ABC of multi-fractal spacetimes and fractional sea turtles

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.

20 pages, 1 figure. v2: typos corrected, minor improvements of the text

References in corpus (14)

• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• Fractal properties of quantum spacetime
• Noncommutative Geometry and the standard model with neutrino mixing
• Thermodynamics of photons on fractals
• Diffusion on $\kappa$-Minkowski space
• Dimensional flow in discrete quantum geometries
• Quantum mechanics in fractional and other anomalous spacetimes
• On the Interpretation of Stratonovich Calculus
• Diffusion in quantum geometry
• Cosmic microwave background and inflation in multi-fractional spacetimes
• Primordial Black Hole Evaporation and Spontaneous Dimensional Reduction
• Lorentz violations in multifractal spacetimes
• Particle-physics constraints on multifractal spacetimes
• Continuum Random Combs and Scale Dependent Spectral Dimension

Cited by in corpus (17)

• Multifractional theories: an unconventional review
• Multifractional theories: an updated review
• Cosmic microwave background and inflation in multi-fractional spacetimes
• Lorentz violations in multifractal spacetimes
• Quantum scalar field theories with fractional operators
• Towards the map of quantum gravity
• Imprint of quantum gravity in the dimension and fabric of spacetime
• Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
• Deformed symmetries in noncommutative and multifractional spacetimes
• Black holes in multi-fractional and Lorentz-violating models
• New Standard Model constraints on the scales and dimension of spacetime
• The geometry of learning
• Log-periodic gravitational-wave background beyond Einstein gravity
• Spacetime Foam Effects on Charged AdS Black Hole Thermodynamics
• Joining Spacetimes on Fractal Hypersurfaces
• Hawking area law in quantum gravity
• Sign Switching Dark Energy from a Running Barrow Entropy