Wave Equations In Black Hole Background And Related Problems

This thesis consists of 5 chapters:In chapter 1,we introduced the concepts and developments of the black hole perturbation.In chapter 2,firstly,we introduced the singularity points of Fuchs type equations that all singularities are regular based on the second-order differential equations.We discussed the ordinary and singular points of Fuchs type equations,and the singular points can be divided into removable singularity,pole and essential singularity,the regular singularity is a kind of the pole.Secondly,we discussed Heun equations and confluent Heun equations as well as some other special differential equations,we discussed the solution of these equations.When the series solution of the equation were given intial conditions,the relationship between coefficients can be seen as expansion of a three-term recurrence relation.So we discussed the characteristics of the three diagonal determinant and the solution of the equation with three diagonal determinant zero.In chapter 3,we discussed the Newman-Penrose formalism of the dynamics equations which contain various field.We gave the definition of the null tetrad,and obtained the spin coefficient equations and the field equations.We investigated the field equations in general spherically symmetric spacetime.Using the Newman-Penrose formalism,we got the uniform expression of the field equations.The field equations were separated into the radial equations and the angular equations.The solution of the angular equation was the spin-weighted spherical harmonics.The radial equation was divided into different classes according to the event horizon.Through the transformation of variables and functions,we converted the equations into a variety of well-known special differential equations,and studied the solution.In chapter 4,we probed into the field equations in spherical symmetric spacetime in conformal gravity,Kerr-Newman spacetime and Grumiller spacetime respectively.We obtained the radial equations in the three kind of spacetime and converted them into Heun equations.In chapter 5,we give the summarization and the prospects.Finally,in the appendix,the Strum theorem is presented to classify the event horizon and get the constraint conditions.