Black hole entropy divergence and the uncertainty principle

TL;DR

This paper links the divergence of black hole entropy to the quantum uncertainty principle, showing that smoothing the boundary between observable and unobservable regions can tame the divergence.

Contribution

It demonstrates that black hole entropy divergence arises from the uncertainty principle and that smoothing the boundary reduces this divergence, extending the concept to quantum field theories.

Findings

Divergence in energy and entropy is due to sharp boundary dividing regions.

Smoothing the boundary reduces the divergence.

Black hole entropy divergence can be explained by the uncertainty principle.

Abstract

Black hole entropy has been shown by 't Hooft to diverge at the horizon. The region near the horizon is in a thermal state, so entropy is linear to energy which consequently also diverges. We find a similar divergence for the energy of the reduced density matrix of relativistic and non-relativistic field theories, extending previous results in quantum mechanics. This divergence is due to an infinitely sharp division between the observable and unobservable regions of space, and it stems from the position/momentum uncertainty relation in the same way that the momentum fluctuations of a precisely localized quantum particle diverge. We show that when the boundary between the observable and unobservable regions is smoothed the divergence is tamed. We argue that the divergence of black hole entropy can also be interpreted as a consequence of position/momentum uncertainty, and that 't Hooft's brick wall tames the divergence in the same way, by smoothing the boundary.