本文主要分为三个部分。第一部分为2-4章，在这部分中我们推广了黑洞内部体积的定义，计算了Kerr、Kerr-Newman(KN)黑洞的内部体积。同时我们对Stefan-Boltzmann公式也进行了推广，使其能够描述同时包含能量、电荷、角动量和宇宙学常数变化的广义Hawking辐射。基于推广后的黑洞体积定义以及Stefan-Boltzmann公式，我们计算了Reissner-Nordstrom (RN)、Kerr、KN和Schwarzschild-(Anti) de Sitter黑洞内部体积中标量场的熵，并构建了标量场的熵与Bekenstein-Hawking熵在Hawking辐射下的演化关系，并根据此关系研究了黑洞在霍金辐射过程中的信息佯谬问题。第二部分为第5章，我们采用热力学几何中外曲率方法研究了3维Einstein-Maxwell-dilaton黑洞的相变。第三部分为第6章，主要检验了RN-AdS黑洞和Banados-Teitelboim-Zanelli (BTZ)黑洞在物质场微扰的二阶近似下是否依然满足弱宇宙监督假设。下面我们将每一章中的具体研究内容以及所得到的结论做详细的阐述。

第1章主要介绍了黑洞物理相关的基础知识，主要包括Schwarzschild黑洞、RN黑洞、Kerr黑洞以及Kerr-Newman (KN)黑洞的时空几何结构，以及黑洞相关的物理性质，如：黑洞热力学、宇宙监督假设以及Hawking辐射。

第2章中，我们将Christodoulou和Rovelli (CR)所给出的球对称黑洞的内部体积定义推广到了轴对称黑洞中。由于轴对称黑洞的时空结构要比球对称黑洞复杂，我们无法解析的求出极大超曲面在轴对称黑洞内的位置。因此根据Schwarzschild黑洞内部体积所对应的类空超曲面的性质，我们直接在Kerr黑洞和KN黑洞内选择一个最大的等r超曲面，并根据极大超曲面外曲率的迹为0这一性质对所选的超曲面进行了验证。结果发现在Kerr黑洞和KN黑洞内，最大等r超曲面并不是CR定义中所指的极大超曲面，但是该超曲面十分接近于极大超曲面。因此我们可以用最大等r面的体积来近似的代替黑洞的内部体积。

第3章中，我们首先回顾并讨论了张保成所提出的计算Schwarzschild黑洞内部体积中标量场熵的方法，以及该方法中存在的一些问题。随后将Stefan-Boltzmann定律推广为能够描述同时包含能量、电荷和角动量的广义Hawking辐射的形式。在此基础上通过计算RN、Kerr、KN黑洞内标量场的熵与Bekenstein-Hawking熵随Hawking辐射的演化关系，研究了黑洞在Hawking辐射过程中信息佯谬问题。结果发现，两类熵之间的演化成近似的正比例关系。同时，该演化关系也可以随着Hawking辐射的进行退化为Schwarzschild黑洞的情况，即两类熵之间的演化关系为精确的正比例关系。当考虑Hawking辐射只包含能量时，所得到的演化关系表达式与考虑广义Hawking辐射时所得到的表达式相同。两类熵之间这种近似正比例的演化关系说明了，黑洞的信息有可能随着Hawking辐射的进行存储在了黑洞的内部体积中。

第4章中，我们计算了Schwarzschild-(Anti) de Sitter黑洞的内部体积，并研究了该体积与宇宙学常数之间的变化关系。同时我们继续推广Stefan-Boltzmann定律，使其能够描述包含宇宙学常数变化的Hawking辐射。利用推广后的Stefan定律，我们依然通过构建标量场的熵与Bekenstein-Hawking熵之间的演化关系来研究Schwarzschild-(Anti) de Sitter黑洞的信息佯谬。结果发现，两类熵随Hawking辐射的演化关系也存在近似正比例的关系。该结果说明对于Schwarzschild-(Anti) de Sitter黑洞来说，其表面的信息也有可能随着Hawking辐射的进行存储在了黑洞的内部体积中。

第5章中，我们采用热力学几何中外曲率的方法研究了3维Einstein-Maxwell-dilaton黑洞的相变。发现外曲率可以很好地反映黑洞的相变点与相变前后的稳定性，但是外曲率曲线中存在额外的发散。当把宇宙学常数当做变量时，外曲率中额外的发散点消失，该曲线能够完全替代热容曲线来反映黑洞的相变点与相变前后的稳定性。

第6章中，我们采用Sorce和Wald所提出的新方法来验证经过物质场微扰后RN-AdS黑洞和BTZ是否还满足弱宇宙监督假设。在计算过程中我们把宇宙学常数当做一个动力学变量来处理。结果发现在物质场微扰的二阶近似下，近极端RN-AdS黑洞和BTZ黑洞仍然满足弱宇宙监督假设。由于BTZ黑洞的奇点是圆锥奇点，该奇点不会引起时空发散，因此上述结果也说明即使不引起时空曲率发散的奇点也要被黑洞的事件视界所包围并隐藏在黑洞内。

