz potential is proved; finally, from a standard decay result for the scalar wave equation, the asymptotic behavior of these higher spin fields is derived. The classical decay results for Maxwell and gravitational fields are recovered.
Abstract: Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which are critical points of a Lagrangian which is a natural geometrical generalization of the free wave Lagrangian. Self-gravitating wave maps are those from an asymptotically flat Lorentzian manifold which evolves according to Einstein's equations of general relativity with the wave map itself as the source. The energy of wave maps is scale invariant if the domain manifold is 2+1 dimensional, hence it is referred to as the critical dimension.
Apart from a purely mathematical interest, the motivation to study critical self-gravitating wave maps is that they occur naturally in 3+1 Einstein's equations of general relativity. Therefore, studying critical self-gravitating wave maps could be a fruitful way of understanding the ever elusive global behavior of Einstein's equations. A few central questions concerning the study of critical self-gravitating wave maps are local and global existence, blow up profile, compactness and bubbling.
In this talk, after a brief discussion on the background and formulation of the Cauchy problem of critical self-gravitating wave maps, we shall present a recent proof of the non-concentration of energy of critical equivariant self-gravitating wave maps before pointing out potential generalizations and applicable methods therein.
Abstract: I give an introduction to quantum field theory on curved spacetimes in the framework of locally covariant field theories, introduced by Brunetti, Fredenhagen, Verch and Hollands, Wald. The main motivation and example will be the covariant definition of the stress-energy tensor of a scalar quantum field.
Abstract: The present work concerns the construction of a lightlike foliation of spacetime which suites the Kerr-Schild framework describing the gravitational field of a massless particle located on the horizon. Despite of being defined only on local grounds, the gained results do not only prove to be consistent to former works of Hayward and Brady, Israel, Droz and Morsink, but fit also former results of Moncrief and Isenberg and, in addition, that of Friedrich, Racz and Wald concerning Gaussian null coordinates. Two simple examples for the construction, describing the situation for a Schwarzschild black hole in Kruskal-Szekeres as well as in Kerr-Schild coordinates, are given. Finally it is explained how the obtained foliation might be used in order to extend the gravitational field of a massless particle off the horizon.
Abstract: I review the basic setup of Kaluza-Klein theory, namely a 5-dim. vacuum with a cyclic isometry (a U(1) fibre bundle over 4-dim. spacetime) which corresponds to Einstein-Maxwell-dilaton theory. I show that the property of compact surfaces of being (stably) marginally trapped is preserved under lift and projection provided the appropriate ("Pauli-") conformal scaling is used for the spacetime metric. I also discuss recently proven area inequalities for stable axially symmetric 2-dimensional and 3-dimensional marginally outer trapped surfaces. This talk is based on joint work with Tim-Torben Paetz, arxiv.org/abs/1302.3052
Abstract: More than 95% of the matter in the Universe is invisible. An overview of our current understanding of abundance and properties of dark energy and dark matter is presented. The first part focusses on issues pertaining to dark matter including observational evidence for its existence and current constraints. MOND is briefly mentioned. The second part focusses on dark energy. Observational strategies to detect and quantify dark energy are reviewed. In particular, recent results from the Planck mission are presented and an overview of the new ESA dark energy mission Euclid is given.
Abstract: I describe ongoing joint work with D. Fajman on this topic. Our inspiration comes from the work arxiv.org/abs/1109.5602 on the pure Einstein-Maxwell case, and from the known strange exact solutions in Einstein-Maxwell-dilaton theory.
Abstract: Newton's Inverse Square Law has been examined in detail from the sub-millimetre scale up to inter-galactic distances. His gravity prediction for these systems is considered valid, but fails to describe important features of cosmology like the accelerating expansion of our universe. While the most straightforward candidate is Einstein's cosmological constant , a plausible alternative is dynamical vacuum energy, or "quintessence", changing over time. Although it is traditional to neglect the couplings of this light scalar to the standard model, some scenarios allow scalar quintessence field to evolve on cosmological time scales today while having couplings to matter, as expected from string theory . Hence the presence of such a field would provide energy changes to Newton's gravity potential of the earth at short distances invisible to electromagnetic interactions. We present a novel direct search strategy with neutrons based on Rabi-spectroscopy of quantum transitions |1> ↔|2>, |1> ↔|3>, |2> ↔ |4>, |2> ↔|3>, and |2> ↔ |4>$ in the gravity potential of the earth. The sensitivity for deviations on Newton's gravity law is right now E = 10-14 eV, providing a severe restriction on quintessence fields and on any possible new interactions on that level of accuracy.
