We investigated the form and implications of the local first law of black hole thermodynamics in relation to an observer located at a finite distance from the black hole horizon. Our study is based on the quasilocal form of the first law for black hole thermodynamics, given by \(\delta {\textsf{E}}=\frac{{\bar{\kappa }}}{8\pi }\delta A\), where \(\delta {\textsf{E}}\) and \(\delta A\) represent the changes in the black hole mass and area, respectively, and \({\bar{\kappa }}\) denotes the quasilocal surface gravity. We show that even at a finite distance, the quasilocal law still holds. It shows how the first law scales with the observer’s location.

我们研究了与位于距黑洞视界有限距离的观察者相关的黑洞热力学局部第一定律的形式和含义。我们的研究基于黑洞热力学第一定律的拟局部形式，由\(\delta {\textsf{E}}=\frac{{\bar{\kappa }}}{8\pi }\delta给出A\)，其中\(\delta {\textsf{E}}\)和\(\delta A\)分别表示黑洞质量和面积的变化，而\({\bar{\kappa }} \)表示准局部表面重力。我们证明，即使在有限距离处，拟局部定律仍然成立。它显示了第一定律如何随观察者的位置变化。