Dynamic Rigid Fractal Spacetime Manifold Theory: PART C

This paper integrates fractal geometry, fractal measures, and their applications across multiple fields of physics to systematically explore fractal spacetime and its modifications to classical physical laws. First, the first law of fractal thermodynamics is introduced, with extended equations incorporating fractal dimensions DH and fractal measures. These equations reveal the energy transfer mechanisms in thermodynamic systems within fractal structures and their theoretical connections to cosmological dynamics. Second, Bloch’s theorem is extended to fractal lattices, introducing the concepts of fractal wavevectors and fractal periodicity, which describe the unique band structures and electron distributions in fractal materials. Third, generalized Maxwell equations for fractal spacetime are constructed using fractal derivatives and non-commutative operators, uncovering the regulatory effects of fractal dimensions on topological phases, electromagnetic wave propagation, and superconductivity. Lastly, by combining fractal dynamics and thermodynamic principles, the study systematically analyzes energy conservation, scaling behaviors of critical parameters, and the applications of fractal materials in high-dimensional physics and cosmological dark energy research. This research expands the classical assumptions of spacetime, providing a unified mathematical framework and theoretical foundation for the physical behavior of fractal structures. It aims to advance interdisciplinary studies in condensed matter physics, quantum gravity, and cosmology