The Study For Exact Solutions, Thermodynamics, Criticality And Other Physical Properties In AdS Spacetime

Ever since general relativity has been presented, the study of exact solutions and their properties have attracted considerable attentions, in order to improve the under-standing of gravity. The solutions of vacuum Einstein equations had been studied exten-sively. This thesis focuses on the exact solutions with negative cosmological constant, and their thermodynamics, criticality and other physical properties. It is generally ex-pected that a thorough understanding of the spacetime thermodynamics will help us to establish the quantum theory of gravity. Moreover, there are some ways to study thermodynamics in AdS spacetime which has abundant features, especially for the crit-icality. Hence, the studies in this thesis have enough physical motivation and feasibility, which is one of frontiers in theoretical physics.We firstly introduce the background and structure of this thesis, and emphasize its importance and physical motivation. Then we present the theoretical basis, including the methods to obtain exact solutions and characterize their fundamental properties es-pecially for the thermodynamics. We present the black hole thermodynamics in (A)dS spacetime solely, and show the difficulty to study thermodynamics in dS spacetime, in order to give the reasons why we mainly study the thermodynamics of AdS black holes.In the following three Chapters, we mainly focus on my origin works.In Chapter3,we discuss the exact solutions and their properties in Einstein gravity. We firstly introduce the BTZ black hole solution and its property, and then study the accelerat-ing BTZ black hole, including its global structure, extrinsic geometric description and thermodynamics. This black hole is another kind beside the BTZ black hole in three dimensions, which evades the no-go theorem, i.e. the black hole with singular horizon topology can have positive and zero cosmological constant. The accelerating solutions contain an accelerating parameter, which corresponds to the proper acceleration in the origin of the spacetime. The global structure becomes much complicated because of the conformal factor, which results in that the conformal infinity and horizon compete for the causal boundary. The thermodynamics of accelerating black hole is still an open question, though the temperature and entropy can be got in some ways, since the mass is still not able to calculate and the present methods lead to a divergent result. Then we study accelerating vacua in higher dimensions, which give us some interesting fea-tures. We discuss the five dimensional accelerating vacua in Einstein gravity, which is the higher dimensional generalization of accelerating vacuum BTZ solution. In this case, the fundamental method to study global structure fails, but one can work along this idea and get the Carter-Penrose diagram by other methods. Besides, we discuss a class of Einstein accelerating vacua, which possess accelerating horizon with the ring topology, thus lead us to find exact solutions with special horizon topology.In Chapter4, we study black holes and their properties in the theory that the scalar field couples to gravity (Einstein-Scalar-AdS Gravity). We give black holes in three dimensions for the theory that the scalar field couples to gravity in a non-minimal way, including a charged hairy black hole and a rotating hairy black hole. They both are AdS black holes, with the cosmological constant appearing naturally as constant term in the self-coupling scalar potential. Under proper choices of parameters, the metric degenerates into some previously known solutions in (2+1)-dimensional gravity. The conditions for the metric to contain0,1,2horizons are identified. We identified the necessary conditions for at least one horizon to be present in the solution, which im-poses some bound on the mass-angular momentum ratio. These hairy solutions extend the family of BTZ black holes and enrich the three dimensional black hole solutions and the understanding of their properties.In Chapter5, we firstly take the four dimensional charged AdS black hole as an example, introduce the P-V criticality of charged AdS black holes in Einstein gravity. Then we study the criticality of charged AdS black holes in Gauss-Bonnet gravity. Tak-ing the (inverse of) Gauss-Bonnet coupling a as a new thermodynamic variable PGB, we revisited the thermodynamics for Gauss-Bonnet-AdS black holes and studied the associated critical behavior at fixed electric charge and bare cosmological constant. It is shown that even for static neutral Gauss-Bonnet-AdS black holes, the corresponding critical point exists only for black holes with spherical topology (i.e. k=+1) in five dimensions, and the set of critical exponents are identical to those of Van der Waals system. This is quite similar to the P-V criticality associated with the cosmological constant at fixed Gauss-Bonnet coupling for the same black holes. However, there is a crucial difference from the case of P-V criticalities:in our case, the phase transi-tion occur only when the temperature is higher than the critical temperature, while the phase transition in P-V criticalities occur only when the temperature is lower than the critical temperature. The situation for static charged Gauss-Bonnet-AdS black holes is much more complicated, and it is shown that there can only be one critical point in five dimensions (for either k=+1or k=-1) when the electric charge Q obeys some appro-priate bound. The corresponding critical exponents are also identical to those for Van der Waals system. In higher dimensions, it is shown that there can be two critical points if the electric charge is taken to be proportional to the (d-3)-th power of the critical radius (Q∞rcd-3). Numerical study also shows that in six dimensions, there can be two different critical points at the same fixed electric charge, and the phase transitions can occur when the temperature is either lower than the lower critical temperature or higher than the higher critical temperature but not in between the two critical temperatures. This situation is not seen in earlier studies on P-V criticalities for the same theory. Therefore, our study indicates that there are still much richer, unexpected structures in the thermodynamics of Gauss-Bonnet-AdS black holes.In the concluding chapter, we summarize the progress that we have made in this paper and outline what deserves to be done in the next step.