The Fractal Nature of Time: A Fractional Calculus Hypothesis Integrating Physics and Philosophy

Time is traditionally modelled as a smooth one-dimensional continuum in physics, yet evidence from complex systems, quantum gravity, and cosmology suggests a richer structure. I propose the fractal time hypothesis, which posits that time exhibits self-similarity and scale invariance, concepts that echo MacTaggart’s argument for time’s infinite regress via self-referentiality. A rigorous mathematical framework for fractal time is developed using fractional calculus, integrating these ideas into physical laws from classical mechanics to quantum physics, relativity, and cosmology. I derive fractional differential formulations of physical laws and scaling laws that link fractal time to observable effects. Key outcomes include modified equations of motion with memory effects, a generalized Schrödinger equation and uncertainty relations under fractional time evolution, corrections to relativistic time dilation, and an extension of cosmological models. These yield testable predictions, such as deviations in high-precision clock measurements, anomalous relaxation dynamics, altered particle decay rates, and potential explanations for cosmological puzzles including the Universe’s expansion rate, Penrose conformal cyclic cosmology and Gödel closed timelike curves. If time is fractal, foundational physics must be rethought, offering a unified framework that connects phenomena such as 1/f noise, anomalous diffusion, and quantum gravity’s dimensional reduction. This interdisciplinary approach also explores the ontological and epistemological implications of fractal time, contrasting it with classical views from McTaggart and Bergson, and demonstrating how a fractal model circumvents traditional paradoxes. Overall, the results suggest that fractal time can reconcile disparate descriptions of temporal reality, introduce scale-invariant laws, and pave the way for new theoretical and experimental advancements.