Stability Analysis On Asymptotically-flat Charged Black Holes

This thesis studies the stability of asymptotically flat charged black hole in four dimension,including the thermodynamic stability and kinetic stability of the charged black hole with non-linear complex scalar,the superradiant instability triggered by the quasinormal modes and quasi-bound states of the complex perturbation under the STU extremal charged black hole,the stable photon sphere as well as some applications of the cosmology associated with the quasi-topological electromagnetism.This thesis consists of six chapters.In the introduction,we begin with a brief history of gravity research.We introduce Einstein’s gravitational field equation and its experiment verifications.We then discuss the three most important black hole solutions in Einstein gravity: the Schwarzschild black hole,the Reissner-Nordstr ¨om(RN)black hole and the Kerr black hole.Using these three black holes as examples,we review some important black hole concepts such as the horizons,black hole thermodynamics,the Euclidean action and the photon sphere.In particular,we illustrate the Penrose Process of the Kerr black hole,which allows one to extract rotational energy from the black hole.We show that there exists an analogous process mediated by the electromagnetic interaction in a system of charged particle and the RN black hole,which allow one to extract charge energy from the black hole.At the end of the introduction,we also briefly review cosmology in General Relativity,illustrating how the different matter in the Universe can influence its evolution.In chapter 2,we firstly study the minimally-coupled scalar perturbation in a generic black hole background and obtain the general equation of motion.Based on the asymptotic behavior of the radial function,we classify three different modes: the scattering modes,the quasinormal modes and the quasi-bound states.For the real scalar perturbation of the Kerr black hole,we study the superradiant scattering process and obtain the superradiant condition from the conserved flux.We then prove that the frequency of the quasinormal modes as well as the quasi-bound states must be complex.Furthermore,we obtain the necessarily conditions for superradiant instability by analysing the conserved flux of the quasinormal modes and quasi-bound states.We discuss some different numerical methods of constructing quasinormal modes and quasi-bound states and illustrate their relative merits.By calculating the associated error,we also analyse the validity of the shooting method for unstable modes.In chapter 3,we consider the Einstein-Maxwell theory minimally coupled to a complex scalar field with non-linear potential and obtain a class of numerical black hole solutions.We firstly prove that for a general potential of the complex scalar,the RN black hole cannot be scalarized by continuous phase transitions.Moreover,we numerically illustrate that in the microcanonical ensemble,the near extremal RN black hole can be thermodynamically unstable and scalarized by the first-order phase transition.In addition,by calculating the quasinormal modes,we found that the scalarized charged black hole can be also kinetically stable against the neutral scalar perturbation.In addition,our numerical results also give a negative answer for Penrose-Gibbons Conjecture and we also suggest two new generalizations of the Penrose Inequality in charged black hole.In chapter 4,we studied the superradiant instability associated with the unstable quasibound states as well as the quasinormal modes of the STU extremal black hole with multiple charges,which can reduce to the RN black hole when all the charges are equal.We firstly review the superradiant condition for the quasi-bound states and the quasi-normal modes,then we investigate the effective potential of the Sch ¨odinger-like radial equation associated with the complex scalar perturbation.We find that the single peak and the double peaks configuration of the effective potential can emerge by choosing appropriate parameters.By adopting the numerical shooting method,we obtain unstable quasi-bound states organized by the overtone number and the charge configuration parameter of the STU model.We found that as long as the charges are not equal,the STU extremal charged black holes are superradiantly unstable for the quasi-bound states,implying that the superradiantly stable RN black hole is a fine-tuning result.As to the double peak configuration,we also numerically found the superradiantly quasinormal modes with high excited frequency of the STU extremal charged black hole.In chapter 5,we introduce a new concept which we name as topological electromagnetism,defined by square norm of the topological 4-form product F ∧ F,whose the energy momentum tensor can be viewed as perfect fluid where the energy density is opposite to its pressure.Therefore,the quasi-topological electromagnetism is a candidate for the dark energy.We then investigate the applications in black hole physics and cosmology.The quasi-topological electromagnetic term has no effect on RN black hole with pure charge and pure magnetic monopole.However,the dyonic black hole will be totally modified.We also found that the dyonic black hole can have at most four horizons.By choosing suitable parameters,we found that there exist four photon spheres of the dyonic black hole and one of them is stable.In addition,we found that the superradiant instability triggered by the low-excited quasi-bound state of the complex field perturbation with two potential well in effective potential in the dyonic black hole background.Due to the interaction between the superradiantly unstable perturbation and the photon in the stable photon sphere,the dyonic black hole in quasi-topological electromagnetism can have an observable so-called light blast effect.In cosmology,we illustrate that the quasi-topological electromagnetism can be equivalently a cosmological constant,which can be interacting with another type of matter.In chapter 6,we give a conclusion of this thesis.