Sub-Planckian black holes and the Generalized Uncertainty Principle

TL;DR

This paper proposes a self-dual, Schwarzschild-like metric for sub-Planckian black holes, linking it to a generalized uncertainty principle and suggesting potential dark matter relics.

Contribution

It introduces a novel self-dual metric model that unifies sub-Planckian and large-mass black holes and connects to a linear form of the Generalized Uncertainty Principle.

Findings

The model reproduces key features of sub-Planckian black holes.

Sub-Planckian black holes exhibit (1+1)-D thermodynamics.

Small black hole evaporation is suppressed, implying dark matter relics.

Abstract

The Black Hole Uncertainty Principle correspondence suggests that there could exist black holes with mass beneath the Planck scale but radius of order the Compton scale rather than Schwarzschild scale. We present a modified, self-dual Schwarzschild-like metric that reproduces desirable aspects of a variety of disparate models in the sub-Planckian limit, while remaining Schwarzschild in the large mass limit. The self-dual nature of this solution under $M \leftrightarrow M^{-1}$ naturally implies a Generalized Uncertainty Principle with the linear form $\Delta x \sim \frac{1}{\Delta p} + \Delta p$. We also demonstrate a natural dimensional reduction feature, in that the gravitational radius and thermodynamics of sub-Planckian objects resemble that of $(1+1)$-D gravity. The temperature of sub-Planckian black holes scales as $M$ rather than $M^{-1}$ but the evaporation of those smaller than $10^{-36}$g is suppressed by the cosmic background radiation. This suggests that relics of this mass could provide the dark matter.