本文主要对经典引力系统中的一些热力学性质(包括黑洞热力学定律和流 体的熵定理)进行了较为系统的研究，并利用Sorce-Wald思想实验对Einstein引 力以及高阶修改引力的弱宇宙监督假设进行了检验和讨论。下面，我们对主 要内容进行简要的阐述：
1. 我们首先计算了耦合电磁场的高曲率引力理论中的Noether荷以及变分恒等式，并在不要求电磁场矢势在分岔面上光滑的情况下，一般性地导出 了渐近平直稳态轴对称黑洞的热力学第一定律。我们的结果表明，即使考虑 了Einstein-Maxwell理论的量子修正，黑洞热力学第一定律也是成立的、具有 普适性的。
2. 黑洞作为一个热力学系统，除了满足第一定律外，也应满足第二定 律。早期研究发现，在高曲率引力中，为了保证线性第二定律成立，我们需 要在Wald熵公式中引入一些额外的修正项去描述黑洞熵。本文首次将相关讨 论推广到含有非最小耦合标量场的理论中，即共形耦合标量场的Lovelock引力 与Horndeski引力，并找到了相应满足线性第二定律的黑洞熵的表达式。通过 与高曲率引力中满足线性第二定律的熵进行比较，我们发现，此时标量场对 黑洞熵有着不可忽略的修正作用，非最小耦合物质场对黑洞热力学的影响是 非平凡的。
3. 除了黑洞以外，时空中也存在一些满足普通热力学的物质场。若要求 物质场的热力学和引力的动力学都成立，时空中的流体应满足熵定理。本文 中，我们利用变分恒等式一般性地证明了微分同胚协变的纯引力系统中静态 自引力理想流体的熵定理：对于在某个紧致区域内满足约束方程的理想流体， 在变分下固定区域内总粒子数和引力场的准局域质量后，该区域内流体的总 熵取极值等价于要求流体分布满足在壳条件，即满足空间部分的运动方程。 我们的研究表明，理想流体的熵定理也蕴含了时空的动力学信息，也显示了 物质场的普通热力学与时空动力学之间的一致性。
4. 我们利用Sorce-Wald思想实验验证了Einstein引力耦合各类物质场时 的弱宇宙监督假设。首先，我们在二阶微扰近似下分别讨论了近极端的静 态dilaton黑洞、 Kerr-Sen黑洞和标量毛Reissner-Nordstrom(RN)黑洞的弱宇宙监督假设。在假定时空满足稳定性条件且物质场满足类光能量条件后，我 们发现，上述黑洞在二阶微扰近似下都不能被破坏。以上结论表明， Einstein引力中的弱宇宙监督假设的有效性可能与背景时空所耦合的物质场的形 式无关。此外，我们还将RN黑洞中的讨论推广到了任意阶球对称微扰近似 下，在利用数学归纳法导出了反映类光能量条件的k阶微扰不等式后，我们发 现，近极端RN黑洞在高阶微扰近似下也满足弱宇宙监督假设。因此可以推断， 近极端RN黑洞的弱宇宙监督假设在微扰水平上是严格成立的。最后，我们 利用Sorce-Wald思想实验重新检验了Li和Bambi的观点：对于不存在奇点的黑 洞，由于不受弱宇宙监督假设的保护，它们的事件视界可以被破坏。我们考 虑了近极端静态带电的无奇点黑洞(regular black hole)在球对称微扰下的Sorce-Wald思想实验，结论表明，即使不存在奇点，它们的视界也不能被破坏。该 结果暗示，除了弱宇宙监督假设之外，可能还存在另外一些基本原理来保证 黑洞不被破坏。
5. 我们利用Sorce-Wald思想实验研究了弱宇宙监督假设对高阶修改引力 的约束作用。对于一个耦合电磁场的高曲率引力理论，我们一般地导出了极端稳态黑洞在一阶Sorce-Wald思想实验中的破坏条件: 。弱宇宙监 督假设作为保证时空因果性的一个基本原理，任何一个合理的经典引力系统 都应满足它。因此，上述黑洞的破坏条件可以对引力理论进行一定的限制。 作为例子，我们考虑了某些二阶和三阶高曲率引力理论，并分别求得了相 应的限制结果。同时，在球对称微扰下，我们也讨论了二阶近似下Lovelock-Maxwell理论的弱宇宙监督假设。结果显示，对于在一阶下不能被破坏的情 况，近极端黑洞在二阶微扰近似下也不能被破坏。我们的研究为弱宇宙监督 假设在经典层面上能约束高阶修改引力理论提供了证据，也为检验天文与宇宙学中的修改引力理论的合理性提供一种新的思路。

This article mainly studies some thermodynamical features of the classical gravitational system (such as thermodynamical law of black hole and the entropy principle of fluid), and examines the weak cosmic censorship conjecture in Einstein gravity as well as higher-order modified gravity by using the Sorce-Wald gedanken experiments. In the following, we briefly explain the content of this article:
1. We first calculate the Noether charge and the variational identity in the higher curvature gravity coupled with electromagnetic fields, and generically derive the first law of thermodynamics for asymptotically flat stationary axisymmetrical symmetric black holes without the requirement that the electromagnetic field is smooth on the bifurcation surface. Our results show that the first law of black hole thermodynamics is valid for any Einstein-Maxwell theory with some quantum corrections.
