| Περίληψη: | The aim of this study is to demonstrate that the thermodynamics of self-gravitating systems in equilibrium is fully specified by variables defined on the system’s boundary. In more precise words, we claim that the thermodynamic state space as well as the thermodynamic properties of the forementioned system can be determined solely by geometric variables defined at the boundary namely, the boundary’s geometry and extrinsic curvature. The presentation that follows may be a restricted sense of holography, but it very important that it arises from the structure of classical gravity theories and the basic principles of thermodynamics with no quantum theory inputs. In particular, we make use of an important relation between Einstein’s equations for static spacetimes and the principle of maximum entropy.
The structure of this project is as follows: Chapter 1 provides a brief review on basic concepts of topology which are necessary for the description of the notion of Global hyperbolicity. In Chapter 2, we present the Hamiltonian formulation of general relativity which requaires a separation of time and space coordinates, known as 3+1 decomposition. In Chapter 3, we briefly describe the proof that the maximum entropy principle leads to Einstein’s equations for static spacetimes and we compute
explicitly the boundary term associated to variations of the appropriate thermodynamic potential. Finally, in Chapter 4, using the results from the previous chapter, we construct the thermodynamic state space as well as some of the basic thermodynamic properties for self-gravitating systems, and we explain in what sense the holography principle is valid.
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