Local first law of black hole

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.