Divergence on the Horizon

TL;DR

This paper explains the divergence of black hole entropy at the horizon as a consequence of quantum uncertainty, showing it parallels entanglement entropy and suggesting a unified understanding of black hole thermodynamics.

Contribution

It demonstrates that the divergence in black hole entropy arises from quantum uncertainty near the horizon, linking it to entanglement entropy and smoothing the 'brick wall' boundary.

Findings

Black hole entropy divergence is due to position/momentum uncertainty.

Smoothing the horizon boundary removes the divergence.

Black hole entropy can be understood as entanglement entropy.

Abstract

Black hole entropy has been shown by 't Hooft to diverge at the horizon, whereas entanglement entropy in general does not. We show that because the region near the horizon is a thermal state, entropy is linear to energy, and energy at a barrier is inversely proportional to barrier slope, and diverges at an infinitely sharp barrier as a result of position/momentum uncertainty. We show that 't Hooft's divergence at the black hole is also an example of momentum/position uncertainty, as seen by the fact that the "brick wall" which corrects it in fact smooths the sharp boundary into a more gradual slope. This removes a major obstacle to identification of black hole entropy with entanglement entropy.