Multifractional spacetimes, asymptotic safety and Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper compares multifractional spacetimes with RG-based quantum gravity theories like asymptotic safety and Hořava-Lifshitz gravity, highlighting how both frameworks describe scale-dependent spacetime dimensionality through different measurement adaptations.

Contribution

It establishes a mapping between multifractional measures and RG-based approaches, offering a new perspective on the fractal structure of spacetime in quantum gravity theories.

Findings

Demonstrates the equivalence of scale-dependent dimensionality in both frameworks.

Shows how measurement tools adapt differently in multifractional and RG-based theories.

Provides a unified view of fractal spacetime structures in quantum gravity.

Abstract

We compare the recently formulated multifractional spacetimes with field theories of quantum gravity based on the renormalization group (RG), such as asymptotic safety and Ho\v{r}ava--Lifshitz gravity. The change of spacetime dimensionality with the probed scale is realized in both cases by an adaptation of the measurement tools ("rods") to the scale, but in different ways. In the multifractional case, by an adaptation of the position-space measure, which can be encoded into an explicit scale dependence of effective coordinates. In the case of RG-based theories, by an adaptation of the momenta. The two pictures are mapped into each other, thus presenting the fractal structure of spacetime in RG-based theories under an alternative perspective.