Coveriance Of Hawking Radiation And Black-hole's Entropy, And Evolution Of Perturbations In Ho(?)ava-Lifshitz Gravity

This thesis is devoted to investigate some trickiness problems in the study of Hawking radiation and the statistical-mechanical entropies of a black hole, and dy-namical evolution of the scalar perturbation in the black hole spacetime of Horava-Lifshitz gravity which is hopeful model for quantization of gravity in small scale. Three results are listed as follows:Firstly, based on physical notion and mathematical technique, we have com-pletely solved two kinds of trickiness issues appeared in the study of Hawking ra-diation, i.e., (1) The Hawking temperature of Schwarzschild black hole can not be obtained in the Kruskal-Szekers and dynamic Lemaitre coordinate representations; (2) The obtained temperature is one half of Hawking temperature of Schwarzschild black hole in the isotropic coordinate representation. To solve these problems, we find that, in the physical field, we can get correct Hawking temperature only by adopting the correct definition of the particle energy. It is easy to see that, from formulismΓ[emission]= e--E/THΓ[absorption], the black hole's Hawking tempera-ture is related to the energy of the radiating particle. Therefore, in order to get correct Hawking temperature, the correct definition of the particle energy must be presented. By means of the Euler-Lagrangian equation and using the fact thatξμpμis a constant in coordinate transformations, we present a definition of the energy of the radiating particles as E=-ξμpμ.Then the first kind of problem can be completely solved by using this definition of the energy. As an example, by using this definition, we study the Hawking radiation of Schwarzschild black hole in the Kruskal-Szekers and dynamic Lemaitre coordinate systems, and find that the Hawking temperature of Schwarzschild black hole is invariant. In the mathematical field, when the radial coordinate transformation is a non-regular or zero function at event horizon, we find that the correct Hawking temperature can be obtained exactly as long as the integral contour is deformed based on the mathematical transform. So the second kind of problem, such as the factor of "(?)" problem, can be ultimately solved. As an example, we study the scalar and Dirac particles tunneling of Kerr-Newman black hole in a general coordinate represen-tation and highlight that the correct Hawking temperature can be obtained when the integral contour is deformed corresponding to the mathematical transform. In a word, using the correct definition of particle energy and deformation of the integral contour, the correct Hawking temperature of a black hole can be obtained in any coordinate representation.Secondly, by using t'Hooft's "Brick wall" model in which the black hole de-grees of the freedom are identified with the ones of a quantum gas of particles and the statistical-mechanical entropy is arisen from a thermal bath of the quantum fields propagating outside the horizon, we calculate the statistical-mechanical en-tropies of Schwarzschild Anti-de Sitter black hole and Schwarzschild-de Sitter black hole in three kinds of coordinate representations, i.e., Schwarzschild-like, Painleve and Lemaitre coordinates. The "Brick wall" model based on the conception of the particle which is dependent on observer in the curved spacetime. This leads to an interesting question, i.e., is the entropy invariant? We find here that (1) although either Painleve or Lemaitre coordinate does not possess the coordinate's singular-ity, the event horizon manifests itself as a singularity in the action function and then there could be particles production, hence we can use the knowledge of the wave modes of the quantum field to calculate the statistical-mechanical entropies; (2) when we construct a vacuum state for the massless arbitrary spin fields in Painleve spacetime we take the condition and then we find that the modes used to calculate the entropies in both the Painleve and Lemaitre coordinates are exactly the same as those in the Schwarzschild-like coordinates since both V and t tend to Schwarzschild time ts as r goes to infinity under this condition. It is interesting to see that though the conception of the particles is dependent on the coordinate representations, the statistical-mechanical entropies are covariant. The result shows that entropy is an intrinsic property of the black hole.Thirdly, by using numerical method, we study the dynamical evolution of the massless scalar perturbation in the background of Horava-Lifshitz black-hole spacetime for the cases of dynamical coupling constants,λ= 1/3,1/2 and 3. We find that there are purely damped modes which are different from those in the usual black hole spacetimes. For the case ofλ= 1/3 and 1/2, the imaginary parts of the frequency are proportional to its Hawking temperature not only for large and intermediate black holes, but also for small black holes. If black holes possess the same Hawking temperature, the decay of the scalar perturbation is much quickly than that in Schwarzschild AdS black hole spacetime. Forλ= 3 case, the imaginary parts of frequency are not proportional to its Hawking temperature any more, but are linear to the radii of the event horizon of the black hole. We think that the purely damped modes depend not only on the shape of potential, but also on its asymptotic form at large distance. By using analytical method, we also study the dynamical evolution of the massless scalar perturbation in the background ofλ= 1/2 Horava-Lifshitz black-hole spacetime and obtain the greybody factor which is valid for any frequency.