Lower bound on the spectral dimension near a black hole

TL;DR

This paper explores how the spectral dimension of spacetime near an evaporating black hole decreases to below three, leading to a halt in evaporation and establishing a universal lower bound on observable dimensions.

Contribution

It demonstrates that black hole evaporation stops at a scale where spacetime becomes effectively 3-dimensional, independent of specific models of spectral dimension variation.

Findings

Evaporation halts when the horizon reaches a scale with spectral dimension below three.

External observers cannot probe scales smaller than a universal bound of spectral dimension greater than two.

The result holds regardless of the details of the spectral dimension's variation with diffusion time.

Abstract

We consider an evaporating Schwarzschild black hole in a framework in which the spectral dimension of spacetime varies continuously from four at large distances to a number smaller than three at small distances, as suggested by various approaches to quantum gravity. We demonstrate that the evaporation stops when the horizon radius reaches a scale at which spacetime becomes effectively 3-dimensional, and argue that an observer remaining outside the horizon cannot probe the properties of the black hole at smaller scales. This result is universal in the sense that it does not depend on the details of the effective dimension as a function of the diffusion time. Observers falling into the black hole can resolve smaller scales, as can external observers in the presence of a cosmological constant. Even in these cases, though, we obtain an absolute bound D>2 on the effective dimension that can be seen in any such attempt to measure the properties of the black hole.