  Quantum entanglement as the one of the most general quantum resources, can be
quantified by von Neumann entropy. However, as we know, the von Neumann entropy
is only statistical quantity or operator, it therefore has fluctuation. The
quantum fluctuation of entanglement (QFE) between Unruh-Dewitt detector modeled
by a two-level atom is investigated in a relativistic setting. The Unruh
radiation and quantum fluctuation effects affect the precise measurement of
quantum entanglement. Inspired by this we present how the relativistic motion
effects QFE for two entangled Unruh-Dewitt detectors when one of them is
accelerated and interacts with the neighbor external scalar field. We find that
QFE first increases by the Unruh thermal noise and then suddenly decays when
the acceleration reaches at a considerably large value, which indicates that
relativistic effect will lead to non-negligible QFE effect. We also find that
the initial QFE (without acceleration effect) is minimum with the maximally
entangled state. Moreover, although QFE has a huge decay when the acceleration
is greater than $\sim0.96$, concurrence also decays to a very low value, the
ratio $\Delta E/C$ therefore still large. According to the equivalence
principle, our findings could be in principle applied to dynamics of QFE under
the influence of gravitation field.
