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