Black hole entropy divergence and the uncertainty principle

Understand

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.