Abstract: I will give an exhaustive description of Killing Initial Data on light-cones, and on transversally intersecting characteristic hypersurfaces, in vacuum space-times.
Abstract: The talk first resumes some recent progress towards the goal to find time functions for a given globally hyperbolic metric for which basic geometric quantities are bounded. Then we conversely fix a time function and ask whether there is a conformal factor such that the corresponding Cauchy surfaces are of bounded geometry which provides us with Sobolev embeddings and denseness results for spaces of initial values. This is done by using a recently developed method called flatzooming which has proven to be powerful in different contexts of Riemannian and Lorentzian geometry.
Abstract: Via the geodesics of the Levi-Civita connection, a pseudo-Riemannian metric on a smooth manifold M determins a projective structure on M. Similarly to the role of the conformal geometry, this projective structure can be used to identify particularly robust properties of pseudo-Riemannian manifolds. Reporting on joint work with A.R. Gover (Auckland) my talk will be devoted to the projective analog of the notion of a conformally compact Riemannian metric. This exhibits a notion of compactification for Ricci flat metrics and non-Ricci-flat Einstein metrics which are similar to - but different from - the ususal notion of conformal compactifications.
Abstract: Geometric inequalities have been of interest in General Relativity in recent years. From them, it is possible to relate physical quantities that have a precise geometric meaning--like mass, area, charge and angular momentum--, and thus be able to predict significant consequences on the evolution and stability of some physical systems. In this talk, I present a conjecture relating the electrical charge to the size of a real object, inspired on the hoop conjecture valid for black holes. First I discuss briefly some relevant aspects of the hoop conjecture and then I state the analogous conjecture for real objects in general. Physical motivation of the inequality is discussed, as well as define with precision what we understand about the "size” of a three dimensional object. As a first approach, I study the spherical problem with ECD wherein this conjecture is precisely formulated and I show that it is true outside and in the bound of the sphere.
Abstract: Light bending, characteristic of geometric descriptions of gravity as spacetime curvature, manisfests dramatically in the existence of black hole spacetimes. Global notions associated with the causal disconnection between spacetime regions, on the one hand, and (quasi-)local concepts related to the convergence of light rays, on the other hand, provide complementary tools for the study of black holes. Here we focus on the latter aspects, namely relying on the notion of trapped surface. More specifically, we discuss the role of the limiting case provided by marginally (outer) trapped surfaces (MOTS) as probes into the geometry of dynamical black holes, placing a special emphasis in their notion of stability. We illustrate the discussion with two examples, the first one dealing with a family of geometric inequalities providing a lower bound for the horizon area, and the second one motivating the role of MOTS as inner "test screens" in a heuristic proposal for a "scattering-like approach" to the a posteriori analysis of dynamical black hole spacetimes.
Abstract: In my talk I will discuss two possible nontrivial scenarios concerning the fate of Lorentz symmetry in the low energy limit of quantum gravity: Lorentz Invariance Violation (LIV) and Lorentz/Poincare symmetry deformation. I will also briefly present some of the experimental bounds on the parameters of the models pertaining to these scenarios
Abstract: There is a strong evidence that anti-de Sitter space is unstable due to small generic perturbations. It is also believed that there might exist solutions that do not lead to the formation of a black hole. I will discuss recent analytical and numerical results concerning time-periodic solutions for Einstein-massless-scalar field system with negative cosmological constant, in particular how to construct such stable configurations. If time permits I will outline the pure vacuum case. The talk will be an extension of joint work with Andrzej Rostworowski presented in the paper arxiv:1303.3186.
Abstract: I will present an alternative to the Dirac quantization of minisuperspaces that admits a time evolution.
