AdS instability: resonant system for gravitational perturbations of AdS5{}_5 in the cohomogeneity-two biaxial Bianchi IX ansatz

We consider five-dimensional, vacuum Einstein equations with negative cosmological constant within cohomogenity-two biaxial Bianchi IX ansatz. This model allows to investigate the stability of AdS without adding any matter to the energy-momentum tensor, thus analyzing instability of genuine gravtational degrees of freedom. We derive the resonant system and identify vanishing secular terms. The results resemble those obtained for Einstein equations coupled to a spherically-symmetric, massless scalar field, backing the evidence that the scalar field model captures well the relevant features of AdS instability problem. We also list recurrence relations for the interaction coefficients of the resonant system, which might be useful in both numerical simulations and further analytical studies.Comment: 40 pages, no figure

Conformal flow on S3S^3 and weak field integrability in AdS4_4

We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime. It turns out that this effective system, which we call the conformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows. Based on these observations and close parallels to the cubic Szego equation, which was shown by Gerard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS$_4$ are integrable as well.Comment: 22 pages, v2: minor revisions, several references added, v3: typos corrected, v4: typos corrected, one reference added, matches version accepted by CM

AdS Robin solitons and their stability

We consider the four-dimensional Einstein–Klein–Gordon–AdS system with the conformal mass subject to the Robin boundary conditions at infinity. Above a critical value of the Robin parameter, at which the AdS spacetime goes linearly unstable, we prove existence of a family of globally regular static solutions (that we call AdS Robin solitons) and discuss their properties

O potoku konforemnym na S3\mathbb{S}^{3} i innych układach rezonansowych

Tematyka pracy dotyczy ograniczonych przestrzennie hamiltonowskich układów rezonansowych. W szczególności omówiony został potok konforemny, który jest układem dynamicznym powstałym przez uśrednienie czasowe konforemnie niezmienniczego nieliniowego równania falowego na cylindrze Einsteina R x S3. Problem jest powiązany z badaniem stabilności czasoprzestrzeni Anty-de Sittera poprzez transformacje konforemna. Potok konforemny posiada kilka charakterystycznych własności, takich jak istnienie symetrii, trójwymiarowych podprzestrzeni rozwiązań, a także stanów stacjonarnych. Główna część pracy poświęcona jest klasyfikacji, stabilności oraz konstrukcji stanów stacjonarnych, bazującej na teorii bifurkacji oraz metodzie Lyapunova-Schmidta. W szczególności, praca zawiera dowód stabilności pewnej rodziny rozwiązań zwanej stanem podstawowym. Ponadto, w pracy omówione są inne przykłady układów rezonansowych. Część obliczeń przeprowadzonych dla potoku konforemnego została powtórzona dla równania opisującego najniższy poziom Landaua dla kondensatu Bosego-Einsteina w harmonicznym potencjale. Na końcu przestawiona została konstrukcja układu rezonansowego dla grawitacyjnych perturbacji czasoprzestrzeni Anty-de Sittera w pięciu wymiarach.This thesis is concerned with spatially confined resonant Hamiltonian systems. In particular, it discusses the cubic conformal flow, which is an effective infinite-dimensional timeaveraged dynamical system accurately approximating the conformally invariant cubic wave equation on the Einstein cylinder R x S3 in the weak field regime. The problem is related to the studies of stability of Anti-de Sitter spacetime by a conformal transformation. The cubic conformal flow exhibits some remarkable features, such as symmetries, low-dimensional invariant subspaces and the existence of stationary solutions. A major part of the thesis is dedicated to the classification, stability and construction of stationary states by bifurcation theory and Lyapunov-Schmidt method. In particular, it contains a proof of stability of a certain family of solutions called the ground state family. Moreover, other examples of spatially confined resonant systems are discussed. A part of the calculations performed for the cubic conformal flow is repeated for the Lowest Landau Level equation, which describes the Bose-Einstein condensate in a harmonic trap. Finally, the thesis presents the construction of a resonant system for gravitational perturbations of Anti-de Sitter spacetime in five dimensions

Równania Maxwella w Ogólnej Teorii Względności

Czasami podobne struktury matematyczne pojawiają się w różnych teoriach fizycznych. Jednym z przykładów są równania Maxwella, które można napotkać w ogólnej teorii względności w formie podobnej do tej znanej z elektrodynamiki. Struktura równań Maxwella może być odtworzona w redukcji Kaluzy-Kleina lub jako pewne przybliżenie równań Einsteina, tak zwany grawitomagnetyzm. Także zachowanie cząstek w czasoprzestrzeni Taub-NUT i własności konformalnego tensora Yano-Killinga stanowią kolejne przykłady opisanej analogii. Celem pracy jest omówienie istniejących wyników tego typu.Sometimes, similar mathematical structures may be found in different physical theories. One of the examples are Maxwell equations that reappear in a very similar form within general relativity. The structure of Maxwell equations may be recovered from Einstein equations via Kaluza-Klein reduction or as a specific limit in the so-called gravitomagnetic approximation. Also behaviour of test bodies in Taub-NUT spacetime and the properties of conformal Yano-Killing tensors contribute to this analogy. The aim of this thesis is to review existing results of this type

Orbits in the gravitational field of black holes

Praca zawiera analizę trajektorii obiektów na orbitach wokół czarnych dziur, położonych blisko horyzontu zdarzeń. Sposób uzyskiwania analitycznych i numerycznych rozwiązań w przypadku symetrii sferycznej został przedstawiony dla czasoprzestrzeni Schwarzschilda. Celem jest odtworzenie znanych wyników i znalezienie warunków, dla których orbity stają się krzywymi zamkniętymi.Rozdział pierwszy stanowi wprowadzenie do problemu, obejmujące charakterystykę ruchu w zakrzywionej czasoprzestrzeni oraz wyprowadzenie opisujących go równań. W drugim rozdziale przedstawione zostały rozwiązania analityczne otrzymane dla różnego rodzaju trajektorii. Rozdział trzeci zawiera opis numerycznego rozwiązywania równań ruchu wraz z porównaniem z wyznaczonymi wcześniej analitycznie krzywymi. Rozdział czwarty rezentuje sposób odnajdywania zamkniętych orbit.The thesis presents the analysis of trajectories of objects orbiting black holes located near the event horizon. The method of obtaining analytical and numerical solutions with spherical symmetry is presented for Schwarzschild spacetime. The aim was to reconstruct known results and find conditions when orbits become closed curves.The first chapter is an introduction to the dilemma including characteristics of motion in curved spacetime and deriviation of its equations. Chapter two introduces analytical solutions obtained for different trajectories. Chapter three presents numerical solutions for given equations and their comparison with analytical ones. The fourth chapter describes the method of finding closed orbits

Discovery of ultra-steep spectrum giant radio galaxy with recurrent radio jet activity in Abell 449

We report a discovery of a 1.3 Mpc diffuse radio source with extremely steep spectrum fading radio structures in the vicinity of the Abell 449 cluster of galaxies. Its extended diffuse lobes are bright only at low radio frequencies and their synchrotron age is about 160 Myr. The parent galaxy of the extended relic structure, which is the dominant galaxy within the cluster, is starting a new jet activity. There are three weak X-rays sources in the vicinity of the cluster as found in the ROSAT survey, however it is not known if they are connected with this cluster of galaxies. Just a few radio galaxy relics are currently known in the literature, as finding them requires sensitive and high angular resolution low-frequency radio observations. Objects of this kind, which also are starting a new jet activity, are important for understanding the life cycle and evolution of active galactic nuclei. A new 613 MHz map as well as the archival radio data pertaining to this object are presented and analyzed

Stationary states of the cubic conformal flow on <inline-formula><tex-math id="M1">\begin{document}S3 \mathbb{S}^3 \end{document}</tex-math></inline-formula>

We consider the resonant system of amplitude equations for the conformally invariant cubic wave equation on the three-sphere. Using the local bifurcation theory, we characterize all stationary states that bifurcate from the first two eigenmodes. Thanks to the variational formulation of the resonant system and energy conservation, we also determine variational characterization and stability of the bifurcating states. For the lowest eigenmode, we obtain two orbitally stable families of the bifurcating stationary states: one is a constrained maximizer of energy and the other one is a local constrained minimizer of the energy, where the constraints are due to other conserved quantities of the resonant system. For the second eigenmode, we obtain two local constrained minimizers of the energy, which are also orbitally stable in the time evolution. All other bifurcating states are saddle points of energy under these constraints and their stability in the time evolution is unknown