  In this study, we investigate the influence of Hawking decoherence on the
quantum correlations of Dirac fields between \textit{Alice} and \textit{Bob}.
Initially, they share a \textit{Gisin} state near the Schwarzschild black hole
(SBH) in an asymptotically flat region. Then, \textit{Alice} remains stationary
in this region, while \textit{Bob} hovers near the event horizon (EH) of the
SBH. We expect that \textit{Bob}, using his excited detector, will detect a
thermal Fermi-Dirac particle distribution. We assess the quantum correlations
in the evolved \textit{Gisin} state using quantum consonance and
uncertainty-induced non-locality across physically accessible, physically
inaccessible, and spacetime regions. Our investigation examines how these
measures vary with Hawking temperature, Dirac particle frequency, and the
parameters of the initial \textit{Gisin} state. Additionally, we analyze the
distribution of these quantum correlation measures across all possible regions,
noting a redistribution towards the physically inaccessible region. Our
findings demonstrate that Hawking decoherence reduces the quantum correlations
of Dirac fields in the physically accessible region, with the extent of
reduction depending on the initial state parameters. Moreover, as Hawking
decoherence intensifies in the physically inaccessible and spacetime regions,
the quantum correlations of Dirac fields reemerge and ultimately converge to
specific values at infinite Hawking temperature. These results contribute to
our understanding of quantum correlation dynamics within the framework of
relativistic quantum information (RQI).
