  Quantum decoherence, which appears when a system interacts with its
environment in an irreversible way, plays a fundamental role in the description
of quantum-to-classical transitions and has been successfully applied in some
important experiments. Here, we study the decoherence in noninertial frames for
the first time. It is shown that the decoherence and loss of the entanglement
generated by the Unruh effect will influence each other remarkably. It is
interesting to note that in the case of the total system under decoherence, the
sudden death of entanglement may appear for any acceleration. However, in the
case of only Rob's qubit underging decoherence sudden death may only occur when
the acceleration parameter is greater than a "critical point."
