黑洞热力学第一定律描述了黑洞演化时，表征黑洞性质的物理量（质量，面积，电量和角动量）之间相应的变化关系，它已经通过许多方式被证明。其中基于Killing视界的物理过程第一定律$\frac{\kappa}{8\pi}\Delta A=\Delta E_{\chi}$，考虑的是一个稳态黑洞中落入少量物质而发生的“无限小”物理过程，形成另一个稳态黑洞。该过程中，沿事件视界上外行Killing矢量场，黑洞视界面积的变化与流入视界的Killing能量相差比例系数$\frac{\kappa}{8\pi}$。证明中，要求加入的物质为少量，但是这个条件是否足够？我们计算给出的结果是否定的。为了验证物理过程第一定律，我们考虑在稳态施瓦希黑洞中分别加入Tolman-bondi尘埃和Vaidya尘埃的情况。通过定性分析和数值计算，我们发现要使第一定律成立，不但加入的物质质量要小，使得膨胀较小，可以忽略二阶项，而且加入的物质要满足一定条件，使得物质密度较小，弱引力条件满足。 另外，我们尝试构造了一个两阶光滑简单时空(simple space-time)，内部为球对称物质，外部为极端RN电磁真空，它满足主能量条件，不存在奇点。 在介绍我们的工作之前，文章还简要介绍了稳态黑洞的热力学四定律和研究非稳态问题时引入的几个局域视界的定义，并对陷俘视界及其上的热力学给出了比较详细的介绍。

When a stationary black hole is perturbed and evolves to another stationary black hole, the changes in area, mass, charge and angular momentum are related by the first law, which has been proved in different ways. The physical process version ofthe first law states that the energy passing through a Killing horizon results in a change in area by$\frac{\kappa}{8\pi}\Delta A=\Delta E_{\chi}$. The first law requires that the mass added to the black hole be tiny and then can be considered as a perturbation. The purpose of our work is toinvestigate whether additional conditions are needed to guarantee the first law. We test the first law in the specific physical processes involving Tolman-Bondi dust and Vaidya dust. By qualitative analysis and numerical calculation, we find that, in order to validate the first law, in addition that the total mass of the matter is small, the density of the matter must be small too so that the weaklyself-gravitating condition is satisfied. We also construct a $C^2$ simple (singularity-free) space-time. It consists of the neutral dust in the interior and the extreme Reissner-Nordstr$\ddot{\mathrm{o}}$m electrovacuum region in the exterior. In the beginning of this thesis, we present a background introduction on black hole thermodynamics in stationary cases as well as local horizons innon-stationary cases . In Particular, works on trapping horizons and their thermodynamical properties are reviewed.