Abstract: Summarising my diploma thesis I will start with introducing the work of Hubert Bray being the paper my thesis is built up upon. The paper deals with a possible explanation of the existence of dark matter by introducing a torsion of space-time. Its basic idea is to derive an extension of General Relativity involving a more general connection from particular axioms for the metric and the connection. According to these axioms the gravitational action functional can only take a specific form. The variation of this action functional leads to Einstein-Klein- Gordon equations. The mass term in the Klein-Gordon equation corresponds to the coupling constant for the torsion. The terms involving the scalar field and its gradient appearing in the Einstein field equations can be interpreted as the effective energy-stress tensor and can be attributed to dark matter. The solution of the Klein-Gordon equation in a spherically symmetric space-time is an oscillating function both in time and space. From the effective energy-stress tensor appearing in the Einstein field equations we derive a Newtonian potential displaying a slowly rotating maximum, which resembles a spoke. In the paper the author performs simulations using this Newtonian potential and obtains results resembling a spiral galaxy. The aim of my diploma thesis is to investigate the measurable effects of the torsion field by analysing the behaviour of a particle with spin-1/2 in the torsion field. The polarization vector of a particle in a torsion field is subject to a torque and hence precesses. To compute the precession two different approaches were chosen: the first one is the supersymmetric approach that enables one to consistently couple a classical spinning particle to the torsion field. The second approach is a quantum mechanical one solving the Dirac equation minimally coupled to the torsion field. The conclusion of my thesis is that the precession of the polarization vector induced by the torsion field results in an oscillatory motion with the deflection of order of magnitude 10^-6 rad. The sense of rotation of the precession changes every half period of the time oscillation of the torsion field.
Abstract: A broad class of theories based on non-linear Lagrangians will be discusssed and their equivalence/nonequivalence with Einstein theory (possibly with additional matter fields) will be analyzed. To simplify technical aspects of such theories, a nonstandard theory of curvature will be used.
Abstract: Following the parts of my thesis, I will first give a brief introduction to the field of spatially homogenous (SH) cosmology with an emphasis on the use of dynamical systems methods to analyse the evolution of these cosmologies qualitatively. After this, I will summarise the results of the central part of my thesis, which deals with the dynamical system analysis of a special class of SH cosmologies (locally rotationally symmetric Bianchi type VIII). The matter content is thereby chosen out of a very general family which allows for anisotropic pressures, and contains physically relevant models like perfect fluids, elastic matter or collisionless matter. The goal was to investigate how the grade of anisotropy of the matter influences the qualitative dynamics, which was achieved via a comparison with the well known results with perfect fluids. It is shown that there are indeed cases where the qualitative dynamics can differ significantly in both, the past and future asymptotics. If time is left I would like to close my talk with a little eye candy, by presenting a Maple document, which allows to plot the solutions to each matter configuration as a flow diagram by a single click on the matter-parameter space.
Abstract: We shall discuss conformally flat hypersurfaces in the realm of Moebius geometry. Particular attention will be paid to the transformation theory and integrable nature of this class of hypersurfaces.
Abstract: I will present a Hamiltonian approach to the definition of mass for a class of asymptotically cylindrical initial data sets. This is based on joint work in progress with Jezierski and Kijowski.
Abstract: We will consider a configuration space of two solids rolling on each other without slipping or twisting, and will identify it with an open subset U of R^{5}. It turns out that U is naturally equipped with a generic distribution D of 2-planes. We will discuss symmetry properties of the pair (U,D) and will mention that, in the case of the two solids being balls, when changing the ratio of their radii the dimension of the group of local symmetries unexpectedly jumps from 6 to 14 . This occurs for only one such ratio, and in such case the local group of symmetries of the pair (U,D) is maximal. It is maximal not only among the balls with various radii, but more generally among all (U,D)s corresponding to configuration spaces of two solids rolling on each other without slipping or twisting. This maximal group is isomorphic to the split real form of the exceptional Lie group G2. In the remaining part of the talk we will argue how to identify the space U defined above with the bundle T of totally null real 2-planes over a 4-manifold equipped with a split signature metric. We call T the twistor bundle for rolling bodies. We show that the rolling distribution D, can be naturally identified with an apropriately defined twistor distribution on T. We use this formulation of the rolling system to find more surfaces which, when rigidly rolling on each other without slipping or twisting, have the local group of symmetries isomorphic to the exceptional group G2
Abstract: Quantum optics provides a high-precision toolbox to enter and to control the quantum regime of the motion of massive mechanical objects. This opens the door to a hitherto untested parameter regime of macroscopic quantum physics. Due to the large available mass range - from picograms in nanomechanical waveguides to kilograms in mirrors for gravitational wave detection - it becomes possible to explore the fascinating interface between quantum physics and (quantum) gravity in table-top quantum optics experiments. I will discuss a few examples.
