ABC of multi-fractal spacetimes and fractional sea turtles

TL;DR

This paper explores the properties of multi-fractal spacetimes in quantum gravity, illustrating how fractal dimensions affect physical phenomena like particle speed and discussing implications for relativistic objects and stochastic spacetimes.

Contribution

It introduces weak and strong definitions of multi-fractal spacetimes, analyzes their physical consequences, and provides novel insights into measure presentations and stochastic spacetime structures.

Findings

Time fractals increase sea turtle speed

Spatial fractals decrease sea turtle speed

Time-like fractals allow superluminal speeds for small objects

Abstract

We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with $q$-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes.