Nonlinear perturbations of Reissner-Nordstr\"om black holes

We develop a nonlinear perturbation theory of Reissner-Nordstr\"om black holes. We show that, at each perturbation level, Einstein-Maxwell equations can be reduced to four inhomogeneous wave equations, two for polar and two for axial sector. Gravitational part of these equations is similar to Regge-Wheeler and Zerilli equations with source and additional coupling to the electromagnetic sector. We construct solutions to the inhomogeneous part of wave equations in terms of sources for Einstein-Maxwell equations. We discuss $\ell=0$ and $\ell=1$ cases separately.Comment: 9 page

Einstein clusters as models of inhomogeneous spacetimes

We study the effect of small-scale inhomogeneities for Einstein clusters. We construct a spherically symmetric stationary spacetime with small-scale radial inhomogeneities and propose the Gedankenexperiment. An hypothetical observer at the center constructs, using limited observational knowledge, a simplified homogeneous model of the configuration. An idealization introduces side effects. The inhomogeneous spacetime and the effective homogeneous spacetime are given by simple solutions to Einstein equations. They provide a basic toy-model for studies of the effect of small-scale inhomogeneities in general relativity. We show that within our highly inhomogeneous model the effect of small-scale inhomogeneities remains small for a central observer. The homogeneous model fits very well to all hypothetical observations as long as their precision is not high enough to reveal a tension.Comment: 23 pages; minor changes to match published versio

Near-horizon orbits

W klasie rozwiązań Kerra istnieją takie czarne dziury, że cząstki z masą, które wokół nich orbitują, mogą znajdować się dowolnie blisko horyzontu zdarzeń. Celem pracy jest analiza możliwości lądowania rakiety na planecie znajdującej się na takiej orbicie, bez potrzeby hamowania z ogromnym przeciążeniem. W tekście przedstawiam podstawowe własności rozwiązania Kerra i na ich podstawie rozwiązuję postawiony problem. Nawiązuję również do filmu „Interstellar”, który był inspiracją do wyboru tematykininiejszej pracy.In the class of Kerr solutions there are such black holes, that massive particles orbiting them can appear arbitrary close to the horizon. In this paper I analyze if there is a possibility for a rocket to land on a planet staying on such an orbit. I present basic features of the Kerr metric and I solve a given problem. I also refer to the movie "Interstallar", which was an inspiration for a subject of this dissertation

Master equations and stability of Einstein-Maxwell-scalar black holes

We derive master equations for linear perturbations in Einstein-Maxwell scalar theory, for any spacetime dimension D and any background with a maximally symmetric n = (D - 2)-dimensional spatial component. This is done by expressing all fluctuations analytically in terms of several master scalars. The resulting master equations are Klein­ Gordon equations, with non-derivative couplings given by a potential matrix of size 3, 2 and 1 for the scalar, vector and tensor sectors respectively. Furthermore, these potential matrices turn out to be symmetric, and positivity of the eigenvalues is sufficient (though not necessary) for linear stability of the background under consideration. In general these equations cannot be fully decoupled, only in specific cases such as Reissner-Nordstrom, where we reproduce the Kodama-Ishibashi master equations. Finally we use this to prove stability in the vector sector of the GMGHS black hole and of Einstein-scalar theories in general

Efekt niejednorodności dla sferycznie symetrycznej cieczy doskonałej w Ogólnej Teorii Względności

W celu zbadania efektu niejednorodności w równaniach Einsteina, zbudowaliśmy prosty model niejednorodnej gwiazdy. Skonstruowaliśmy statyczne, sferycznie symetrycznie rozwiązania bazując na znanych rozwiązaniach: Wewnętrznym Schwarzschildzie oraz Tolmanie VII. Zbadaliśmy efekt niejednorodności i zinterpretowaliśmy go. Pokazujemy również nieprzydatność schematu Greena--Walda do naszego modelu w pewnych przypadkach.In this thesis we develop a toy model of inhomogeneous stars to study backreaction in Einstein equations. We construct static, spherically symmetric, inhomogeneous solutions, on the basis of Schwarzschild Interior and Tolman VII solutions. We observe backreaction and interpret it. Finally, we show non–applicability of Green–Wald framework to spherical symmetry in some cases

Ultracompact rotating gravastars and the problem of matching with Kerr spacetime

A number of authors provided arguments that a rotating gravastar is a good candidate for a source of the Kerr metric. These arguments were based on the second order perturbation analysis. In the following paper, we construct a perturbative solution of the rotating gravastar up to the third perturbation order and show that once we demand finiteness of the Kretschmann scalar expansion, it cannot be continuously matched with the Kerr spacetime