Fractal temporal dynamics in black hole accretion and quasi-periodic oscillation scaling

Accretion processes around black holes play a central role in high-energy astrophysics by governing compact object growth and driving luminous emissions across the electromagnetic spectrum. In this work, we propose a theoretical framework in which the local flow of time near the event horizon acquires a fractal structure, potentially arising from quantum gravitational fluctuations at Planckian length scales. This fractal temporal behavior, characterized by a scaling relation with , modifies the time evolution of surface density in viscous accretion disks. We derive a generalized diffusion equation incorporating the fractal-time exponent and perform numerical simulations across a wide range of black-hole masses ( to ) and accretion rates (0.01– ). Our results reveal enhanced modulation in the accretion luminosity and the emergence of quasi-periodic oscillations (QPOs) in the X-ray light curves. The predicted QPO frequencies, which range from 4– for stellar-mass black holes, scale with the fractal exponent and follow (classical limit ), remaining within the detection capabilities of current instruments such as NICER, XMM-Newton, and NuSTAR. These findings indicate that microscopic spacetime fluctuations may leave macroscopic imprints on accretion dynamics, thereby providing an observational probe of fractal-time behavior near black holes. This work offers a testable connection between quantum gravitational microphysics and observable X-ray variability.