  We study the dynamics of steering between two correlated Unruh-Dewitt
detectors when one of them locally interacts with external scalar field via
different quantifiers. We find that the quantum steering, either measured by
the entropic steering inequality or the Cavalcanti-Jones-Wiseman-Reid
inequality, is fragile under the influence of Unruh thermal noise. The quantum
steering is found always asymmetric and the asymmetry is extremely sensitive to
the initial state parameter. In addition, the steering-type quantum
correlations experience "sudden death" for some accelerations, which are quite
different from the behaviors of other quantum correlations in the same system.
It is worth noting that the domination value of the tight quantum steering
exists a transformation point with increasing acceleration. We also find that
the robustness of quantum steerability under the Unruh thermal noise can be
realized by choosing the smallest energy gap in the detectors.
