  We investigate the ultimate estimation precision, characterized by the
quantum Fisher information, of a two-level atom as a detector which is coupled
to massless scalar field in the Minkowski vacuum. It has been shown that for an
inertial detector moving with a constant velocity, its quantum Fisher
information is completely unaffected by the velocity, however, it still decays
over time due to the decoherence caused by the interaction between the atom and
the field. In addition, for a uniformly accelerated detector ($w=0$) moving
along spatially straight line, the accelerated motion will reduce the quantum
Fisher information in the estimation of state parameters. However, when the
detector trajectory is generated by a combination of the linear accelerated
motion and a component of the four-velocity $w=dy/d\tau$, we find quite unlike
the previous results that, for the non-relativistic case $(w\ll1)$, the
acceleration could degrade the quantum Fisher information, while the
four-velocity component will suppress the degradation of the quantum Fisher
information, and thus could enhance the precision of parameters estimation.
Furthermore, in the case for ultra-relativistic velocities
$(w\rightarrow\infty)$, although the detector still interacts with the
environment, it behaves as if it were a closed system as a consequence of
relativity correction associated to the velocity, and the quantum Fisher
information in this case can be shield from the effect of the external
environment, and thus from the relativistic motion.
