  Dirac equation in Newman-Penrose formalism is comprehensively introduced to
prove the non-existence of bound states for {\omega} < me in Schwarzschild
geometry. The proof by contradiction is drawn in the context of finding
polynomial solutions for the generalized Heun equation, thus the solutions are
resonant states decaying over time. This work is divided into five parts. The
first part is about the Dirac equation and the circumstances in which this
equation emerged. Followed by two brief subsections on the mathematical
substratum of the Dirac equation and its solution, i.e., Dirac algebra and
Dirac spinor respectively. For the sake of elaborating on the mathematical
tools used in this subject, the second part consists of a brief motivation for
the general theory of relativity and its mathematical methods. The third part
is dedicated to treating the Dirac equation in a more advanced mathematical
framework, the Newman-Penrose formalism, which is a special case of the tetrad
formalism of general relativity. The proof is elaborated in part 4, followed by
conclusions in the fifth part.
