Theoretical Research On Thermodynamics Of Quasilocal Gravitational System

Exploring the thermodynamical phenomena in gravity theory including the ther-modynamic properties of gravitational field and self-gravitating matter can help us to understand the quantum theory of gravity. On the one hand, as a typical object in gravity theory, black hole is a thermodynamic system with temperature and entropy. By properly choosing the energy of gravitational field, we find that the laws of black hole thermodynamics are extremely the same as those in the thermodynamic system by usual matter. There exist no classical limit of black hole temperature and entropy, be-cause they are the results of macroscopic quantum phenomena. So we can study such macroscopic quantum effect by using the methods which we are familiar with in ther-modynamics. The black hole thermodynamics opens a window for us to look into the quantum gravity. On the other hand, the gravitational field can affect the distribution of matter in spacetime, and matter distribution can also change the gravitational field in turn. They are a combined unit. For a self-gravitating system, the thermodynamics of usual matter must be influenced by the gravitational field. We can start from the laws of gravity to reveal the laws of thermodynamics by usual matter. And we can also deduce the gravitational field equations by using the laws of thermodynamics of usual matter. This proves that there is a certain equivalence between gravitational laws and the laws of thermodynamics. Also, the gravitational phenomena may be thermodynamic phe-nomena in essence. There may exist a more fundamental microscopic quantum theory behind.Black branes are the solutions of higher dimensional Einstein gravity theory. They also have horizons, and thus can emit particles through Hawking radiation with temper-ature like black holes. So black brane system is also a thermodynamic system. Unlike black holes, the spacetime singularities of black branes behind horizons take place on some higher dimensional extended objects which are called branes. A typical exam-ple black brane is black D-brane, which is the non-perturbative solution of low energy effective string theory. The D stands for Dirichlet. D-branes are attachment surfaces of open string endpoints with Dirichlet boundary condition. Study the thermodynamic properties of black branes can give us a well understanding of quantum gravity.One way to study the black hole thermodynamics is the traditional one which mainly focus on the global thermodynamic quantities for some thermodynamically sta-ble system like AdS black hole system. Another way is called the quasi-local ensemble method by setting an environment. For black holes or black branes with properties of asympotic flatness, the Hawking radiation will make the system unstable. For such system, we shall set environment to replenish the lost of degree of freedom. We con-sider the thermodynamical phase transitions of black branes with asympotic flatness. We take the quasi-local canonical ensemble by putting the black brane in a cavity as environment, and fix the surface area, surface temperature of the cavity, the brane vol-ume and charge density of black brane are also fixed in the cavity. We find that when the co-dimension of brane is greater than four, there exist a van der Waals type phase transition between two locally stable black branes. While when the co-dimension of brane is less than or equal to four, the van der Waals type phase structure disappear. The co-dimension of D5brane is four, so the phase structure of a single black D5brane is trivial. But when we add many black D1branes parallelly on black D5brane, we recover the van der Waals type phase structure in a region of parameter space.The Einstein gravity is a special kind of gravity theories. When the spacetime di-mensions are greater than four, a more natural theory of gravity is called the Lovelock gravity. The action of Lovelock gravity contains higher order terms of spacetime cur-vature, but the derivatives of spacetime metric in equations of motion are not greater than second order. So this is a ghost free modified gravity theory. We consider a self-gravitating perfect fluid in D dimensional Lovelock gravity theory with a maximally symmetric D-2dimensional space. We deduce the generalized TOV equation by using the time-time and radial-radial components of the gravitational equations. We can also use the maximal entropy principle of perfect fluid together with the time-time component of the gravitational equations to get the generalized TOV equation. This imply the equivalence between gravitational laws and the laws of thermodynamics. For a static self-gravitating perfect fluid, we show that when the Hamiltonian constraint and evolution equation of system are satisfied, the entropy of the perfect fluid will take an extremal value. The role of evolution equation and entropy extremum can be exchanged in our proof. The fluid system is quasi-local, so we set some conditions in our proof, like the Tolman law, the fixed particle number and some boundary conditions for space-time metric. Finally, we find that the boundary conditions for spacetime metric together with the fixed particle number condition can be realized as the isolation condition of the system quasi-locally.