本文的主要工作为将Painlevé-Gullstrand（PG）坐标系推广为一个单参坐标系族，并将其应用在霍金隧穿辐射的计算中。全文将分为四章对此问题进行研究讨论。第一章介绍此课题的研究背景。首先以史瓦西度规为例展示了几种不同坐标系的优缺，并给出PG 坐标系的原始形式；之后对霍金隧穿辐射的历史由来和机制做一简要介绍，说明此坐标系族将被应用的环境。第二章将介绍PG 坐标系在史瓦西时空中的推广以及在隧穿计算中的具体应用。第一节首先详细地展示了如何通过史瓦西时空在史瓦西坐标系下的守恒量表达式建立PG 坐标系，然后运用WKB 近似的结论计算视界处的隧穿辐射率，其中类光测地线法为M.Parikh 和F.Wilczek 首次提出此机制时采用的方法；第二节讲述了类PG 坐标系族的建立过程，并原创性地用统一的方式推导了类时和类光的测地线；第三节讲述了我们将Parikh和Wilczek 的的方法推广到类PG 坐标系族下，并分别用统一测地线法和Hamilton-Jaccobi（HJ）方程法计算了隧穿辐射率，其中首次在HJ 方程中运用了此坐标系族。第三章将介绍PG 坐标系在克尔时空中的推广以及在隧穿计算中的应用。首先我们用复坐标延拓的方式将类PG 坐标系族推广到克尔时空中，并验证了参数的两个极限。之后我们将其运用到克尔外视界处隧穿辐射率的计算中。由于本章内容大多为作者多次尝试后的新计算，所以其中有着诸多诸如可行性之类的分析以及延拓和所选计算方法的讨论。第四章将总结全文工作，明确列出文中主要的新工作，同时提出了本文的有待改进之处，另外讨论了文中的方法在未来工作中可以推广的方向。附录给出作者对弯曲时空中的HJ 方程的简洁推导方法。全文采用几何单位制c = G = 1。

The main work of this thesis is to generalize the Painlevé-Gullstrand（PG）coordinatesystem into a one-parameter family of coordinate systems, and to apply this family of coordinatesinto the calculation of Hawking radiation as tunneling. We will use four chapters to discuss theissue.In the first chapter, the research background is introduced. Firstly we set Schwarzschildmetric as an example to show the advantages and disadvantages of different coordinates, and givethe original form of PG coordinates explicitly; then we give a brief introduction of the historicalevolution and mechanics of Hawking radiation as tunneling, and explain the environment thecoordinates are applied.In the second chapter, we introduce the generalizations of PG coordinates in Schwarzschildspacetime and its applications in computations of tunneling in detail. In the first section we showhow to construct the PG coordinates through the conservative quantity in Schwarzschild coordinatesof Schwarzschild spacetime, and calculate the tunneling rate at the horizon via WKB approximation,where the null geodesics approach was the one that M.Parikh and F.Wilczek’s usedwhen they came up with the idea in the first time. In the second section, we construct the Painlevé-Gullstrand-like(PG-like) coordinates, and derive thetimelike and null geodesics in a unifying newway; In the third section we generalize Parikh and Wilczek’s approach into cases in PG-like coordinatefamily, and calculate the tunneling rate of radiation via the general geodesics’ approach andHamilton-Jaccob（HJ）formulas’ approach respectively, where the applications of this coordinatefamily in the HJ approach is for the first time.In the third chapter, we we introduce the generalizations of PG coordinates in Kerr spacetimeand its applications in computations of tunneling. We will firstly use complex extension togeneralize the PG-like into the cases in Kerr spacetime. Due to the fact that the majority of contentsin this chapter are new calculations after the author’s several trials, there are numbers of analysisIIlike the feasibilities and several discussions of why we use certain calculation methods.In the fourth chapter, we make a conclusion of the thesis. We list the main works in thisthesis, and figure out the points to be improved. And we discuss the possibilities of extensions ofthis approach in future works.In the Appendix the author’s derivation of the HJ equation in curved spacetime is given.Geometric units c = G = 1 are used throughout the thesis.