Black Hole Thermodynamical Phase Transition And Its Correspondence To The Stability Of Dynamical Perturbation

In this thesis, we investigate the thermodynamical properties and dynamical perturbations for several black hole models. The thesis is com-posed of three parts and includes the thermodynamical phase transitions of Born-Infeld AdS (BI-AdS) black hole and Gauss-Bonnet AdS (GB-AdS) black hole in the so-called extended phase space, and the correspondence of 4D RN-AdS black hole and 3D hairy black hole thermodynamical phase transitions to the stability of dynamical perturbation against black holes.We firstly discuss thermodynamics of D-dimensional BI-AdS black hole in the so-called extended phase space, where the cosmological con-stant is regarded as a variable and also identified with thermodynamic pressure. Then, temperature expression T of black hole can be rewritten to equation of state which describes the system analogy with the Van de Waals liquid-gas system. Using a constraint equation that determines the critical points of Van der Waals analogy thermodynamical phase transi-tion, we find that different from the four-dimensional case, in the sys-tem of higher dimensional BI-AdS black holes there is no reentrant phase transition. For the three-dimensional BI-AdS black hole, there does not exist critical phenomena. Later,we also diseuss thermodynamics in the grand canonical ensemble of D-dimensional charged GB-AdS black holes in the extended phase space,and shows that the usual Van de Waals like small-large black hole phase transition only holds in five-dimensional spherical charged GB-AdS black when its potential is fixed within the range 0<Φ<(?)/4.As to the Reissner-Nordstrom-AdS(RN-AdS)black holes,with vanishing Gauss-Bonnet parameter,there is no critical behav-ior in the grand canonical ensemble.This result holds independent of the spacetime dimensions and topologies.In Chapter 3,we focus on the correspondence of 4D RN-AdS black hole thermodynamical phase transition to the stability of dynamical per-turbation against black hole.Considering that quasinormal modes of dy-namical perturbations are characteristic sounds of black holes,it is ex-pected that black hole phase transitions can be reflected in the dynamical perturbations in the surrounding geometries of black holes through fre-quencies and damping times of the oscillations.We found that when the Van der Waals analogy thermodynamical phase transition happens,no matter in the isobaric process or in the isothermal process,the slopes of the quasinormal frequency change drastically different in the small and large black holes as we increase the value of the horizon radius.This fur-ther supports that the quasinormal mode can be a dynamic probe of the thermodynamic phase transition.By introducing a new form of scalar potential V(φ),we construct a proper form of the rotating black hole solution in three-dimensional Ein-stein gravity with nonminimally coupled sealar field,and find that first law of thermodynamics of this new rotating hairy black hole can be protected. Then we further study the quasinormal modes(QNM) of electromagnetic perturbations of this black hole,and find that the imaginary parts of quasi- normal frequencies of dynamical perturbations can reflect the properties of thermodynamical stability disclosed by examining the free energy. This result is also valid for static black hole in the three-dimensional Einstein gravity with nonminimally coupled scalar field.