Abstract: The perturbations of black hole spacetimes, when decaying, show characteristic (damped) oscillations called quasi-normal modes. The asymptotically highly damped modes are widely suspected to carry information about certain black hole quantum properties in the semi-classical limit. We analyse the behavior of asymptotic quasi-normal frequencies of static black hole spacetimes and interpret the meaning of the results, linking them to possible quantum properties of spacetime. We analyse our suggestions in the broader context of spacetime thermodynamics and discuss some open questions.
Abstract: I will discuss some conformal properties of the extremal Reissner-Nordström spacetime ---in particular in what concerns the behaviour of the spacetime close timelike infinity. I will show how Friedrich's construction of the "cylinder at spatial infinity" can be used, together with a conformal discrete symmetry of the spacetime, to show that there exists a conformal representation of timelike infinity in this spacetime for which the various conformal field quantities and equations regular. I will also discuss some numerical evidence of this conformal representation.
Abstract: A compact Einstein metric is called Linearly stable if the second variation of the Einstein-Hilbert functional is nonpositive on TT-tensors.
We will discuss curvature conditions which ensure stability. Then we will show that under certain conditions on the spectrum of the Laplacian, linear stability implies that the given Einstein manifold is an attractor of the Ricci flow.
Abstract: An introduction is given to some recent developments in Yang-Mills matrix models, focusing on the effective geometry of brane solutions and their possible relevance to gravity in a brane-world picture.
Abstract: Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points couple the equation for harmonic maps with spinor fields. At present, a general existence result for Dirac-harmonic maps is not available.
In the first part of the talk we will introduce the notion of Dirac-harmonic maps and explain their basic properties. We will also summarize what is currently known about the existence of Dirac-harmonic maps. In the second part of the talk we present an approach to the existence question by the so-called heat flow method and explain how far this idea can be pushed.
Abstract: I provide an introduction to 3-dimensional higher spin gravity, review some of the recent developments with particular emphasis on holography and point out some of the puzzling open questions, especially those concerning a geometric interpretation of the field configurations.
Abstract: I consider a spherically symmetric SU(2) Yang-Mills field on the exterior of extreme Reissner-Nordstrom black hole. The problem is equivalent to a Yang-Mills field propagating on a regular asymptotically flat spacetime. Infinitely many non-trivial static solutions are shown to exist. I analyze linear perturbations of the solutions and find their spectrum (unstable modes and quasinormal modes). Then I show the dynamics of the field and the approach to a static solution.
Abstract: Oliver Rinne (AEI) and I developed, a few years ago, a fully constrained method for integrating the vacuum Einstein field equations out to Scri. Oliver subsequently implemented this proposal numerically for the case of axially symmetric metrics and showed that it gave stable evolutions, reproducing in particular (in a fully nonlinear code) the well-known quasi-normal ringing modes characteristic of black holes. In this talk I will describe some very recent work with Oliver in which we have extended the theoretical developments to include conformally invariant matter sources, including Yang-Mills fields and implemented these numerically in the case of spherical symmetry. The extra resolution available in this case permits us not only to recover the ringing but also the (Price law) tails in the various radiation fields.
Abstract: We study the scalar wave equation on the open exterior region of an extreme Reissner-Nordstr\"om black hole and prove that, given compactly supported data on a Cauchy surface orthogonal to the timelike Killing vector field, the solution, together with its $(t,s,\theta,\phi)$ derivatives of arbitrary order, $s$ a tortoise radial coordinate, is bounded by a constant that depends only on the initial data. Our technique does not allow to study transverse derivatives at the horizon, which is outside the coordinate patch that we use. However, using previous results that show that second and higher transverse derivatives at the horizon of a generic solution grow unbounded along horizon generators, we show that any such a divergence, if present, would be milder for solutions with compact initial data.
This talk is based on http://lanl.arxiv.org/abs/1209.0213, and it is joint work with G. Dotti.
