Mimimal Length Uncertainty Principle and the Transplanckian Problem of Black Hole Physics

TL;DR

The transplanckian problem in Hawking radiation arises from extreme localization near the horizon, yielding unphysical blue-shifts of outgoing modes. By adopting the Kempf–Mangano–Mann (KMM) minimal length uncertainty principle via a mutilated commutator, the paper derives a nonlocal framework in which horizon-adjacent modes do not remain confined to arbitrarily small distances; the relevant scale is set by $|r-2M| \simeq \beta_H \omega /(2\pi)$. The analysis shows that modes acquire a nonlocal character, with wave packets spreading over a region of width $\Omega_H$ and exhibiting non-conservation of the Klein–Gordon current within the nonlocal zone, interpreted as the Hawking mode interacting with a reservoir of other modes. This reservoir interpretation naturally leads to a non-unitary effective evolution and motivates a conjecture that Bekenstein–Hawking entropy may originate from fluctuations of fields across the nonlocal region. The work provides a potential route to resolve the transplanckian problem and links horizon physics to entropy through the dynamics of nonlocal quantum mechanics, while underscoring the need for a quantum gravity framework to fully identify the reservoir and its implications.

Abstract

The minimal length uncertainty principle of Kempf, Mangano and Mann (KMM), as derived from a mutilated quantum commutator between coordinate and momentum, is applied to describe the modes and wave packets of Hawking particles evaporated from a black hole. The transplanckian problem is successfully confronted in that the Hawking particle no longer hugs the horizon at arbitrarily close distances. Rather the mode of Schwarzschild frequency $ω$ deviates from the conventional trajectory when the coordinate $r$ is given by $| r - 2M|\simeq β_H ω/ 2 π$ in units of the non local distance legislated into the uncertainty relation. Wave packets straddle the horizon and spread out to fill the whole non local region. The charge carried by the packet (in the sense of the amount of "stuff" carried by the Klein--Gordon field) is not conserved in the non--local region and rapidly decreases to zero as time decreases. Read in the forward temporal direction, the non--local region thus is the seat of production of the Hawking particle and its partner. The KMM model was inspired by string theory for which the mutilated commutator has been proposed to describe an effective theory of high momentum scattering of zero mass modes. It is here interpreted in terms of dissipation which gives rise to the Hawking particle into a reservoir of other modes (of as yet unknown origin). On this basis it is conjectured that the Bekenstein--Hawking entropy finds its origin in the fluctuations of fields extending over the non local region.