Local first law of black hole

TL;DR

This paper explores the local first law of black hole thermodynamics from the perspective of an observer at a finite distance, demonstrating its validity and scaling behavior in a quasilocal context.

Contribution

It extends the understanding of black hole thermodynamics by analyzing the quasilocal first law at finite distances from the horizon, revealing its persistence and scaling properties.

Findings

The quasilocal first law holds at finite distances from the black hole horizon.

The law's form scales with the observer's location.

The study confirms the applicability of the quasilocal first law beyond the horizon.

Abstract

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 $\delta E=\frac{\bar{\kappa}}{8\pi}\delta A$, where $\delta E$ 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.