Abstract: In many cases the mathematical structures which we use in applications (computer science, dynamical systems, general relativity) present both a topology and an order. There is a beautiful but little known topological theory which unifies these concepts into that of 'quasi-uniformity'. In practice one has simply to drop an axiom in topology to find that an order naturally arises. Most of topology can be still developed, leading to concepts such as normally preordered spaces or completely regularly preordered spaces. I wish to introduce and comment on this generalization of topology which allows us to prove, among the other results, the existence of time functions in stably causal spacetimes.
Abstract: In recent years there were renewed interest in extending the black hole uniqueness theorems to space-times which are neither real-analytic nor axially-symmetric. Thus far the results obtained have either been conditional on an additional rigidity assumption of the black hole event horizon, or on an additional smallness assumption of the space-time being suitably "close" to being Kerr(-Newman). I will describe a result of the latter class: that a weighted point-wise control of local space-time geometry yields topological constraints on the domain of outer communications. This provides a rigorous formulation for the intuitively obvious fact that "if on every patch the space-time looks similar to a Kerr-Newman solution, it cannot contain more than one black hole".
Abstract: Solutions to the Einstein-Vlasov system describe spacetimes with collisionless matter. The nonlinear stability problem for the Einstein-Vlasov system with symmetries has been considered in a series of works starting with Rein and Rendall in 1992. Recently, the first result for the Einstein-Vlasov system without symmetry assumptions has been established by Ringström, considering a positive cosmological constant. In the talk, we present the proof of future nonlinear stability of the Einstein-Vlasov system in 2+1 dimensions without symmetry assumptions and no cosmological constant. Due to the slow expansion and low spatial dimension in that situation, it is essential to prove strong decay properties of the energy momentum tensor. We obtain these decay rates, by introducing geometric Vlasov energies using a specific metric on the tangent bundle of spacelike hypersurfaces - the Sasaki metric. We present energy estimates for those energies and their application in the proof of nonlinear stability. Finally, we give an outlook to applications and related work in progress on the corresponding higher dimensional problem.
Abstract: In the first part of the talk a Schwarzschild black hole is considered. We assume that light sources are distributed on a (big) sphere of radius R that emit, at an instant of time, photons isotropically. We calculate the resulting photon distribution and find that in the long-time limit the density becomes infinitely large near the photon sphere at r=3m. This suggests that every Schwarzschild black hole in nature should be surrounded by a shell of very high photon density which could be detrimental to the health of any observer who comes close to this region. In the second part we discuss how the situation changes if a Kerr black hole is considered. -- The first part is based on the Bachelor Thesis of Dennis Philipp and the second part is ongoing work with Arne Grenzebach.
Abstract: We investigate accreting disk systems with polytropic gas in Keplerian motion. Numerical data and partial analytic results show that the self-gravitation of the disk speeds up its rotation -- its rotational frequency is larger than that given by the well known strictly Keplerian formula that takes into account the central mass only. Thus determination of central mass in systems with massive disks requires great care -- the strictly Keplerian formula yields only an upper bound. The effect of self-gravity depends on geometric aspects of disk configurations. Disk systems with a small (circa 10^{-4}) ratio of the innermost radius to the outermost disk radius have the central mass close to the upper limit, but if this ratio is of the order of unity then the central mass can be smaller by many orders of magnitude from this bound.
Abstract: I discuss the classical motion of electromagnetically bound systems in an external gravitational field and associated quantum effects.
Wednesday, October 24, 14:15, Ettore Minguzzi (Pisa): Lightlike lines and time functions in general relativity
Abstract: "The causal ladder of spacetimes is introduced and the role of stable causality is commented. Some details are given of the recent solution to the problem of the equivalence between stable causality and K-causality. In particular this result is used to show that under reasonable conditions the absence of a cosmological time implies the null geodesic singularity of spacetime."
Abstract: "I will present a class of diagrams, that we call projection diagrams, as a tool to visualise the global structure of space-times, and show how they can be used for the Kerr-Carter family of metrics with cosmological constant. A seemingly new class of overspinning such solutions with negative cosmological constant and unusual global properties will be presented."
Abstract: "I will discuss old and new well posed sets of conformally covariant versions of the vacuum Einstein equations."
Abstract: "About twenty years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, self-similar solution at the threshold of black hole formation. We give a rigorous, computer assisted proof of the existence of Choptuik's spacetime, and show that it is real analytic. This is joint work with E. Trubowitz."