Fractal Dynamics and Quantum Gravity

Exploring the Fractal Structure of Space-Time at the Sub-Planckian Scale: New Experimental Perspectives Author: Christophe Duplan, France Abstract: This study introduces a novel fractal approach to understanding spacetime at the sub-Planckian scale, challenging the conventional perception of continuous spacetime. Using iterated function systems (IFS) and Banach's fixed-point theorem, the research suggests that spacetime exhibits self-similar fractal structures below the Planck length, with significant implications for quantum gravity, cosmology, and particle physics. By applying experimental protocols using NbSe₂ superconducting sheets encapsulated in h-BN, observable deviations in spacetime dynamics are investigated, contributing towards the unification of quantum mechanics and general relativity. Introduction The unification of general relativity and quantum mechanics remains elusive. Current theories such as string theory and loop quantum gravity rely on the Planck scale as a fundamental limit of spacetime. However, the lack of experimental evidence suggests the need for innovative methodologies to explore spacetime at sub-Planckian scales. This study proposes that spacetime is granular at these scales and adopts a self-similar fractal metric. Methodology Utilizing the Banach fixed-point theorem and iterative function systems (IFS), the study models spacetime as a fractal structure. The experimental setup involves superconducting sheets of NbSe₂ and h-BN, enabling precise measurements of spacetime granularity and self-similarity. Experimental Procedure Experiments involve controlled conditions: Materials: NbSe₂ sheets with h-BN encapsulation for stability. Measurement Techniques: Doppler interferometry and torsion balance are employed to observe deviations from classical predictions. Environmental Control: Cryogenics and ultra-high vacuum settings are essential for maintaining precise experimental conditions. Results The study shows that spacetime at sub-Planckian scales behaves fractally, with self-similar structures that redefine fundamental constants. Observed deviations in expected movement durations support the fractal metric hypothesis, suggesting a framework for understanding sub-Planckian spacetime. Discussion Key findings include: Fractal Metrics and Black Hole Singularities: Fractal structures could provide insights into black hole interiors, avoiding infinite densities. Implications for Quantum Gravity: Self-similar spacetime structures offer a new approach to unify quantum mechanics and general relativity. Experimental Challenges: Although technically demanding, further refinement of the setup could validate these findings. Conclusion This work proposes a fractal structure for spacetime at the sub-Planckian scale, with broad implications for theoretical physics. Future experiments could confirm the fractal hypothesis, potentially advancing our understanding of the universe's fundamental structure. Keywords: Planck scales, fractal, quantum gravity, Banach fixed-point theorem, NbSe₂ superconducting sheets, self-similarity, sub-Planckian