Entropy of Horizons, Complex Paths and Quantum Tunneling

TL;DR

This paper explores the entropy associated with horizons in spacetime, analyzing quantum tunneling across horizons and deriving a relation between entropy, surface gravity, and energy leakage, with implications for horizon thermodynamics.

Contribution

It introduces a general framework linking quantum tunneling probabilities to horizon entropy and provides a formula relating entropy change to variations in mass and surface gravity.

Findings

Tunneling probability is proportional to exp[-(2π/κ)E]

Horizon entropy S(M) relates to energy leakage and surface gravity

Conditions for horizon area contributions to entropy are discussed

Abstract

In any spacetime, it is possible to have a family of observers following a congruence of timelike curves such that they do not have access to part of the spacetime. This lack of information suggests associating a (congruence dependent) notion of entropy with the horizon that blocks the information from these observers. While the blockage of information is absolute in classical physics, quantum mechanics will allow tunneling across the horizon. This process can be analysed in a simple, yet general, manner and we show that the probability for a system with energy $E$ to tunnel across the horizon is $P(E)\propto\exp[-(2\pi/\kappa)E)$ where $\kappa$ is the surface gravity of the horizon. If the surface gravity changes due to the leakage of energy through the horizon, then one can associate an entropy $S(M)$ with the horizon where $dS = [ 2\pi / \kappa (M) ] dM$ and $M$ is the active gravitational mass of the system. Using this result, we discuss the conditions under which, a small patch of area $\Delta A$ of the horizon contributes an entropy $(\Delta A/4L_P^2)$, where $L_P^2$ is the Planck area.