The Reaction Of The Static Spherically Symmetric Space-time, Statistical Thermodynamics And Its De Broglie-bohm, Quantum Of,

The spherically symmetric space-time, which is widely studied, is one of the simple and important models in gravitational theory. The contents of the present thesis are mainly focus on some effects of quantum and thermodynamical properties for Reissner-Nordstrom (RN) space-time, de Sitter space-time and Reissner-Nordstrom-de Sitter (RNdS) space-times. In addition we try to study the quantum effect of the spherically symmetric black hole in Brans-Dicke theory in view of the de Broglie-Bohm (dBB) interpretation. The thesis is composed of three parts as follows.In Part one, we study that the back-reaction of the thermal radiation to the black hole geometry. By consideration of quantum field theory in curved spacetime, Hawking in 1974 found that the black hole should emit a black body radiation. However, the radiation will alter the background geometry according to the semiclassical Einstein equation. In 1985 York introduced a perturbed method to solve the back-reaction problem, which regard the renormalized energy-stress tensoe of a vacuum quantum matter field as the source of gravity and then solve the semiclassical Einstein equation to obtain the perturbed geometry. As an example, York found the back-reaction of the conformally invariant scalar field on the Schwarzschild black hole. The most important thing in this method is obtained a renormalized energy-stress tensor of the vacuum quantum matter field.Part one includes one Chapter and two appendixes (Appendix A and B). In chapter one, by using the renormalized energy-stress tensor of the conformally invariant scalar field given by Huang, we study the back-reaction on RN space-time and then the properties of the perturbed RN spacetime. We also discuss the stability of the Cauchy horizon in RN spacetime and find that it will become spacelike hypersurface in perturbed RN spacetime. In Appendix A and B. we give a brief introduction on the renormalized energy-stress tensor and its calculation.In Part two we study the statistical thermodynamics of the black hole by using the Euclidean action of the gravity. This Part includes three chapters (Chapter two,Chapter three and Chapter four). In Chapter two, we briefly introduce Hawking's path-integral approach to quantum gravity and York's formalism developed from the Hawking's path-integral approach. In York's formalism, the black hole is surrounded by a cavity of a boundary with a finite radius. By fixing various data on the boundary we can get various thermodynamical ensemble. We find that if the boundary is located inside the Cosmological horizon, the York's formalism can be generalized to the asymptotically de Sitter sapcetime. So using the York's formalism we study the de Sitter spacetime (the first section of the Chapter three) and luckwarm black hole ( the second section of the Chapter four) and then obtain their first law of the thermodynamics.In Second two of the Chapter three, considering the universe is not "empty" we investigate the thermodynamics of the de Sitter universe by using a grand canonical ensemble, and gives the first law of thermodynamics including the vacuum energy and pressure contributed by the cosmological constant.In Second three of the Chapter three, using the canonical ensemble and anti-Wick rotation we study again the thermodynamics of the de Sitter spacetime. Many years ago, Hawking conjectured that the spacetime foam may be formed from the S2 x S2 bubble. Recently Liu shows that the S2 x S2 bubble can be created really from vacuum fluctuation in one-loop approximation of both steady state universe and closed de Sitter universe. Using this result, we calculate the one-loop correction of the entropy for de Sitter sapcetime in the Section three of the Chapter three.In chapter four we first briefly introduce some special UN spacetime, and then using the York's formlism we study the thermodynamics of the luckwarm black hole which is a RNdS spacetime in thermal equilibrium.In Part three (the Chapter five), using the canonical quantum approach of gravity, an spherically symmetry black hole in BD theory is studied in view of the dBB interpretation. The Wheeler-DeWitt(WD) equation is solved in a minisuperspace model. The dBB interpretation of the wave function is different from the Copenhagen interprata-tion, the latter is the usual probability interpretation of the wave function. According to the dBB interpretation the wave function is considered to have a " trajectory ", which gives the quantum trajectory and the quantum potential. When the quantum effect can be negligible, the quantum potential will tends to zero and the quantum trajectory will reduced to the classical trajectory. Kenmoku et al investigated the quantum effects of the Schwarzschild black hole by using the canonical quantum approach in view of the dBB interpretation. We generalize their method to the BD theory and study the quantum effect of the Brans type I black hole, which is a non-trivial BD black hole solution different from the usual Schwarzschild solution. In addition we also discuss some thermodynamical properties of the quantum Brans type I.