2. As a thermodynamical system, in addition to the first law, the black hole should also satisfy the second law. Early investigations have shown that we need to introduce some additional term to the Wald entropy in the higher-curvature gravity to ensure the validity of the linearized second law. For the first time, we extend the relevant discussion to the theories with a nonminimally coupled scalar field, such as the scalar-hairy Lovelock gravity and the Horndeski gravity, and find the corrected expressions of the black hole entropy satisfying the linearized second law. By comparing with the black hole entropy which satisfies the second linear law in higher curvature gravity, we show that the scalar field has a significant correction to the black hole entropy. Our result shows that the effect of non-minimally coupled matter fields on the thermodynamics of black holes is non-trivial.
3. In addition to black holes, there also exists some matter field that satisfies the ordinary thermodynamical laws. If we believe that the thermodynamics of the matter field and the dynamics of spacetime are both reliable, the fluid should satisfy the entropy principle. By performing the variational identity, we proved the entropy principle of static self-gravitating fluid in a general diffeomorphism-covariant purely gravitational theory: the extrema of the total entropy inside a compact region for fixed total particle number demands that the static configuration is an on-shell solution after we introduce some appropriate boundary conditions, i.e., it also satisfies the spatial gravitational equations. Our result implies that the entropy principle of fluid contains the dynamic information of gravity, and also shows the consistency between the ordinary thermodynamics of the matter field and the spacetime dynamics.
4. We perform the Sorce-Wald gedanken experiment to verify the weak cosmic censorship conjecture of Einstein gravity coupled with various matter fields. Firstly, we examine the weak cosmic censorship conjecture for static dilaton black holes, Kerr-Sen black holes, and scalar-hairy Reissner-Nordstrom (RN) black holes under the second-order perturbation approximation. Assuming that the spacetime satisfies the stability condition and the matter field satisfies the null energy condition, we find that none of the above black holes can be destroyed under the second-order approximation. These results suggest that the validity of the weak cosmic censorship conjecture in Einstein gravity may be independent of the form of the matter field coupled to the background spacetime. Moreover, we also extend the Sorce-Wald gadanken experiment into the high-order approximation in RN black holes. Under the perturbation of spherically symmetric matter, after deriving the kth order perturbation inequality reflecting the null energy condition, we find that the nearly extremal RN black hole also satisfies the weak cosmic censorship conjecture under the high-order approximation. Therefore, it can be inferred that the weak cosmic censorship conjecture of nearly extremal RN black holes is strictly valid at the perturbation level. Finally, we reexamine the idea of Li and Bambi using the Sorce-Wald gedanken experiment: the event horizon of a regular black hole can be destroyed because these objects have no gravitational singularity and therefore they are not protected by the weak cosmic censorship conjecture. we perform the new version of the gedanken experiments proposed by Sorce and Wald to overcharge a static electrically regular black hole and find that the nearly extremal charged regular black hole cannot be destroyed even if there is no center singularity. These results indicate that there might be some deeper mechanisms to protect the event horizon of the black holes.
5. We use the Sorce-Wald method to investigate the constraint from the weak cosmic censorship conjecture to the high-order modified gravity. For a high-order theory coupled with an electromagnetic field, we generally derived the destruction condition of an extremal black hole under the first-order approximation:. If we require all reasonable theories to satisfy the weak cosmic censorship conjecture, the above conditions can constrain the gravitational theory. As an example, we consider some second- and third-order curvature gravitational theories and obtain the corresponding constraints respectively. At the same time, we also discuss the weak cosmic censorship conjecture of the Lovelock-Maxwell theory under the second-order approximation in spherically symmetric perturbation. The results show that the nearly extremal black holes cannot be destroyed in the second-order approximation for the case that they cannot be destroyed in the first order. Our research provides evidence that the weak cosmic censorship can serve as a constraint to the gravitational theory, and may also provide a new way to test the rationality of modified gravity theories in astronomy and cosmology.