The Instability And Spontaneous Scalarizaion Of Schwarzschild-de Sitter Black Hole

Black hole(BH)solutions in modified gravity(MG)have attracted considerable attention in recent years.One important reason is that BH in MG can escape from the restriction imposed by the no-hair theorem which is characterized by BH in general relativity(GR),and can be equipped with a new degree of freedom(DoF)besides of mass,angular momentum and electric charge.As a result,black hole with additional DoF are usually known as "hairy BHs".Especially,the new DoF of hairy BH in scalar-tensor gravity is dubbed a scalar hair.The process in which a no-hair BH spontaneously transits into a hairy BH under the dynamic perturbations of certain scalar field,is known as the spontaneous scalarization.Spontaneous scalarization is one of the importars to obtain scalarized BH and it closely connects with the BH(in)stability analysis.In details,the dynamic instability mode caused by the scalar field perturbation is the necessary condition of scalarization.Usually,in order to ensure the exitence of unstable modes,one needs to investigate the quasi-normal modes(QNMs)in the frequency domain in probe limit,which can be simultaneously reflected from the evolution in the time domain.Then some unstable modes could lead to a non-trivial scalar field configuration that admits the evasion of the no-hair theorem,and its formation denotes the occurrence of unstable modes in reverse.In short,we can say that the BH spontaneous scalarization starts with unstable modes and ends with the formation of a hairy BH solution.This thesis consists of the following four parts:In Chapter 1,we briefly review the development and challenges of general relativity,and then introduce various modified gravitation theory which tends to cure the defects of general relativity.In Chapter 2,we introduce BH solutions and its uniqueness theorem(i.e.,the no-hair theorem that restricts the maximum DoF of a BH)in different gravitational theories.Then we demonstrate the evasion of no-hair theorem in Einstein-scalar-Gauss-Bonne(EsGB)gravity which is a special case in scalar-tensor gravity.In Chapter 3,we introduce a wide spread mechanism of the formation of hairy BHs,namely the spontaneous scalarization of BHs.Then we discuss the tachyonic&superradiance instability both of which are important mechanism triggering the spontaneous scalarization.For the tachyonic instability,we review three examples,including static and rotating BHs in EsGB gravity,and static neutral BH in Einstein-Maxwell-Scalar(EMS)theory,to discuss three different types of tachyonic instability.While for the superradiance instability,we take a RN-dS BH as an example to exhibit the instability induced by a charged complex scalar field.In Chapter 4,we compute the QNMs,dynamical evolution and static scalar field configuration in the context of the EsGB-Λ model,and then we construct a numerical scalarized BH solution.Through those analysis,we present a whole spontaneous scalarization.