Applications of Dirac equation in curved spacetime

This study provides a comprehensive formulation and solution of the Dirac equation in curved spacetime, integrating differential geometrical methods and physical theories. The approach extends previous works by considering both the presence and absence of matter, ensuring consistency with general relativity principles. Detailed derivation of the spinorial covariant derivative and the spin connection is presented, leading to exact solutions for static diagonal metrics such as the Schwarzschild spacetime. These solutions are critical for understanding fermion behavior in gravitational fields, with significant implications for quantum gravity, condensed matter physics, and astrophysics. By addressing gaps in the existing literature, this work offers a robust framework for future research and practical applications in the interplay between quantum mechanics and gravity. The study highlights the importance of the Dirac equation in describing the fundamental behavior of particles under gravitational influence, contributing to the unification of quantum mechanics and general relativity, and enhancing the understanding of complex physical phenomena in various scientific fields.