最后在第7章我们对前面所得到的结论进行了总结，并分析和讨论了在这些研究中需要进一步解决和完善的问题。

This doctoral dissertation is divided into three parts. The first part contains Chapters 2-4. In this part, we generalize the definition of the interior volume from a spherically symmetric black hole to an axially symmetric black hole. While using the definition, the interior volume of Kerr and Kerr-Newman (KN) black holes is calculated respectively. Besides, the Stefan-Boltzmann law is also generalized. The generalized Stefan-Boltzmann law can describe the Hawking radiation carrying energy, charge, angular momentum and the varying of the cosmological constant. Based on the definition of the interior volume and the general Stefan-Boltzmann law, the scalar filed entropy in the interior volume of Reissner-Nordstrom (RN), Kerr, KN and Schwarzschild-(Anti) de Sitter black holes is calculated respectively. Furthermore, the evolution relation between the scalar field entropy and Bekenstein-Hawking entropy under Hawking radiation is constructed. According to the evolution relation, the information paradox of black holes is investigated. The second part is in Chapter 5. In this part, we use the method of the extrinsic curvature in the thermodynamic geometry to investigate the phase transition of three-dimensional Einstein-Maxwell-dilaton black holes. The third part contains Chapter 6. In this part, we examine whether the Weak Cosmic Censorship Conjecture (WCCC) of RN-AdS and Banados-Teitelboim-Zanelli (BTZ) black holes under the second-order perturbation of the matter filed is still valid respectively. The main content of each chapter is summarized one by one as follows.

In Chapter 1, we primarily introduce the relevant background knowledge about black holes, such as the spacetime geometries of Schwarzschild, RN, Kerr, and KN black holes, the black hole thermodynamics, the Cosmic Censorship Conjecture, and Hawking radiation.

In Chapter 2, we generalize the definition of the interior volume of a spherically symmetric black hole proposed by Christodoulou and Rovelli (CR) to an axially symmetric black hole. Since the spacetime geometry of axially symmetric black holes is more complicated than the geometry of a spherically symmetric black hole, we cannot use the definition to calculate the position of the largest hypersurface in an axially symmetric black hole. However, according to the property of the largest hypersurface in Schwarzschild black holes, we choose a maximal hypersurface at constant r inside a Kerr and a KN black hole directly. Based on the theorem, the extrinsic curvature of the largest hypersurface is zero, we demonstrate that the hypersurface is not the largest hypersurface derived from the definition, but the hypersurface extremely approaches to the largest hypersurface. It means that the hypersurface can be regarded as the largest hypersurface approximately, and its volume is the interior volume of Kerr and KN black holes approximately.

In Chapter 3, we first review the method of calculating the scalar field entropy in the interior volume of Schwarzschild black holes proposed by Baocheng Zhang and discuss some relevant problems in this method. Subsequently, the Stefan-Boltzmann law is promoted to the general case, which can describe Hawking radiation carrying energy, charge and angular momentum. After that, the information paradox of RN, Kerr, KN black holes is investigated respectively. Utilizing the method and general Stefan-Boltzmann law, we calculate the scalar field entropy in the interior volume of black holes and construct the evolution relation between the scalar field entropy and Bekenstein-Hawking entropy under general Hawking radiation. It is shown that the two types of entropy are proportional to each other approximately, and the evolution relation can degenerate into the Schwarzschild case with general Hawking radiation. Considering Hawking radiation carrying only energy, the evolution relation is the same as that under general Hawking radiation. The results show that the lost information with Hawking radiation may be stored in the interior volume of black holes.

In Chapter 4, we calculate the interior volume of Schwarzschild-(Anti) de Sitter black holes and investigate the changing behavior of the interior volume with the cosmological constant. Besides, we generalize the Stefan-Boltzmann law to contain the varying cosmological constant. Using the law, we find that the scalar field entropy inside the black hole is still proportional to Bekenstien-Hawking entropy under Hawking radiation.

In Chapter 5, we use the thermodynamics geometry to investigate the phase transition of the 3-dimensional Einstein-Maxwell-dilaton black hole. It is shown that the extrinsic curvature can locate the critical point of the black hole and reflect the stability of the black hole near the critical point. However, an extra divergence appears in the curve of the extrinsic curvature. If the cosmological constant is regarded as a dynamical variable, the extra divergence disappears, and the extrinsic curvature can replace the capacity to reflect both the critical point and the stability of the black hole.

In Chapter 6, based on the new version of the gedanken experiment proposed by Sorce and Wald, we examine the WCCC of RN-AdS and BTZ black holes under the second-order perturbation of the matter filed. The cosmological constant is regarded as a dynamical variable. The results show that after the perturbation of the matter field, the nearly extremal RN-AdS and BTZ black holes cannot be destroyed under the matter field perturbation. It means that the WCCC of RN-AdS and BTZ black holes are still valid under the perturbation. Meanwhile, it is also shown that even if the singularity is conical, not canonical, it also satisfies the WCCC and should be surrounded by the event horizon.

In Chapter 7, we summarize the conclusions firstly, and then we introduce some remaining problems to be resolved and improved in these researches.